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\begin{document}
\title{Bound States for Schr\"odinger Hamiltonians:
Phase Space Methods and Applications}
\author{Ph. Blanchard\\
Theoretische Physik and BiBoS,
Universit\"at Bielefeld\\
D-33615 Bielefeld
\and
J. Stubbe\\
Dept.~de Physique Th\'eorique,
Universit\'e de Gen\`eve\\
CH-1211 Geneve\\ and\\
Theory Division, CERN, CH-1211 Gen\`eve 23}
\date{}
\maketitle
\bigskip
\bigskip
\bigskip
\begin{abstract}
Properties of bound states for Schr\"odinger operators are reviewed. These
include: bounds on the number of bound states and on the moments of the
energy levels, existence and nonexistence of bound states, phase space bounds
and semi-classical results, the special case of central potentials, and
applications of these bounds in quantum mechanics of many particle systems
and dynamical systems. For the phase space bounds relevant to these
applications we improve the explicit constants.
\end{abstract}
\bigskip
\bigskip
\bigskip
\begin{center}
{\large{\bf Dedicated to Andr\'e\ Martin}},\\
\smallskip
{\large{\bf an old friend, who ''retired" recently}}\\
\smallskip
{\large{\bf in order to accomplish even more!}}
\end{center}
\thispagestyle{empty}
\newpage
\section{Introduction}
Consider the Schr\"odinger operator
\begin{equation}
H = - \Delta + V(x)
\end{equation}
on $L^2({\bf R}^d)$. It describes either the motion of a quantum
particle of mass $1/2$ in an external potential $V$ or the relative
motion of two particles having reduced mass $1/2$
interacting via the potential $V$. Units have been chosen such that
Planck's constant $\hbar$ equals to one. In the present review we want to
report about
estimates for the number of bound states and their energies $E_j \le 0$.
A bound state is a stationary state with a negative eigenvalue for the
Schr\"odinger operator $H$; the bound states are the quantum states
corresponding to the closed orbits of the classical theory. Such
estimates are of obvious importance in quantum mechanical problems and they
play a fundamental role in the proof of the stability of matter.
This involves a problem first raised by L.~Onsager in the 1930's. A basic fact
of astrophysics claims that bulk matter in the absence of nuclear forces
undergoes gravitational collapse. Onsager asked why bulk matter does not undergo
electrostatic collapse. The Pauli exclusion principle for the electrons
is the basic mechanism which prevents bulk matter from collapse. The
first rigorous proof of this fact was given by F.~Dyson and A.~Lenard
[DL67, DL68]
in the sixties. Pioneering work was done ten years after the Dyson-Lenard
result by E.~Lieb and W.~Thirring who gave a rather elementary proof of
the stability of bulk matter using new principles which provided much better
physical insights. Lieb and Thirring relate stability problems for the
quantum many particle system to the Thomas-Fermi model [LT75]. Certain
properties of the Thomas-Fermi model are then used to prove the stability
of the quantum system. As a byproduct one obtains quite accurate bounds on the
energy of atoms and molecules. The basic new idea is a Sobolev type
inequality for fermions, i.e. a lower bound on the kinetic energy of N-fermions
in terms of an appropriate functional of the corresponding one particle
density. The proof of this new principle relies on the methods and results
which will be reviewed in the present article.
A beautiful presentation of the main physical and
mathematical aspects of this problems has been given by E.~Lieb
[L76b, L89, L90a]. However,
such estimates are also useful in different fields of mathematical physics
as in the theory of infinite dimensional dynamical systems where by means of
such bounds one can estimate the Hausdorff dimension of attractors
[Ru82, L84].
Physically, the number of bound states $N_0(V)$ of $H$ is given by the number
of cells of size $h^d$, where $h$ denotes Planck's constant, in the phase
space where the classical Hamiltonian $H(x,p) = p^2 + V(x)$ is negative.
Since $h = 2\pi$ in our units one should have
\begin{equation}
N_0 (V) \sim (2\pi)^{-d} \int\!\!\int_{H(x,p) \le 0} dx\,dp = (2\pi)^{-d}
B_d \int_{{\bf R}^d} |V_-(x)|^{d/2} dx \ .
\end{equation}
Here $B_d = \frac{\pi^{d/2}}{\Gamma(1+d/2)}$ denotes the volume of the unit
ball in ${\bf R}^d$, $\Gamma(\alpha)$ for $\alpha > 0$ being the
Gamma function $\Gamma(\alpha) = \int^{\infty}_0 t^{\alpha -1}
e^{-t} dt$ and $V_-(x)$ denotes the negative part of $V(x)$, i.e.~
$V_-(x) = \sup (-V(x),0)$. Mathematically, $N_0 (V)$ is the dimension of the
spectral projection $P_{(-\infty,0)}$ for the operator $H$.
It turns out that (1.2) is true in the strong coupling limit, or alternatively
in the semiclassical limit. This was first proven by A.~Martin [M72] in 1972
for H\"older continuous potentials of compact support and later by H.~Tamura
[Ta]
under milder restrictions on $V(x)$. However, there have been earlier
attempts. K.~Chadan informed us about a paper by J.S.~De Wet and F.~Mandl from
1949 where the asymptotic validity of (1.2) was proven for dimensions 2 and 3
for a restrictive class of potentials going to infinity at large distances
[WM50].
To be more precise, (1.2) is valid in the following sense [RSIV,
Theorem XIII.79].
%\newpage
\bigskip
\bigskip
\noindent
\underline{\bf Theorem 1.1.} \enspace Let $V:{\bf R}^d \to {\bf R}$ be a
continuous function of compact support. For any $\lambda > 0\ , N_0
(\lambda V)$
denotes the dimension of the spectral projection $P_{(-\infty ,0)}$
of $-\Delta + \lambda V$. Then
\begin{equation}
\lim_{\lambda \to\infty} \frac{N_0 (\lambda V)}{\lambda^{d/2}} =
(2\pi)^{-d} B_d \int_{{\bf R}^d} |V_-(x)|^{d/2} dx\ .\ \Box
\end{equation}
The proof is rather elementary and uses the Dirichlet-Neumann bracketing
method, a tool which can be already found in the book by Courant and Hilbert
[CH].
The above asymptotic result suggests to look for bounds of $N_0 (V)$ in terms
of the classical phase space volume given by the r.h.s.~of (1.2) and (1.3),
respectively. We should remark that on the other hand such bounds will
allow us to extend the asymptotic formula (1.3) to all $V\in L^{d/2}
({\bf R}^d)$. This will be the case if $d\ge 3$. [RSIV, Theorem XIII.80]
These considerations will be generalized in the following way.
For $\gamma \ge 0$ let
\begin{equation}
S_\gamma (V) \equiv \sum_{E_j \le 0} |E_j |^\gamma
\end{equation}
where $E_j$ denote the (negative) eigenvalues of $H$. Obviously $S_0(V) =
N_0 (V)$. Then we should look for bounds on $S_\gamma(V)$ in terms of the
corresponding classical phase space expression, i.e.~for bounds of the form
\begin{equation}
S_\gamma(V) \le {\bf C}_{\gamma, d}(2\pi)^{-d} \int\!\!\!\int_{\{H(x,p)\le 0\}}
|H(x,p)|^\gamma dxdp
\end{equation}
or after performing the $p$-integration
\begin{equation}
S_\gamma (V) \le L_{\gamma, d} \int_{{\bf R}^d} |V_-(x)|^{d/2 + \gamma} dx
\end{equation}
In fact, using the techniques of Theorem 1.1., one can prove the following
\bigskip
\bigskip
\noindent
\underline{\bf Corollary 1.1.} \enspace Let $V: {\bf R}^d \to {\bf R}$ be a
continuous function of compact support. Then for $\gamma\ge 0$,
\begin{equation}
\lim_{\lambda\to\infty} \frac{S_\gamma (\lambda V)}{\lambda^{d/2 + \gamma}}
= L^{class}_{\gamma, d} \int_{{\bf R}^d} | V_-(x)|^{d/2 + \gamma} dx
\end{equation}
where $L^{class}_{\gamma , d}$ denotes the "classical constant" given by
$L^{class}_{\gamma , d} \equiv (4 \pi)^{-d/2}
\frac{\Gamma(\gamma +1)}{\Gamma (\gamma + 1 + \frac{d}{2})}$
\hfill $\Box$.
\bigskip
The integrating theme of this article is the study of the bound (1.5). In the
next section we shall review some technical tools and some general properties
of the function $N_0 (V)$ and its generalisation $N_E(V)$ for $E\le 0$. The
main technical tool will be the so-called Birman-Schwinger principle
[Bir61, Schw61],
which relates the problem of counting eigenvalues of $H$ to the problem of
counting eigenvalues of a compact operator associated to $H$. The solution of
boundary and eigenvalue problems by variational methods are essentially based
on the existence of suitable continuous and compact embeddings of Sobolev
spaces. We wish therefore to discuss briefly the most important resulting
inequalities. In section 3 we discuss conditions
of the form (1.5) which imply the nonexistence of bound states, i.e. $N_0
(V) = 0$, and estimates for the ground state eigenvalue of a Schr\"odinger
operator. We also show that such conditions cannot exist in dimensions 1 and
2 by proving that any attractive potential has at least one bound state.
We present some generalisations to conditions guaranting $N_E (V) = 0$
for $E < 0$.
Since central potentials are much simpler to treat some bounds on $N_0 (V)$
existed already since the early fifties when R.~Jost and A.~Pais [JP51] and
V.~Bargmann [Ba52] proved a bound on the number of s-states in a central
potential. Since the partial differential equation
decomposes into an infinite family of
one-dimensional problems we may apply well-known results about ordinary
differential equations. In particular,
there is a strong relation between the nonexistence conditions of section 2
and the bounds on the number of bound states in a given sector. The major
problem in this approach
is then to sum over all sectors. This will be discussed in Section 4.
Finally, in Section 5 we discuss the general case for phase space estimates
and present bounds on $S_\gamma(V)$ which are of the form (1.5). For
$\gamma =0$ we separately review the results and techniques by Cwickel
[Cw77], Lieb [L76a] and Li and Yau [LY83]. These bounds imply bounds on
$S_\gamma (V)$ for all $\gamma \ge 0$. The first bounds on $S_\gamma(V)$
for $\gamma > 0$, however, were obtained by Lieb and Thirring [LT75, LT76].
By combining these ideas with Lieb's method [L76a] we are able to improve
the constants in the estimates for $S_\gamma (V)$ [BS94]. The particular
case of dimensions 1 and 2 will be considered in Section 5.5. In Section 5.6
we present some general properties of phase space bounds leading to some
interesting conjectures and open problems. In Section 5.7 we mention
recent results about Schr\"odinger operators with magnetic fields.
In Section 6 we present some applications of phase space bounds. We sketch
the Lieb-Thirring proof [LT75] of Sobolev's inequality for fermions
leading to the proof of stability of matter and to estimate on the number
of characteristic exponents for the Navier Stokes equations. A second
application we use the nonexistence condition of Section 3 to obtain
nonbinding results for many particle systems. This example is due to [GGMT76].
In an Appendix we discuss the best numerical values for different relevant
constants available in the literature so far and give the improvements
detailled in the present paper which are relevant for the applications.
\section{General Methods and Principles}
In this section we give few proofs, our purpose being merely to list and recall
some general results and technical tools which will be needed to derive the
bounds on $S_\gamma (V)$. We do not discuss the problem of selfadjointness of
$H = -\Delta + V$ where we refer to the wide literature on the subject
[RSI-IV]. Generally, we suppose
that $V$ is a measurable function on ${\bf R}^d$ which belongs to a suitable
$L^p$ space.
As we shall see later almost all bounds on $S_\gamma (V)$ for $\gamma > 0$ are
derived from bounds for the counting function $N_E (V)$. Therefore we start
with discussing some properties of this function.
\subsection{Properties of $N_E (V)$}
For $E \le 0$ we denote by $N_E(V)$ the dimension of the spectral projection
$P_{(-\infty,E)}$ for $H$.
\bigskip
\bigskip
\noindent
\underline{\bf Lemma 2.1.}\enspace Let $V \le W$ pointwise. Then
\begin{equation}
N_E(W) \le N_E(V)
\end{equation}
for all $E\le 0$. In particular,
\begin{equation}
N_E(V) \le N_E (-V_-)\ . \Box
\end{equation}
\bigskip
The proof follows immediately from the comparison Theorem for operators
[RSIV].
Evidently, $N_E(V)$ is also a monotone function of $E$. The next result
is a little bit more subtle, although its proof is still elementary.
\bigskip
\bigskip
\noindent
\underline{\bf Lemma 2.2.} [LT75,76]
\enspace For all $\alpha \in [0,1]$, and $E \le 0$
\begin{equation}
N_E (V) \le N_{\alpha E} (-|V(x) - (1 - \alpha) E|_- )
\end{equation}
where $|V(x) - (1-\alpha)E|_-$ denotes the negative part of the potential
$V(x) - (1-\alpha )E\ . \Box$
\bigskip
\bigskip
Looking at the graph of $N_E (V)$ one sees that integrating $N_E(V)$ with respect
to $E$ yields (minus) the sum of the negative eigenvalues. More generally,
since
$$
\frac{\partial}{\partial E} N_E = \sum_j \delta (E - E_j)
$$
where $\{ E_j\}_j$ denote the eigenvalues of $H$, we have
\bigskip
\bigskip
\noindent
\underline{\bf Lemma 2.3.} [LT75,76]
\enspace Let $S_\gamma (V) = \sum_{E_j < 0}
|E_j|^\gamma$. Then
\begin{equation}
S_\gamma (V) = \gamma \int^\infty_0 d|E| |E|^{\gamma -1} N_{-|E|} (V)\ .
\end{equation}
\bigskip
\subsection{The Birman-Schwinger principle}
In 1961, M.~Birman [Bir61] and J.~Schwinger [Schw61], independently proved a bound on
$N_E(V)$ by introducing a bounded integral operator whose eigenvalue are
related to the bound states of the Schr\"odinger operator $H$. Here we follow
the presentation given in the review article by B.~Simon [Si76a].
First of all we note the following
\bigskip
\bigskip
\noindent
\underline{\bf Lemma 2.4.} \enspace Consider the operator $-\Delta + \lambda V$
with $V \le 0$, and $\lambda > 0$. $E$ is an eigenvalue, $E < 0$, if and
only if $\lambda^{-1}$ is an eigenvalue of
\begin{equation}
K_E = |V|^{1/2} ( - \Delta + E)^{-1} |V|^{1/2}
\end{equation}
and their multiplicities are equal. $\Box$
\bigskip
The proof of the Lemma 2.4. is nothing but a careful analysis of the formal
correspondance that $(-\Delta + \lambda V) \psi = E\psi $ holds if and only if
$K_E \phi = \lambda^{-1} \phi $ where $\phi = |V|^{1/2} \psi$\ .
\bigskip
\noindent
\underline{\bf Remark:}\enspace We shall call $K_E$ given by (2.5) the
Birman-Schwinger operator.
\bigskip
The second step is
\bigskip
\bigskip
\noindent
\underline{{\bf Lemma 2.5.}} \enspace The number $N_E(V)$
of eigenvalues of $-\Delta + V$ less than $E < 0$ is equal to the number of
$\lambda \in (0,1)$ for which $E$ is an eigenvalue of $-\Delta + \lambda V \ .$
\enspace $\Box$.
\bigskip
The proof follows from the fact that the eigenvalues of $-\Delta + \lambda
V$ are monotone and continuous in $\lambda$ and converge to zero as $\lambda$
tends to zero.
\bigskip
These two lemmas imply the Birman-Schwinger principle, which is of considerable
importance.
\bigskip
\bigskip
\noindent
\underline{{\bf Theorem 2.1.}}
\noindent
Let $V \le 0, E < 0$.
The number of eigenvalues of $-\Delta + V$ in $(-\infty , E), N_E (V)$, is
the same as the number of eigenvalues of the Birman-Schwinger operator $K_E$
in $(1,\infty)$ counting the multiplicities. \ $\Box$
\bigskip
On ${\bf R}^3$, for suitable $V$, $K_E$ is a
Hilbert-Schmidt operator. Since $(-\Delta + E)^{-1}$ has an integral kernel
given by $(4\pi |x-y|)^{-1} \exp (-\sqrt{-E} |x-y|)$ one has
\begin{eqnarray}
\lefteqn{N_E (V) \le Tr (K_E K_E^\ast)} \\
& & = \frac{1}{(4\pi)^2} \int_{{\bf R}^3}
\int_{{\bf R}^3} | V(x)| |V(y)| |x-y|^{-2} \exp (-2\sqrt{-E} |x-y|)dxdy
\nonumber \ .
\end{eqnarray}
\bigskip
As $E$ approaches $0$ we obtain the Birman-Schwinger bound. The theorem
has been proved for $V \in {\bf R}$, the set of
Rollnik potentials. We recall that a measurable function $V$ on ${\bf R}^3$ is
called a Rollnik potential if
$$
\| V \|^2_R \equiv \int_{{\bf R}^3} \frac{|V(x)| |V(y)|}{| x - y|^2}
dx\, dy < + \infty\ .
$$
We next present a more general version of the Birman-Schwinger principle
by introducing a one-parameter family of bounded operators whose number of
eigenvalues bigger than one yields a bound on $N_E(V)$. To our knowledge
this generalisation was introduced by Lieb and Thirring [LT76] to obtain
the celebrated bound on the sum $S_1 (V)$ over the eigenvalues of
$-\Delta + V$. We shall see in Section 5 that by means of this generalisation we will
obtain the currently best numerical constants in bounds of the form (1.5).
For $E\le 0, \alpha \in [0,1]$ we define the generalised Birman-Schwinger
operator
\begin{eqnarray}
K_{\alpha E} &\equiv& | V(x) - (1-\alpha)E|_-^{1/2} (-\Delta - \alpha
E)^{-1} |V(x) - (1-\alpha) E|_-^{1/2} \nonumber\\
&\equiv& |V_\alpha |^{1/2} (-\Delta + \alpha |E|)^{-1} |V_\alpha |^{1/2}
\end{eqnarray}
where $V_\alpha \equiv |V(x) - (1-\alpha) E|_-$ denotes the negative part of
the function $V(x) - (1-\alpha)E$.
\bigskip
Using Lemma 2.2. we obtain the following
\bigskip
\bigskip
\noindent
\underline{\bf Theorem 2.2.} \enspace The number $N_E(V)$ of eigenvalues
of $-\Delta + V$ in $(-\infty , E)$, is bounded above by the number of
eigenvalues of the generalised Birman-Schwinger operator $K_{\alpha E}$
in $(1,\infty)$ counting multiplicities. \ $\Box$
\bigskip
\bigskip
\noindent
Alternatively, we can formulate the Birman-Schwinger principle in momentum
space [Si71]. For a function $f(x)$ let us define its Fourier transform
${\hat f}(p)$ by
\begin{equation}
{\hat f} (p) = \int_{{\bf R}^d} e^{ipx} f(x)\ dx \ .
\end{equation}
The eigenvalue problem for the Schr\"odinger equation in momentum space reads
\begin{equation}
T_E (p) {\hat \psi} (p) + (2\pi)^{-d} \int_{{\bf R}^d} {\hat V}(p-p^\prime)
{\hat \psi}(p^\prime) dp^\prime = 0
\end{equation}
with $T_E (p) = p^2 - E,\ E\le 0$. Defining $\phi (p) = T_E (p)^{-1/2}
{\hat \psi} (p)$ we see that the operator ${\hat K}_E$ given by
\begin{equation}
({\hat K}_E\phi)(p) \equiv -(2\pi)^{-d} \int_{{\bf R}^d} T_E (p)^{-1/2}
{\hat V} (p - p^\prime) T_E^{-1/2} (p^\prime) \phi
(p^\prime)\ dp^\prime
\end{equation}
has an eigenvalue $1$.
By mimicking the x-space proof we obtain the Birman-Schwinger principle
in its $p$-space version.
\bigskip
\bigskip
\noindent
\underline{{\bf Theorem 2.3.}}\enspace
The number of eigenvalues $N_E(V)$ of $-\Delta +V$
in $(-\infty,E),\linebreak
\noindent E\le 0$ equals the number of eigenvalues of the Birman-
Schwinger operator ${\hat K}_E$ in $[1,\infty)$ counting the multiplicities.
\hfill $\Box$
\bigskip
It is interesting to remark that for $d=3$ the bound corresponding to (2.6) will be
\begin{equation}
N_E (V) \le tr ({\hat K}_E^\ast {\hat K}_E)\ .
\end{equation}
It is elementary to check that
\begin{eqnarray}
tr (K^\ast_E K_E) &=& tr ({\hat K}^\ast_E {\hat K}_E)\\
&=& \frac{1}{32} \pi^{-4} \int_{{\bf R}^3}
\frac{|{\hat V}(p)|^2}{|p|} \arctan
\frac{|p|}{2\sqrt{|E|}} \ . \nonumber
\end{eqnarray}
As in configuration space, one may introduce a one parameter family of
operators ${\hat K}_{\alpha E},\ \alpha \in [0,1]$, by using $T_{\alpha E}(p)$
and the Fouriertransform of $V_\alpha = |V(x) + (1-\alpha)E|_- $.
\bigskip
\noindent
\underline{\bf Remark:}\enspace While in this article only stated for
Schr\"odinger-type operator the Birman-Schwinger principle holds in
general for an operator of the form $H = A + \lambda B$ for
$A,B$ selfadjoint and $B$ relatively $A$-compact. For general results
in this abstract situation and some results about existence of eigenvalues
we refer to [Kl82].
\subsection{Some classical inequalities}
To make this review accessible to as wide a readership as possible we recall
now classical inequalities playing a major role in various areas of applied
mathematics and mathematical physics. This short section is devoted to
such questions. Beside local inequalities we will recall also integrated
ones. The term "Sobolev inequality" properly applies to convolution inequalities
of the form $|< R_\alpha f, g>| \le C \| f\|_p \| g\|_{q^\prime}$, where
$R_\alpha = \frac{a}{|x|^{d-\alpha}}$ with $0 < \alpha < d$ is the Riesz
kernel in ${\bf R}^d$. By extension however any inequality that a function
is in a certain class if its derivatives are in another class is called a
Sobolev inequality. The constants given below will be the best possible
ones for the particular inequality, in the sense that it cannot be replaced
by a smaller one.
A useful result is the convolution inequality by Young.
\bigskip
\bigskip
\noindent
\underline{\bf Young's inequality\ [Be75, BL76]}
\medskip
\noindent
Let $p, q, r \ge 1$ such that $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 2$.
Let $\| f\|_p = [\int_{{\bf R}^d} |f(x)|^p dx]^{1/p}$ and $\| f\|_\infty
= {\mbox{ess}\sup}_{x\in \int_{{\bf R}^d}} | f(x) |$ . By $p^\prime$ we denote
the dual exponent of $p$, i.e. $\frac{1}{p} + \frac{1}{p^\prime} = 1$.
Then
\begin{equation}
| \int_{{\bf R}^d}\int_{{\bf R}^d} f(x) g(x-y) k(y) dx\ dy | \le [C_pC_qC_r]^d
\| f\|_p \| g\|_q \| h \|_r
\end{equation}
with
\begin{equation}
C^2_p = p^{1/p}\ (p^\prime)^{-1/p^\prime}\ .
\end{equation}
We note that $C_pC_qC_r \le 1$.
\bigskip
\bigskip
\noindent
\underline{\bf Sobolev's inequality and its generalisations}\par
\noindent
\underline{\bf [So38, GGMT76, L83, O, Ta76]}
\medskip
\noindent
In 1938 Sobolev found an inequality relating the $L^q$ norm of a function in
${\bf R}^d$ to the $L^p$-norms of the functions and its derivations
up to order $k$ if $\frac{1}{q} = \frac{1}{p} - \frac{k}{d}$.
In the present paper we shall extensively use the following generalised
Sobolev inquality [GGMT76, L83].
Let $d \ge 3, \ 0 \le b\le 1$, and $p = 2d/(2b + d - 2)$. Then the
following inequality holds
\begin{equation}
K_{n,p} \| \nabla f\|_2 \ge \| |x|^{-b} f\|_p
\end{equation}
with
\begin{eqnarray}
K_{n,p} &=& \omega^{-1/2 r}_{d-1} (d-2)^{1/2 r -1}
\left(\frac{r -1}{r}\right)^{\frac{r-1}{2r}} \\
& & \cdot \left(\frac{\Gamma(2r)}{\Gamma(r + 1)\Gamma(r)}
\right)^{1/2r} \nonumber
\end{eqnarray}
where $r = \frac{p}{p-2} = \frac{d}{2} \frac{1}{1-b}$ and
$\omega_{d-1} = \frac{2\pi^{d/2}}{\Gamma (d/2)}$ is the area of the unit sphere
in ${\bf R}^d$.
In particular, if $b=0$, then $p = \frac{2d}{d-2} = 2^\ast$ (the dual
Sobolev exponent) and
\begin{equation}
K_{d,2^\ast} = [\pi d(d-2)]^{-1/2} \left( \frac{\Gamma (d)}{\Gamma(d/2)}
\right)^{1/d}\ .
\end{equation}
If $1 - d/2 \le b < 0$, the above inequality holds only for spherically
symmetric functions. $K_{n,p}$ is given by the same expression. \ $\Box$
\bigskip
The first calculation of the sharp constant $K_{3,2^\ast}$ is due to
Rosen [Ro71].
\bigskip
\noindent
In [St90] the following generalisation was obtained. Let $0 \le \alpha < 1$, then
\begin{equation}
K^2_{n,p} (\alpha) (\| \nabla f\|^2_2 - \alpha (\frac{d-2}{2})^2
\| \frac{1}{|x|} f\|^2_2 ) \ge \| |x|^{-b} f \|^2_2
\end{equation}
with $p$ and $b$ as above and
\begin{equation}
K_{n,p} (\alpha) = (1-\alpha)^{-\frac{d-1}{2d}} K_{n,p}\ .
\end{equation}
If one considers only spherically symmetric function the inequality holds
for all $\alpha < 1$.
By using a series of reduction procedures (see [L83] for the details) the
above inequalities can be related to the following one-dimensional Sobolev
inequality:
For $2 < p < \infty$
\begin{equation}
M_p (\| f^\prime\|^2_2 + \| f \|^2_2) \ge \| f \|^2_p
\end{equation}
with
\begin{equation}
M_p = 2^{\frac{1}{r}-2} \left( \frac{r - 1}{r}
\right)^{\frac{r-1}{r}} \left\{
\frac{\Gamma (2r)}{\Gamma (r) \Gamma(r +1)} \right\}^{1/r}
\end{equation}
where $r = \frac{p}{p-2}$.
\bigskip
\begin{flushleft}
\underline{\bf Hardy-Littlewood-Sobolev inequalities and the weak Young}
\\
\underline{\bf inequality [HL28, HL30, L83, So38]}
\end{flushleft}
\smallskip
\noindent
Let $1 < p, t < \infty,\ 0 < \lambda < d$ and $\frac{1}{p} + \frac{1}{t} +
\frac{\lambda}{d} = 2$. Then the following inequality holds
\begin{equation}
| \int\int f(x) |x-y|^{-\lambda} g(y) dx\ dy | \le N_{p,\lambda ,d}
\| f\|_p \| g\|_t \ .
\end{equation}
If $p = t = \frac{2d}{2d-\lambda}$, the constant $N_{p,\lambda,d}$
is explicitly given by
\begin{equation}
N_{p,\lambda ,d} = \pi^{\lambda/2}
\frac{\Gamma(d/2 - 1/2)}{\Gamma (d - \lambda/2)} \left\{\frac{\Gamma(d/2)}
{\Gamma(d)}\right\}^{\lambda/d -1}\ .
\end{equation}
For $p = 2$ or $p = \frac{2d}{2d-\lambda}$ (i.e. $t=2$), and $d/2 < \lambda
< d$, one has
\begin{equation}
N_{p,\lambda , d} = \pi^{\lambda/2}
\frac{\Gamma (\frac{d}{2} - \frac{\lambda}{2})}{\Gamma (\lambda/2)} \left\{
\frac{\Gamma (\lambda - \frac{d}{2})}{\Gamma (\frac{3d}{2} - \lambda)}
\right\}^{1/2} \left\{ \frac{\Gamma (\frac{d}{2})}{\Gamma (d)}
\right\}^{\frac{\lambda}{d}-1}\ .
\end{equation}
The sharp constants in (2.23) and (2.24) are due to [L83]. A very nice
alternative derivation was found in [CL90].
The generalisation of the Hardy-Littlewood-Sobolev inequality is the weak Young
inequality:
\begin{equation}
| \int\int f(x) h(x-y) g(y) dx\ dy | \le N_{p,\lambda,d} \| f\|_p \| g\|_t
\| h\|_{d/\lambda,\omega}
\end{equation}
where as before $\frac{1}{p} + \frac{1}{t} + \frac{\lambda}{d} = 2$ and
$N_{p,\lambda,d}$ denotes the same constant as in (2.17) and $\| h \|_{q,\omega}$
denotes one of the two following definitions of the weak $L^q$-norm:
\begin{equation}
\| h \|^\ast_{q,\omega} = \left( \frac{1}{B_d} \right)^{1/q} \sup_{\alpha >0}
\mbox{Vol} \{ x\in{\bf R}^d | |h(x)| > \alpha\}^{1/q}
\end{equation}
or
\begin{equation}
\| h \|_{q,\omega}^{\ast\ast} = \frac{q-1}{q} \left( \frac{1}{B_d}
\right)^{1/q} \sup_{\mbox{Vol}\ A < \infty} (\mbox{Vol}\ A)^{1/q - 1}
\int_A |h(x)| \ dx \ .
\end{equation}
In both definitions the weak-$L^q$-norm of $|x|^{-\lambda}$ equals $1$,
for $q = d/\lambda$.
\bigskip
\bigskip
\noindent
\underline{\bf Hausdorff-Young inequality [Be75]}
\smallskip
\noindent
Let $p^\prime \ge 2$, then
\begin{equation}
\| {\hat f}\|_{p} \le (2\pi)^{d/p^\prime} C^d_p \| f \|_{p^\prime}
\end{equation}
with $C^2_p = p^{1/p} (p^\prime)^{-1/p}$ as in Young's inequality.
$p^\prime$ and $p$ are dual exponents, i.e. $\frac{1}{p} +
\frac{1}{p^\prime} = 1$.
Lieb has proved that equality in (2.28) occurs only for Gaussians [L90b].
(2.28) expresses the fact that the Fourier transformation is a continuous
linear transformation for $L^{p^\prime} ({\bf R}^d)$ to $L^p ({\bf R}^d)$.
\section{Nonexistence of Bound States}
In this section we discuss conditions on a potential which imply that the
operator $-\Delta +V$ has no bound state, ore more generally no bound state
below a given energy $E\le 0$.
These conditions will then imply lower bounds on $\inf \mbox{spec}(H)$
which is in fact the principal application of these results. For $d=3$
several conditions guaranting the absence of bound states have been known
for a long time. For instance, Jost and Pais [JP51], Bargmann [Ba52], Birman
[Bir61] and Schwinger [Schw61] have shown that a spherically symmetric potential
$V(x) = v(r), r = |x|,$ which satisfies
\begin{equation}
\int^\infty_0 v_- (r) r\ dr < 1
\end{equation}
has no bound states. Another example is that if for a potential $V(x)$
the following inequality holds
\begin{equation}
\sup_{x\in{\bf R}^3} |x|^2 V_- (x) < 1/4
\end{equation}
then $N_0 (V) = 0\ $ [CH].
Finally, making use of the Birman-Schwinger bound (2.6) or (2.11) we see
that there is no bound state below $E < 0$ if
\begin{equation}
\frac{1}{(4\pi)^2} \int_{{\bf R}^3}\int_{{\bf R}^3}
\frac{V(x)\ V(y)}{|x-y|^2} e^{-2\sqrt{-E} |x-y|} dx\ dy < 1\ .
\end{equation}
In particular, by letting $E \to 0$, this will be the case if
\begin{equation}
\frac{1}{(4\pi)^2} \int_{{\bf R}^3}\int_{{\bf R}^3}
\frac{V(x)\ V(y)}{|x-y|^2} dx\ dy\ < 1
\end{equation}
or equivalently
\begin{equation}
\frac{1}{(4\pi)^3} \int_{{\bf R}^3} \frac{|{\hat V}(p) |^2}{|p|} dp < 1\ .
\end{equation}
On the other hand, in dimensions one and two any "reasonable" purely
attractive potential has at least one bound state. We shall give a more
precise statement below. However, for any $E < 0$ there will be similar
bounds as in higher dimensions.
The most important result on the absence of bound states is due to V.~Glaser,
H.~Grosse, A.~Martin and W.~Thirring [GGMT76]. For $d=3$ they computed the
sharp constants $K_{3,p}$ in the Sobolev inequalities (2.14) and by means
of these inequalities they obtained a family of optimal integral conditions
on the potential $V$ which guarantee the nonexistence of bound states.
Its generalisation to all space dimension $d \ge 3$ is straight-forward
and will be presented in the following subsection.
\bigskip
\noindent
\underline{\bf Remark:} \enspace The occurence of a bound state at $E = 0$
has nothing exceptional. Although eigenvalues at $E > 0$ are classical
unexpected,
explicit examples of potentials having positive energy bound
states are known. These potentials are characterized by the fact that they
fall off slowly as $\frac{1}{r}$ and oscillate rapidly at infinity
[RSIV XIII.13].
\subsection{The GGMT-Bounds}
We consider $H = - \Delta + V(x)$ on $L^2({\bf R}^d), d \ge 3$, where
$V : {\bf R}^d \to {\bf R}$. For simplicity, we assume that $V$ is
$C^\infty_0$. For more general potentials such that one of the integrals
over $V$ given below is finite the bounds will follow by a standard limiting
argument.
\bigskip
\bigskip
\noindent
\underline{\bf Theorem 3.1.} \enspace Let $b,r,p$ satisfy the conditions
of the Sobolev inequalitites (2.14) i.e. $r \ge 1$, $p \ge 2$, $1-d/2 \le
b\le 1$ and $p = \frac{2d}{d-2+2b}\ ,\ r= \frac{p}{p-2}$. Let $K_{d,p}$ be the
constant given in (2.14b).
\bigskip
\noindent
{\bf (a)}\enspace If $r \ge d/2$, then $H$ has no bound state if
\begin{equation}
K^{2r}_{d,p} \inf_{x\in{\bf R}^d} \int_{{\bf R}^d} |x-y|^{2r-d} |V_-(y)|^r
dy < 1 \ .
\end{equation}
\bigskip
\noindent
{\bf (b)}\enspace If $1 \le r < d/2$, then for a spherically symmetric
potential $V(x) = v(r)$ there is no bound state if
\begin{equation}
K^{2r}_{d,p} \int_{{\bf R}^d} |x|^{2r-d} v_- (|x|) dx < 1 \ .\hfill \Box
\end{equation}
\bigskip
\noindent
\underline{\bf Remark:}\enspace In the two limiting cases $r = 1$ and
$r \to \infty$ we obtain the d-dimensional versions of conditions (3.2) and
(3.1), respectively. More precisely there is no bound state, if
\begin{equation}
\frac{1}{d-2} \int^\infty_0 v_- (r) r\ dr < 1
\end{equation}
which is the generalisation of Bargmann's bound and
\begin{equation}
\inf_{y\in{\bf R}^d} \sup_{x\in{\bf R}^d} |x-y|^2 V_-(x) < \left(
\frac{d-2}{2}\right)^2\ .
\end{equation}
For $r = d/2$ the left hand side of (3.6) is proportional to the
corresponding classical phase space expression.
\bigskip
\noindent
\underline{\bf Remark:}\enspace The conditions (3.5) are optimal in the
following sense. If $1 < r < \infty$ there are potentials having a zero
energy bound state solution and the left hand side of (3.6) is equal to
one. This follows from the fact that in Sobolev's inequality there is
an optimizing function. For $r=1$ and $r\to\infty$ we conclude by a limiting
argument as for Sobolev's inequality. Said more briefly, Sobolev-type
inequalities imply sharp conditions on the absence of bound states (and
vice versa!).
\bigskip
\bigskip
\noindent
\underline{\bf Proof of Theorem 3.1.}
\noindent
If we can show that the quadratic form associated with $H$
\begin{equation}
= \int_{{\bf R}^d} |\nabla\psi|^2 dy + \int_{{\bf R}^d}
V |\psi|^2 dy
\end{equation}
is nonnegative for all $\psi \in C^\infty_0 ({\bf R}^d),\ \psi$ real valued,
then it will follow that $H$ has no bound state.
\noindent
The first step is to prove part (a). Let $r \ge d/2$. Then $0\le b \le 1$.
Obviously $ \ge \int_{{\bf R}^d} |\nabla \psi |^2 dy -
\int_{{\bf R}^d}
V_-(y) |\psi |^2 dy$.
For any $x \in {\bf R}^d$ we rewrite the potential term as
$$
\int_{{\bf R}^d} |x-y|^{2b} V_- (y) |x-y|^{-2b}|\psi |^2 dy\ .
$$
Applying H\"older's inequality with exponents $r$ and $p/2$ and Sobolev's
inequality (2.14a), respectively we obtain
\begin{equation}
< H \psi ,\psi > \ge \| \nabla \psi \|^2_2 (1 - K^2_{n,p} \| |x-y|^{2b}
V_- \|_r)\ .
\end{equation}
With this we have established that the right hand side of (3.9) is positive
for some $x\in{\bf R}^d$ if $V$ satisfies (3.5a).
To prove part (b) we note that it is sufficient to consider only
spherically symmetric functions $\psi$ since the ground state, if it
exists, has necessarily spherical symmetry. Then we follow the same lines
except that we have to choose $x=0$. \enspace $\Box$
For spherically symmetric potentials GGMT also obtained conditions for the
absence of bound states of a given angular momentum $\ell$. In [St90] these
bounds were put into a more general context in connection with the
generalised Sobolev inequality (2.15).
Consider the Schr\"odinger operator defined as follows
\begin{eqnarray}
H(\alpha) &=& - \Delta - \alpha \left( \frac{d-2}{2}\right)^2
\frac{1}{|x|^2} + V(x) \\
&=& H - \alpha \left( \frac{d-2}{2}\right)^2 \frac{1}{|x|^2}\nonumber
\end{eqnarray}
where $0 \le \alpha < 1$ in the general case and $\alpha < 1$ for spherically
symmetric potentials $v(r)$.
The problem of finding a condition for the absence of bound states for given
angular momentum $\ell$ of the operator $-\Delta + v(r)$ is then equivalent
to the question how to find a condition such that $H(\alpha)$ has no bound
state for
$$
\alpha = - \left( \frac{2}{d-2}\right)^2 \ell (\ell + d - 2)\ .
$$
By using the generalised Sobolev inequality (2.15) we obtain the following
conditions implying the nonexistence of bound states for $H(\alpha)$, [St90].
\bigskip
\bigskip
\noindent
\underline{\bf Theorem 3.2.} \enspace Let $b,r,p, d$ and $K_{d,p}$ be as in
Theorem 3.1.
\begin{description}
\item{(a)} If $r \ge d/2$ and $0 \le \alpha < 1$, then $H(\alpha)$ has no bound
state if
\begin{equation}
K^{2r}_{d,p} (1-\alpha)^{\frac{1}{2}(1-2r)} \inf_{x\in{\bf R}^d}
\int_{{\bf R}^d} |x-y|^{2r-d} |V_-(y)|^r dy < 1 \ .
\end{equation}
\item{(b)} If $V(x) = v(r)$ and $\alpha < 1$, then $H(\alpha)$ has no bound
state if
\begin{equation}
K^{2r}_{d,p} (1-\alpha)^{\frac{1}{2}(1-2r)}
\int_{{\bf R}^d} v_-(|x|)^r |x|^{2r-d} dx < 1 \ . \hfill \Box
\end{equation}
\end{description}
The following corollary gives sufficient conditions for the absence of bound
states of a given angular momentum $\ell$.
\bigskip
\bigskip
\noindent
\underline{\bf Corollary 3.1.}\enspace Let $V$ be spherically symmetric,
i.e. $V(x) = v(r)$. Then $H$ has no bound state of angular momentum $\ell$
if
\begin{equation}
K^{2r}_{d,p} \left(\frac{d-2+2\ell}{d-2}\right)^{1-2r} \int_{{\bf R}^d}
|v_- (|x|)|^r |x|^{2r-d} dx < 1 \ . \hfill \Box
\end{equation}
\bigskip
\noindent
To obtain a condition such that $N_E (V) = 0$ for some $E < 0$ we may replace
$V_-$ by $|V(x)-E|_-$. However, such conditions are no more optimal. We study
improved conditions for $E < 0$ in the next subsection.
We conclude this subsection by discussing a conjecture by A.~Martin
(private communication) about
a possible extension of Theorem 3.1 for non-spherically symmetric potentials.
The conjecture is that part (b), in the case $1 \le r < d/2$ should extend to
non-spherically symmetric potentials in the following sense.
If, for \newline\hfill
$1 \le r < d/2$ the following inequality holds
\begin{equation}
K^{2r}_{d,p} \sup_{x\in{\bf R}^d} \int_{{\bf R}^d} |x-y|^{2r-d}
|V_-(y)|^r dy < 1 \ .
\end{equation}
Then $H = - \Delta + V$ has no bound state.
It turns out from a paper by Hunziker [Hu61] on the convergence of the Born
series, that $H$ has no bound state for $d=3$, if
\begin{equation}
\sup_{x\in{\bf R}^3} \frac{1}{4\pi} \int_{{\bf R}^3} \frac{V_-(y)}{|x-y|}
dy < 1
\end{equation}
which is the natural extension of the Bargmann bound (3.1) to nonradial
potentials. To prove (3.17) we consider the integral equation
\begin{equation}
\psi (x) = - \int_{{\bf R}^3} G_E (x-y) V(y) \psi(y)\ dy
\end{equation}
with $G_E (x-y) = \frac{1}{4\pi} \frac{1}{|x-y|} e^{-\sqrt{-E}|x-y|}$.
Taking the sup-norm on both sides of (3.16) yields that if (3.16) has a
bounded solution then $\sup_{x\in{\bf R}^3} \int_{{\bf R}^3} G_E (x-y)
|V_- (y)| dy \ge 1$.
For $r\in (1,d/2)$ Martin and Sabatier [MS77] could prove the following. If
\begin{equation}
\sup_{x\in{\bf R}^3} \int_{{\bf R}^3} |x-y|^{r-2} |x|^{r-1} V_-(x)^r <
\frac{1}{4\pi}\left( \frac{\pi^2}{4} \right)^{1-r}
\end{equation}
then $H$ has no bound state.
\subsection{Absence of bound states and lower bounds on Hamiltonians}
This subsection is concerned with some lower bounds on $H = - \Delta + V$
in terms of integral expression of $V$ and conditions which guarantee
$N_E(V) = 0$ for negative energies.
We start from the nonlinear functional
\begin{equation}
G_p (u) = \frac{\| \nabla u \|^2_2 + \| u \|^2_2}{\| u \|^2_p}\ ,
\end{equation}
defined on $H^1({\bf R}^d)$, where $p \in (2,2^\ast)$ if $d\ge 3$ and
$p \in (2,\infty)$ if $d=1,2$. We define
\begin{equation}
g_p = \inf_{u\in H^1({\bf R}^d)} G_p (u)\ .
\end{equation}
It is now of interest to introduce the related functional $F_p(u)$ given by
\begin{equation}
F_p (u) = \frac{\| \nabla u \|^\alpha_2 \| u \|^{2-\alpha}_2}{\| u \|^2_p}
\end{equation}
with $\alpha = d(1-\frac{2}{p})$. Similarly, we define
\begin{equation}
f_p = \inf_{u\in H^1({\bf R}^d)} F_p (u)\ .
\end{equation}
We need one more prepatory well known result, e.g.~[W83, BB]:
\bigskip
\noindent
\underline{\bf Proposition 3.1.}\enspace The minima $g_p$ and $f_p$ are attained
by some function \hfill\newline
$u\in H^1 ({\bf R}^d)$. By scaling we have
\begin{equation}
g_p = (\frac{\alpha}{2})^{-\alpha/2}
(\frac{2-\alpha}{2})^{-\frac{2-\alpha}{2}} f_p\ .
\end{equation}
\bigskip
\noindent
\underline{\bf Remark:}\enspace If $d\ge 3$, we see by using H\"older's and
Sobolev's inequality, respectively
\begin{equation}
f_p > K^{-\alpha}_{d,2^\ast} \qquad\qquad \mbox{if}\quad d \ge 3
\end{equation}
and for $d=1$ we see from (3.16)
\begin{equation}
g_p = \frac{1}{M_p} \qquad\qquad \mbox{if}\quad d=1
\end{equation}
where $M_p$ is the constant given in (3.16b). \hfill $\Box$
\bigskip
We prove the following:
\bigskip
\bigskip
\noindent
\underline{\bf Theorem 3.3.}\enspace Let $p > 2$ and $p < 2^\ast,$ if
$d \ge 3$. Let $r = \frac{p}{p-2}$ and $\gamma = \frac{p}{p-2} - \frac{d}{2}$.
Then for $\gamma > 0$ we have
\begin{equation}
H \ge - g_p^{-r/\gamma} \| V_- \|_r^{r/\gamma}\ .
\end{equation}
Equivalently, if for $E < 0$
\begin{equation}
g^{-r}_p \int_{{\bf R}^d} V_- (x)^{\frac{d}{2} + \gamma} dx < |E|^\gamma
\end{equation}
then $N_E (V) = 0$. \hfill $\Box$
\bigskip
\bigskip
\noindent
\underline{\bf Remark:}\enspace From Theorem 3.2.~one concludes that the
Sobolev-type inequality
$G_p (u) \ge g_p$ (or $F_p (u) \ge f_p$) implies the lower bound (3.22) on the
Hamiltonian $H$. On the other hand (3.22) implies
$$
\ \ge - g_r^{-r/\gamma} \| V_-\|_r^{r/\gamma} \| u\|^2_2
$$
for all suitable $u$ and $V_-$. Taking $V_- = \lambda |u|^{p/r} = \lambda
|u|^{p-2}$ we obtain
$$
\| \nabla u\|^2_2 - \lambda \| u \|^p_p + g_p^{-r/\gamma} \lambda^{r/\gamma}
\| u \|_p^{p/\gamma} \| u \|^2_2\ .
$$
Optimizing with respect to $\lambda$ yields $F_p (u) \ge f_p$. So the lower
bound (3.22) implies a Sobolev-type inequality.
\bigskip
\bigskip
\noindent
\underline{\bf Proof of Theorem 3.3.}
For all $\psi \in C^\infty_0 ({\bf R}^d)$ we obtain by using H\"older's
inequality and the Sobolev-type inequality $F_p (u) \ge f_p$, respectively
\begin{eqnarray}
&\ge& \| \nabla\psi\|^2_2 - \| V_-\|_r \| \psi\|^2_p\\
&\ge& \| \nabla\psi\|^2_2 - f^{-1}_p \| V_-\|_r \|\nabla\psi
\|^\alpha_2 \|\psi\|^{2-\alpha}_2\nonumber
\end{eqnarray}
with $\alpha = d(1-\frac{2}{p}) = d/r$.
\smallskip
\noindent
Optimizing with respect to $\|\nabla\psi\|_2$ and using relation (3.20) between
$f_p$ and $g_p$ we deduce the desired result. \hfill $\Box$
\bigskip
We turn now to similar lower bounds in momentum space. For an eigenvalue
$E < 0$, the Schr\"odinger equation in $p$-space is given by (2.9) and the
quadratic form associated to $H-E$ for any $E < 0$, can be written as
\begin{eqnarray}
\lefteqn{<(H-E) \psi,\psi>} \nonumber\\
&=& \int T_E (p) | \hat{\psi} (p) |^2 + (2\pi)^{-d} \int\!\!\!\int \hat{\psi}
(p)^\ast \hat{V} (p-p^\prime) \hat{\psi} (p^\prime) dp\ dp^\prime
\end{eqnarray}
with $T_E(p) = p^2 - E$.
The first result is derived using Hardy-Littlewood Sobolev's inequality (2.18)
\bigskip
\bigskip
\noindent
\underline{\bf Theorem 3.4.} \enspace Let $2\ge q > \max (1,\frac{2d}{d+2})$
and $r^\prime$ given by $\frac{1}{r^\prime} = 2(1 - \frac{1}{q}) =
\frac{2}{q^\prime}, \frac{1}{r} + \frac{1}{r^\prime} = 1$ and $r = \frac{d}{2}
+ \gamma$. If $\hat V$ satisfies
\begin{equation}
(2\pi)^{-d} N_{q, \frac{d}{r^\prime},d} ( \int_{{\bf R}^d} T_{-1}
(p)^{-\frac{q}{2-q}})^{\frac{2-q}{q}} \| \hat{V} \|_{r^\prime,w}
\le |E|^{\gamma/r}
\end{equation}
for $E < 0$, where $N_{q,\frac{d}{r^\prime},d}$ is the constant in the
Hardy-Littlewood-Sobolev inequality (2.18) and $T_{-1} (p) = p^2 +1$,
then $N_E (V) = 0$.
In particular
\begin{equation}
H \ge - (\mbox{l.h.s.\ of\ (3.31)})^{r/\gamma}\ . \hfill \Box
\end{equation}
This statement is reinforced by the fact that these estimates are sharp and
saturated for Coulomb potentials $V(x) = - \frac{Z\alpha}{|x|}$ for $d \ge 2$.
\bigskip
\noindent
\underline{\bf Remark:}\enspace Note that
\begin{equation}
\left( \int_{{\bf R}^d} T_{-1} (p)^{-\frac{d}{2-d}} dp\right)^{\frac{2-q}{q}}
= \left[ \frac{\pi^{d/2} \Gamma (\frac{q}{2-q} - \frac{d}{2})}{\Gamma
(\frac{q}{2-q})} \right]^{\frac{2-q}{q}}\ .
\end{equation}
Replacing Hardy-Littlewood-Sobolev's inequality by Young's inequality
we obtain
\bigskip
\bigskip
\noindent
\underline{\bf Corollary 3.2.} Let $q,r^\prime, r$ as in Theorem 3.4.
If $\hat{V}$ satisfies for some $E < 0$
\begin{equation}
(2\pi)^{-d} [C^2_q C_{r^\prime} ]^d \| T^{-1}_{-1} \|_{\frac{q}{2-q}}
\| \hat{V} \|_{r^\prime} \le |E|^{\gamma/r}
\end{equation}
then $N_E(V) = 0$.
\bigskip
If $r^\prime \ge 2$ (i.e. $q \le 4/3$ which gives a non-empty condition if
$d \le 3$) we may apply the Hausdorff-Young inequality (2.20) which gives
a condition of the form (3.23) which, however, is not sharp, but has an
explicit constant for $d=2$.
\subsection{About dimensions one and two}
We want to give a precise version of the statement that any reasonable
potential in one or two-dimensions has a bound state.
\bigskip
\noindent
It is easy to see that, if $V\in C^\infty_0 ({\bf R}^d) , d = 1,2, V \le 0,
V \not\equiv 0,$ then the Birman-Schwinger operator $K_E$ or ${\hat K}_E$
given in (2.5) and (2.10) respectively, is Hilbert-Schmidt for all $E < 0$
but that its operator norm diverges as $E \to 0$. Hence $K_E$ has at
least one eigenvalue bigger than one for all $E$ sufficiently close to zero.
By the Birman-Schwinger principle we conclude that H has a bound state.
\bigskip
\noindent
For $d = 1$, a more general result is proven in [AHS]. We can do more and
extend this to $d = 2$.
\bigskip
\noindent
\underline{{\bf Proposition 3.2.}}\enspace
Let $V \in L^1 ({\bf R}^d) + (L^\infty
({\bf R}^d))_\epsilon,\ d = 1, 2.$ Then $H = - \Delta + V$ has a negative
eigenvalue if any of the following conditions holds:
\begin{description}
\item{(i)} $V \le 0 \qquad V \not\equiv 0 $
\item{(ii)} $\int |V| < \infty \qquad \int V < 0$
\item{(iii)} $\int V_+ < \infty \qquad \int V_- = \infty \qquad where\ V_\pm
= \sup (0, \pm V)$
\end{description}
\bigskip
\bigskip
\noindent
\underline{{\bf Proof:}}\enspace With the help of $L^1({\bf R}^d) + L^\infty
({\bf R}^d)_\epsilon$-assumption we find immediately that the essential spectrum
of $H$ is $[0,\infty)$. Therefore it is sufficient to find a trial function
$\psi$ in $L^2({\bf R}^d)$ such that $\ <0$.
\bigskip
\noindent
Note that by increasing $V$, we can suppose that we are in the second case.
\bigskip
\begin{description}
\item{\underline{d=1:}} Taking $\psi_a = e^{-a|x|}, a > 0,$ then
\[ \int_{\bf R} |\nabla \psi_a |^2 = a^2 \int_{{\bf R}^d} e^{-2a|x|}
= a
\]
and
\[
\lim_{a\to 0} \int_{\bf R} V (x) |\psi_a (x) |^2 = \int_{\bf R} V(x) < 0
\]
by the dominated convergence theorem.
\item{\underline{d=2:}} Taking $\psi_a = e^{-\frac{1}{2} |x|^a}, \ a > 0,$
then
\[
\int_{{\bf R}^2} \|\nabla \psi_a\|^2 = \frac{\pi}{2} a
\]
and
\[
\lim_{a\to 0} \int_{{\bf R}^2} V(x) |\psi_a (x) |^2 = \int_{{\bf R}^2}
V(x) < 0 \ . \qquad \Box
\]
\end{description}
\bigskip
\noindent
\section{Central Potentials - Bounds on $N_0 (V)$}
If $V(x) = v(r), r = |x|$ then $H = - \Delta + v(r)$ splits into an
infinite family of radial Hamiltonians $h_\ell$ which read in its reduced
form
\begin{equation}
h_\ell = - \frac{d^2}{dr^2} - \frac{d-1}{r} \frac{d}{dr} + \frac{(\ell +
\frac{d-2}{2})^2}{r^2} + v(r)
\end{equation}
$\ell = 0, 1, \ldots$.
\bigskip
\noindent
Indeed, if $\psi$ is an eigenfunction of $H$, the eigenvalue problem reduces
to an eigenvalue problem for $h_\ell$ by introducing the reduced wave function
\begin{equation}
\psi (x) = r^{\frac{1-d}{2}} u_\ell (r) Y_\ell (x/r).
\end{equation}
\noindent
The degeneracy of each level is determined by the number $H_\ell$ of
harmonic polynomials of degree $\ell$ in $d$ dimensions
\begin{equation}
D_\ell = H_\ell - H_{\ell -2} \qquad \mbox{with}\ \ H_\ell = {d+\ell -1 \choose \ell}\ .
\end{equation}
We quote the following examples as an illustration:
\noindent
For $d=2, D_\ell = 2$ if $\ell \ge 1$, if $d=3, D_\ell = 2\ell + 1$ and
if $d=4, D_\ell = (\ell + 1)^2$.
\bigskip
\noindent
By $n_\ell (v)$ we denote the dimension of the spectral projection
$P_{(-\infty,0)}$, of $h_\ell$. Obviously $n_\ell$ satisfies Lemma 2.1, and,
in particular $n_\ell$ is monotone in $\ell$. The total number of bound
states $N_0 (v)$ is related to the $n_\ell's$ by
\begin{equation}
N_0 (v) = \sum^\infty_{\ell = 0} D_\ell n_\ell (v)\ .
\end{equation}
\bigskip
\noindent
For our purposes the following two results are important
\bigskip
\bigskip
\noindent
\underline{{\bf Theorem 4.1.}}\enspace Let $V \in C^\infty_0 ({\bf R}^d)$
be the spherically symmetric, i.e. $V(x) = v(r)$.
\noindent
Let $u_\ell$ be the zero energy solution for $h_\ell$ satisfying Dirichlet
boundary conditions at zero, i.e.
\[
h_\ell u_\ell = 0 \qquad \qquad u_\ell (0) = 0\ .
\]
\noindent
Then $n_\ell (v)$ is the number of zeros of $u_\ell$ in $(0,\infty)$.
\bigskip
\bigskip
\noindent
\underline{{\bf Theorem 4.2.}}\enspace Let $I_\ell (v) = \int^\infty_0
\rho_\ell (r) |v(r)|^p dr, \ l \ge 0,$ for $\rho_\ell \ge 0$. If
$I_\ell (v) < 1$ implies $n_\ell (0) = 0$ then
\begin{equation}
n_\ell (v) \le I_\ell (v)\ .
\end{equation}
\bigskip
\noindent
The proof of Theorem 4.1 follows from the mini-max principle and the Sturm-
Liouville oscillation theorem for ordinary differential equations [RSIV].
\bigskip
\noindent
Theorem 4.2.~is a consequence of Theorem 4.1.
\smallskip
\noindent
Indeed let $u_\ell$ be the zero energy solution for $h_\ell$ and let
$r_0 = 0$, $r_1 \ldots r_{n_{\ell}(v)}$ be its zeros.
\bigskip
\noindent
Then on each interval $(r_{i-1}, r_i) (i=1, \ldots, n_\ell (v)$, the operator
$h_\ell$ restricted to $(r_{i-1}, r_i)$ with Dirichlet boundary conditions
has a zero energy eigenstate given by $u_\ell \chi_{(r_{i-1},r_i)}$. Hence
$I_\ell (V \chi_{(r_{i-1}, r_i)} ) \ge 1$. Summing over all intervals yields
$I_\ell (v) \ge n_\ell (v)$ [GGMT76, Si76a].
\bigskip
\noindent
Hence we may take the conditions of Corollary 3.1.~about the nonexistence
of bound states in the sector $\ell$ to obtain
\bigskip
\bigskip
\noindent
\underline{{\bf Corollary 4.1.}}\enspace Let $d\ge 3$, $p \ge 2$,
$1 \in r \le \infty$ with $r = \frac{p}{p-2}$. If $K_{d,p}$ denotes the
Sobolev constant in (2.14 b) then
\begin{equation}
n_\ell (v) \le \left( \frac{d-2}{2\ell + d-2}\right)^{2r-1}
K^{2r}_{d,p} \int_{{\bf R}^d} | v_- (|x|)^r |x|^{2r-d} dx \ .
\end{equation}
\bigskip
\noindent
Corollary 4.1.~extends to $d=2$ by noting that the constant on the
r.h.s.~of (4.6) has a finite limit for $l > 0$ as $d \to 2$. For $\ell = 0$
the bound is divergent reflecting the fact that a negative potential
always has a bound state in $d=2$.
\bigskip
\noindent
\underline{{\bf Remark:}} We note in passing that for $r=1$ we recover
Bargmann's bound in $d$ dimensions
\begin{equation}
n_\ell (v) \le \frac{1}{2\ell + d - 2} \int^\infty_0 v_- (r) r\ dr\ .
\end{equation}
\bigskip
\noindent
However, the original proof given by Bargmann [Ba52] is different.
\bigskip
\bigskip
\noindent
To obtain a bound on the total number of bound states we have, according
to (4.4), to sum over all sectors up to a maximal angular momentum
$\ell_{\max}$ which is the biggest integer $\ell$ such that the r.h.s.~of
(4.6) is less or equal than one.
\bigskip
\noindent
We define $I_r (V) = (d-2)^{2r-1} K^{2r}_{d,p} \int_{{\bf R}^d} V_-
(|x|)^r |x|^{2r-d} dx$.
\bigskip
\noindent
The interesting case $r = d/2$ which corresponds to the classical
phase space volume expression
gives a bound which always contains a logarithm of I.
\bigskip
\noindent
For simplicity, we consider only the case $d=3$. Then we obtain, replacing
sums by integrals,
\begin{equation} N_0 (V) \le I_{3/2} (V) [1 + \frac{1}{4} \ell n I_{\frac{3}{2}}
(V)]
\end{equation}
for $I_{\frac{3}{2}} (V) \ge 1$.
\bigskip
\noindent
It is worth noting that although in each sector $\ell$ the bound is
optimal, it is saturated for different potentials. Hence suming over
various values of $\ell$ we overestimate the number of bound states.
On the other hand the first term in (4.7) is optimal since in view of the
nonexistence condition (Theorem 3.1) the best one can hope to prove is
$N_0 (v) \le I_{3/2} (V)$.
\bigskip
\noindent
The problem of overestimating the number of bound states by the summation
procedure was partially overcome in [GGM78] by making a different use of Theorem 4.1.~and
relating bounds on $N_0 (V)$ in $d$ dimensions for a spherically symmetric
potential $v(r)$ to estimates on $S_\gamma (v)$ in one dimension.
\bigskip
\noindent
For $d=3$ they obtained $N_0 (V) \le C I_{\frac{3}{2}} (V)$ with $C > 1$ while for
$d=4$ they proved $N_0 (V) \le I_2 (V)$ for $V$ spherically symmetric.
\bigskip
\noindent
There is another bound on $n_\ell (V)$ due to Calogero which requires
a careful analysis of the zero energy solutions $u_\ell$ [Ca65]. More
precisely, if $v(r) \le 0, v^\prime (r) \ge 0$ then
\begin{equation}
n_\ell (v) \le \frac{2}{\pi} \int^\infty_0 |v(r)|^{1/2} dr\ .
\end{equation}
\noindent
The bound is optimal in the sector $\ell = 0$ and saturated by a square
well potential.
\bigskip
\noindent
Recently, a sharper bound for $\ell > 0$ was found [CMS95a]
\begin{equation}
n_\ell (v) \le \frac{2}{\pi} \int^\infty_0 |v(r)|^{\frac{1}{2}} dr + 1 -
\sqrt{1 +(\frac{2}{\pi})^2 \ell (\ell + 1)}\ .
\end{equation}
\bigskip
\noindent
The bound (4.9) looks like an extension of the bounds (4.6) to $r = \frac{1}{2}$.
This suggested to look for bounds containing integrals of powers of the
potential between $1/2$ and $1$. [GGMT76]
\bigskip
\noindent
Recently, this gap was closed in [CMS95b] where it was shown that if
$v(r) \le 0$ and $\frac{d}{dr} r^{1-2p}(-v)^{1-p} \le 0$ for some
$p \in [\frac{1}{2},1)$ then
\begin{equation}
n_\ell(v) \le p(1-p)^{p-1} (2\ell +1)^{1-2p} \int^\infty_0 (-r^2v)^p
\frac{dr}{r}\ .
\end{equation}
It should be stressed that the constants in (4.11) except in the limit
$p \to 1$ are not sharp.
Finally, Chadan and Kobayashi extended Calogero's bound to a larger class
of potentials. [CK95, CKMS95]. More precisely, if $v(r) \le 0$ and
$\frac{d}{dr} r^{1-2p} (-v)^{1-p} \le 0$ for some $p \in [\frac{1}{2},1)$
then
\begin{equation}
n_\ell (v) \le \frac{1}{2\sqrt{1-p}} \int^\infty_0 (-v)^{1/2} dr\ .
\end{equation}
\section{Phase Space Bounds: The General Case}
In this section we present bounds on $S_\gamma (V) = \sum |E_j|^\gamma$
in terms of the corresponding classical phase space expression, i.e.~bounds
of the form (1.5) and (1.6), respectively. For $\gamma > 0$, the first
bounds have been obtained by Lieb and Thirring in 1975 by using the one
parameter family of generalised Birman-Schwinger operators
$K_{\alpha E}$ defined in (2.7), and the Birman-Schwinger principle
as formulated in Theorem 2.2. In particular, they were interested in the case
$\gamma = 1, d = 3$ since this estimate played an important role in a simple
proof of the "stability of matter" (see also Section 6 of this review).
\noindent
The case $\gamma = 0$ was still an open problem. The first bounds on $N_0 (V)$
having the correct large coupling behaviour are due to Simon [Si69] and
Martin [M75], however they are not in terms of the classical phase space
expression. The break through was independently achieved in 1976
by Cwickel [Cw77], Lieb [L76a] and Rosenbljum [Ro72]. We shall discuss these
methods seperately. In 1983 a different, simple method has been proposed
by Li and Yau [LY83] and later be generalised by Blanchard, Rezende and
Stubbe [BRS87] yielding an elementary proof of the bound on $N_o (V)$. Another approach
has been proposed by Conlon [Co85].
\bigskip
\noindent
Once such bounds have been established the question for best constants
in these estimates was raised [LT76, L89]. To be more precise we would like
to know what is
\begin{equation}
R(\gamma, d) \equiv \sup_V \left\{
\frac{\sum_{E_j \le 0} |E_j|^\gamma}{(2\pi)^{-d} {\int\int}_{H(x,p)\le 0}
| H (x,p) |^\gamma dx\ dp} \right\},
\end{equation}
or equivalently
\begin{equation}
R(\gamma, d) \equiv \sup_V \left\{
\frac{S \gamma (V)}{L^{class}_{\gamma, d} \int_{{\bf R}^d}
| V_- |^{d/2 + \gamma} dx} \right\}\ .
\end{equation}
\bigskip
\noindent
In view of the asymptotic results presented in Theorem 1.1, and Corollary
1.1, respectively, we always have $R(\gamma,d) \ge 1$. We shall
discuss these problems together with some general properties of
$R(\gamma, d)$ in Section 5.5.
\bigskip
\noindent
In this article we shall obtain the currently best constants in the case $\gamma =1$
by combining Lieb's method on Wiener integrals and the generalised Birman-Schwinger
operators $K_{\alpha E}$. [BS93]
\subsection{Cwickel's bound}
Cwickel proved the following result:
\noindent
Let A be an integral operator defined by
\[
(A\phi) (x) = \int_{{\bf R}^d} f (x-y) g(y) \phi(y) dx
\]
with $f\in L^{p^\prime}_{weak} ({\bf R}^d)$ and $g \in L^p ({\bf R}^d)$
where $2 < p < \infty$.
\noindent
By $\mu_k$ we denote its singular values.
\bigskip
\bigskip
\noindent
\underline{{\bf Theorem 5.1.}} Under the above assumptions A is a bounded
operator on $L^2({\bf R}^d)$ with singular values $\mu_k$ obeying
\[
|\mu_k | \le c_{p,d} \| f \|^\ast_{p^\prime} \|g \|_p k^{-1/p}
\]
where $c_{p,d}$ is a constant depending only on $p$ and $d$.
\bigskip
\noindent
In particular
\[
c_{p,d} \le \frac{p}{2} \left( \frac{4}{\frac{p}{2} - 1} \right)^{1-2/p}
(B_d)^{1/p} \pi^{d/p}. \hfill \Box
\]
\bigskip
\noindent
The Theorem is applied to the operator given by $(-\Delta)^{-\frac{1}{2}}
| V_- |^{\frac{1}{2}},\ d \ge 3$ and $p=d$.
\bigskip
\noindent
For $f(x) = \frac{1}{2} \pi^{-\frac{d+1}{2}} \Gamma \left( \frac{d-1}{2}
\right) | x |^{1-d}$ with $\| f \|^\ast_{d-1} = \frac{1}{2}
\pi^{-\frac{d+1}{2}} \Gamma \left( \frac{d-1}{2} \right)$ and $g(x) = V_- (x)$
we find by the Birman-Schwinger principle
\begin{equation}
N_0 (V) \le c^d_{d,d} 2^{-d} \pi^{-\frac{d+1}{2}d} \Gamma
\left( \frac{d-1}{2} \right)^d \int_{{\bf R}^d} V_-^{\frac{d}{2}} dx
\end{equation}
which yields the constant $\frac{27}{2\pi^2}$ if $d=3$ which is about
17 times bigger than the best possible value $K^3_{3,2^\ast} =
\frac{4}{3\sqrt{2} \pi^2} \ .$
\subsection{Lieb's method}
Lieb's derivation of the phase space bound on $N_E (V)$ makes use of the
Wiener Integral representation of functions $F$ of the Birman-Schwinger
operator $\kappa_E$ given in (2.5).
\bigskip
\noindent
Let $d\mu_{x,y,t}$ be the conditional Wiener measure on paths $\omega(t)$
with $\omega (0) = x$ and $\omega (t) = y$. This measure allows
a representation of the Green's function for the semigroup $e^{t\Delta}$
by
\begin{equation}
\int d\mu_{x,y,t} (\omega) = (4\pi t)^{-d/2} \exp \left( -
\frac{|x-y|^2}{4t} \right)\ .
\end{equation}
\bigskip
\noindent
By means of the Feynman-Kac formula Lieb proved the following
\bigskip
\bigskip
\noindent
\underline{{\bf Theorem 5.2.}} [Lieb's trace formula] Let $V \le 0$ and
$V \in L^p ({\bf R}^d) + L^q ({\bf R}^d),$
with $p = \frac{d}{2}$ (if $d \ge 3$), $p > 1 (d=2), p = 1 (d=1),$
and $p < q < \infty$.
Let $f$ be a nonnegative lower semicontinuous function on $[0,\infty)$
satisfying $f(0) = 0$ and $x^r f(x) \to 0$ as $x \to \infty$ for some
$r < \infty$. Define $F$ by
\begin{equation}
F(x) = \int^\infty_0 f(x,y) e^{-y} \frac{dy}{y}\ .
\end{equation}
Then
\begin{equation}
Tr F (K_E) = \int^\infty_0 \frac{dt}{d} e^{-|E|t} \int_{{\bf R}^d}
\int d\mu_{x,x,t} f (\int^t_0 V_- (\omega(s)) ds)\ .
\end{equation}
\bigskip
\noindent
If $f$ is in addition a convex function we may apply Jensen's inequality
to the last term in Lieb's trace formula. Then we have
\begin{equation}
f ( \int^t_0 V_- (\omega(s)) ds) \le \frac{1}{t} \int^t_0 f(tV_- (\omega
(s))) ds \ .
\end{equation}
\bigskip
\noindent
By Fubini's Theorem we may change the order of integration $ds$ and $d\mu$ with $dx$
and thereby obtain the inequality
\begin{equation}
T_r f(K_E) \le \int^\infty_0 \frac{dt}{t} e^{-|E|t} \int d\mu_{0,0;t}
\int^t_0 \frac{ds}{t} \int_{{\bf R}^d} dx\ f(V_- (\omega (s) + x)\ .
\end{equation}
\noindent
Since $dx$ is translation invariant we may drop the $\omega-$term for
any field path $t \to \omega (t)$ and we are now in position to perform
the s-integration and the integration with respect to $d\mu_{0,0,t}$
using the representation (5.4).
\bigskip
\noindent
We find [L76a, L80]
\bigskip
\bigskip
\noindent
\underline{{\bf Corollary 5.1.}} Let $f$ be convex and satisfy the conditions
of Theorem 5.2. Then
\begin{equation}
Tr F(K_E) \le (4\pi)^{-d/2} \int^\infty_0 \frac{dt}{t} t^{-d/2} e^{-|E|t}
\int_{{\bf R}^d} dx f (t V_- (x)).
\end{equation}
\bigskip
\bigskip
Letting $E \nearrow 0$ and doing a change of the $t$ variable we obtain
using the Birman-Schwinger principle.
\bigskip
\bigskip
\noindent
\underline{{\bf Corollary 5.2.}} Let $f$ be convex and satisfy the conditions
of Theorem 5.2. Then
\begin{equation}
N_0 (V) \le \frac{\int^\infty_0 \frac{ds}{s} s^{-d/2} f(x)}{\int^\infty_0
\frac{ds}{s} e^{-s} f(s)} (4\pi)^{-d/2} \int_{{\bf R}^d} | V_- (x)|^{d/2} dx
\end{equation}
or
\begin{equation}
R (0,d) \le \Gamma (\frac{d}{2} + 1) \frac{\int^\infty_0 \frac{ds}{s}
s^{-d/2} f(s)}{\int^\infty_0 \frac{ds}{s} e^{-s} f(s)}\ .
\end{equation}
\bigskip
\noindent
>From this we get that the best constant is given by
\begin{eqnarray}
\ell_d &\equiv & \inf \left\{ \Gamma (\frac{d}{2} + 1 ) \int^\infty_0
\frac{ds}{s} s^{-d/2} f(s) / \int^\infty_0 \frac{ds}{s} e^{-s} f(s) \right.
\nonumber \\
& & \left. f \not\equiv 0, f (0) = 0, f\ \mbox{nonnegative and convex}\right\} .
\end{eqnarray}
\noindent
To make use of this inequality we choose
\[
f(s) = \left\{ \begin{array}{ll}
0 & b \le x\\
s-x & b \ge x
\end{array} \right.
\]
for some $x > 0$ and for $d > 2$ we arrive at
\begin{equation}
\ell_d \le \min_{x > 0} \frac{\Gamma (\frac{d-2}{2})}{x^{d/2} \Gamma
(-1,x)}
\end{equation}
where $\Gamma (a,x) = \int^\infty_x t^{a-1} e^{-t} dt $ denotes the incomplete
$\Gamma$-function.
If $d=3$ we obtain $\ell_3 \le 0.116$\ .
\bigskip\noindent
We note that the above choice of $f$ is the best possible as can be seen from the
following formal calculation.
Rewriting the right hand side of (5.11) as
\[
I_{a,b} (f) = \frac{\int^\infty_0 a^{\prime\prime} (s) f(s)}{\int^\infty_0
b^{\prime\prime} (s) f(s)}
\]
with $a(s) = \Gamma (\frac{d-2}{2} ) s^{+1-d/2}$
and
\[
b(s) = \int^\infty_s \int^\infty_t \frac{e^{-t^\prime}}{t^\prime} dt^\prime dt =
s \Gamma (-1,s)
\]
we find by integration by parts
\[
I_{a,b} (f) = \frac{\int^\infty_0 a (s) f^{\prime\prime}(s)}{\int^\infty_0
b (s) f^{\prime\prime}(s)} \ge \min \frac{a(s)}{b(s)}
\]
which is attained if $f^{\prime\prime} (s) \simeq \delta (s-x)$ where $x$
is the position of the minimum of $\frac{a(s)}{b(s)}$.
Bounds on $R(\gamma,d)$ can be obtained in the following way [L84].
Since $N_E(V) \le \frac{Tr F(K_E)}{F(1)}$ Corollary 5.1 yields a
bound on $N_E(V)$. Using the integration formula (2.4) we do the
$E$-integration. Dividing both sides by the classical constant
$L^{\mbox{class}}_{\gamma,d}$ given in (1.7) we obtain
\bigskip
\bigskip
\noindent
\underline{{\bf Corollary 5.3.}}\enspace Let $\gamma \ge 0$ and $d + 2\gamma > 2$.
Then
\begin{equation}
R(\gamma,d) \le \ell_{d+2\gamma}
\end{equation}
with $\ell_d$ defined in (5.12). In particular
\begin{equation}
R(\gamma,d) \le \min_{x>0} \frac{\Gamma (\frac{d+2\gamma-2}{2})}{x^{d/2+\gamma}
\Gamma (-1,x)}. \hfill \Box
\end{equation}
\bigskip
\noindent
Evidently, we also have
\begin{equation}
R(\gamma,d) \le \ell_d
\end{equation}
provided $\gamma \ge 0$ and $d > 2$. Note that the range of $\gamma$ and
$d$ in Corollary 5.3 is bigger and we shall see later that this condition
is optimal since there is no such bound for $\gamma = 0$ and $d=2$ (by
the result of Section 3) and for $0 \le \gamma < \frac{1}{2}$ and $d=1$.
In Section 5.6 we shall prove a result by Aizenman and Lieb [AL78]
which states that $R(\gamma,d)$ is nonincreasing in $\gamma$. (Theorem 5.5)
Therefore we have
\bigskip
\bigskip
\noindent
\underline{\bf Corollary 5.4.}\enspace Let $\gamma \ge 0$ and $d +
2\gamma > 2$.
Then
\begin{equation}
R(\gamma ,d) \le \min_{0\le\epsilon\le\gamma} \ell_{d+2\epsilon}\ .
\hfill \Box
\end{equation}
\bigskip
\bigskip
\noindent
Setting $y = \frac{d}{2} + \gamma$ we consider the function
\[
F(x,y) = \frac{\Gamma (y-1)}{x^y \Gamma (-1,x)}\ .
\]
Numerically, we find its minimum at $x_0 = 0.8984, y_0 = 2.3563$ and
$F(x_o, y_o) = 5.95884$ which shows that for all $d\le 4$ and $\gamma$
such that $d/2 + \gamma \ge y_0$
\begin{equation}
R(\gamma,d) \le 5.95884\ .
\end{equation}
\bigskip
\bigskip
\noindent
\underline{\bf Remark:}\enspace An extension of Lieb's method to more
general kinetic energy terms $T(p)$, including in particular the
relativistic expression $T(p) = \sqrt{p^2 + m^2} - m$ where $m$ denotes
the mass of the particle can be found in [D83].
\subsection{The method of Li and Yau}
A different way of exploiting the Birman-Schwinger principle and connecting
it with Sobolev inequalities was presented in a beautiful paper by Li and Yau
[LY]. The problem of counting the bound states of a Schr\"odinger operator
will be related to the eigenvalues of the following Dirichlet boundary
value problem
\begin{eqnarray}
- \frac{1}{q(x)} \Delta f & = & \lambda f \qquad\qquad \mbox{on}\quad \Omega
\subset {\bf R}^d \\
f|_{\partial \Omega} & = & 0 \nonumber
\end{eqnarray}
in the space $L^2(\Omega , q(x) dx)$ where $\Omega$ denotes some bounded
domain in ${\bf R}^d$ and $q(x)$ is a strictly positive function on
$\bar{\Omega}$. If one takes
$$
q(x) = V_-(x)
$$
then the number of eigenvalues below 1 of the boundary value problem is equal
to the number of bound states of the Schr\"odinger operator $- \Delta_D - q(x)$
where $-\Delta_D$ denotes the Dirichlet Laplacian on $\Omega$. By some
technical steps, which we shall describe below, we may perform the limit
$"\Omega \to{\bf R}^d"$ to the operator $- \Delta - q(x)$ on ${\bf R}^d$.
The main result is the following.
\bigskip
\bigskip
\noindent
\underline{\bf Theorem 5.3.} \enspace Let $d\ge 3$. Let $\lambda_k$ denote the
$k$-th eigenvalue of the Dirichlet boundary value problem (5.17). Then
\begin{equation}
\lambda_k^{d/2} \int_\Omega q_{(x)}^{d/2} dx \ge k K^{-d}_{d,2^\ast}
\dot e^{1-d/2} \ .
\end{equation}
We recall that $K_{d,2^\ast}$ denotes the best constant of the Sobolev
inequality (2.15) given by
$$
K_{d,2^\ast} = [\pi d(d-2)]^{-1/2} \left( \frac{\Gamma(d)}{\Gamma (d/2)}
\right)^{1/d}\ .
$$
\bigskip
\bigskip
\noindent
\underline{\bf Remark:}\enspace The constant given by Li and Yau differs
apart from the numerical error in [LY83] by a factor of $e$. The improvement
of this constant is due to a refinement of the Li-Yau method given in a
paper by Blanchard, Stubbe and Rezende [BSR87]. To prove the Theorem we
therefore follow the proof given in [BSR87].
\bigskip
\bigskip
\noindent
\underline{\bf Proof of Theorem 5.3.}
Let $\{f_i\}_{i\in {\bf N}}$ denote the set of orthonormal eigenfunctions
(in $L^2(\Omega, q(x) dx$) with eigenvalues $\{\lambda_i\}_{i=1}^\infty$.
The key idea is to study the associated heat kernel
\begin{equation}
H_0 (x,y,t) = \sum^\infty_{i=1} e^{-\lambda_i t} f_i(x) f_i(y)
\end{equation}
and the "weighted" heat kernel
\begin{eqnarray}
H_{-1} (x,y,t) &\equiv& \frac{1}{2} \int_t^\infty H_0 (x,y,s) ds \\
&\equiv& \sum^\infty_{i=1} (2\lambda_i)^{-1} e^{-\lambda_i t} f_i(x) f_i(y)
\ . \nonumber
\end{eqnarray}
The function $H_{-1}$ was introduced in [BSR87] and yields the improvement of the
Li-Yau method. We note that both $H_0$ and $H_{-1}$ are positive in the
interior of $\Omega\times \Omega$ and vanish on $\partial\Omega\times\partial
\Omega$ for all $t > 0$. Furthermore they satisfy the heat equation
\begin{equation}
\left( \frac{\partial}{\partial t} - \frac{1}{q(y)} \Delta_y \right)
H_\alpha (x,y,t) = 0 \ , \qquad\qquad \alpha = 0, -1 \ .
\end{equation}
Next starting from
\begin{eqnarray}
h_{-1} (t) &\equiv& \sum^\infty_ {i=1} (2\lambda_i)^{-1}
e^{-2\lambda_i t} \\
&\equiv& \int_\Omega \int_\Omega H_{-1} (x,y,t) H_0 (x,y,t) q(x) q(y) dx\,dy
\nonumber
\end{eqnarray}
we arrive, after some simple steps involving H\"older's inequality and
Sobolev's inequality at the following differential inequality
\begin{equation}
h_{-1} (t) \le K^2_{d,2^\ast} (\int_\Omega q(x)^{d/2} )^{2/d} \left(
-\frac{dh_{-1} (t)}{dt} \right)^{\frac{d-2}{d}}\ .
\end{equation}
To show (5.25), we rewrite $h_{-1} (t)$ as
\begin{eqnarray}
h_{-1} (t) &=& \int_\Omega \int_\Omega H_{-1} (x,y,t) \cdot [H_0 (x,y,t)
q(y)^{\frac{d+2}{4}}]^{2/d}\\
& & \cdot [H_0 (x,y,t)^2 q(y)]^{\frac{d-2}{2d}} dy\ q(x)\ dx\nonumber
\end{eqnarray}
and apply H\"older's inequality with exponents $\frac{2d}{d-2}, \frac{d}{2},
\frac{2d}{d-2}$ for the $y$-integration and then with exponents
$2, d, \frac{2d}{d-2}$ for the $x$-integration. We obtain therefore
\begin{eqnarray}
h_{-1}(t) &\le& \left[ \int_\Omega q(x) \left( \int_\Omega H_{-1}
(x,y,t)^{\frac{2d}{d-2}}\ dy\right)^{\frac{d-2}{d}} dx\right]^{1/2}\\
&\cdot & \left[ \int_\Omega q(x) \left( \int_\Omega H_0 (x,y,t)
q(y)^{\frac{d+2}{4}}\ dy\right)^2 dx\right]^{1/d}\nonumber\\
&\cdot & \left[ \int_\Omega q(x) \int_\Omega H_0 (x,y,t)^2 q(y)
dy\ dx \right]^{\frac{d-2}{2d}}\ . \nonumber
\end{eqnarray}
At this stage it may be remarked that the last term equals $\left( -
\frac{dh_{-1}}{dt} \right)^{\frac{d-2}{2d}}$. To estimate the second
term we observe
that the function $Q(x,t) = \int_\Omega H_0 (x,y,t)\linebreak
q(y)^{\frac{d+2}{2d}}\ dy$
also satisfies the heat equation with $Q(x,t) = 0$ on $\partial\Omega$ for
$t>0$ and initial value $Q(x,0) = q(x)^{\frac{d-2}{4}}$. We compute
\begin{eqnarray}
\frac{\partial}{\partial t} \int_\Omega Q^2(x,t)\ q(x)\ dx &=& - 2 \int_\Omega
|\nabla_x Q(x,t)|^2dx\\
&\le & 0\ . \nonumber
\end{eqnarray}
Hence
\begin{eqnarray}
\int_\Omega Q^2 (x,t)\ q(x)\ dx &\le & \int_\Omega Q^2(x,0) q(x) dx\\
&=& \int_\Omega q(x)^{d/2} dx\ .\nonumber
\end{eqnarray}
There remains the question of estimating the first term. Applying Sobolev's
inequality (2.14) and the identity
\begin{eqnarray}
\frac{1}{2} h_{-1} (t) &=& -\int_\Omega \int_\Omega H_{-1} (x,y,t)
\frac{\partial}{\partial t} H_{-1} (x,y,t)\ q(x)\ q(y)\ dx\ dy \nonumber\\
&=& - \int_\Omega \int_\Omega H_{-1} (x,y,t)
\Delta_y H_{-1} (x,y,t)\ q(x)\ dx\ dy \nonumber \\
&=& \int_\Omega q(x) \int_\Omega |\nabla_y H_{-1} (x,y,t)|^2dy\ dx
\end{eqnarray}
where we have used the definition of the functions $h_{-1} (t)$ and $H_{-1} (x,y,t)$
and the heat equation (5.21) we conclude that the first term is bounded by
$$
K_{d,2^\ast} \cdot (\frac{1}{2} h_{-1}(t))^{1/2}\ .
$$
Combining these estimates we obtain inequality (5.25). Integrating with respect
to $t$ yields
\begin{equation}
h_{-1} (t) \le \left( \frac{d-2}{2}\right)^{\frac{d-2}{2}} \left( \frac{1}{2}
K^2_{d,2^\ast} \right)^{d/2} \int_\Omega q(x)^{d/2} dx\ t^{\frac{2-d}{2}} \ .
\end{equation}
In the following we derive an estimate for the sum of the first $k$ eigenvalues
which is slightly stronger than the inequality presented in the theorem.
Using the fact that $\frac{1}{2\lambda} e^{-2\lambda t}$ is strictly convex
in $\lambda$ we may replace in (5.29) $h_{-1}(t)$ by
$$
k \frac{1}{\frac{1}{k} \sum^k_{l=1} (2\lambda_i)} \exp \left(- \frac{1}{k}
\sum^k_{l=1} (2\lambda_i) t\right)\ .
$$
Choosing $t = \frac{d-2}{2} \frac{1}{\frac{1}{k} \sum^k_{l=1} (2\lambda_i)}$
we obtain
\begin{equation}
\left( \frac{1}{k} \sum^k_{l=1} \lambda_i \right)^{d/2} \int_\Omega
q(x)^{d/2}\ dx \ge k\ K^{-d}_{d,2^\ast} e^{1-d/2}
\end{equation}
which implies the estimate stated in the Theorem. $\Box$
\bigskip
We will now apply the Birman-Schwinger principle and the Theorem to estimate
the number of bound states of the Schr\"odinger operator $H = - \Delta +
V(x)$.
\bigskip
\bigskip
\noindent
\underline{\bf Corollary 5.4.}\enspace Let $N_0 (V)$ denote the number of bound
states of the Schr\"odinger operator $H = - \Delta + V$ on ${\bf R}^d,\,
d \ge 3$, where the negative part $V_-(x)$ of the potential is an element
of $L^{d/2} ({\bf R}^d)$. $N_0 (V)$ satisfies the estimate
\begin{equation}
N_0 (V) \le e^{\frac{d-2}{2}} K^d_{d,2^\ast} \int_{{\bf R}^d} |V_- (x)
|^{d/2} dx
\end{equation}
where $K_{d,2^\ast} = [\pi d(d-2)]^{-1/2} \left(
\frac{\Gamma (d)}{\Gamma(\frac{d}{2})} \right)^{1/d}$.\hfill $\Box$
\bigskip
\bigskip
\noindent
The proof of the Corollary consists of a series of reduction procedures to
apply the Theorem.
\begin{description}
\item{(i)} We use first the fact that $N_0 (V) \le N_0 (-V_-)$.
\item{(ii)} We may next suppose that $V_-(x) > 0$ for all $x\in {\bf R}^d$.
Otherwise we approximate $V_-(x)$ by a sequence of strictly positive
functions in $L^{d/2} ({\bf R}^d)$. To apply the theorem we then take
$q(x) = V_-(x)$.
\item{(iii)} Evidently the number $k$ of negative eigenvalues of $-\Delta_D
- q(x)$ on \linebreak $\Omega \subset {\bf R}^d$ bounded, is nothing else as the number
of eigenvalues of $-\frac{1}{q(x)} \Delta_D$ less than one from which it
follows that
\begin{eqnarray*}
k &\le & e^{\frac{d-2}{2}} K^d_{d,2^\ast} \int_\Omega q(x)^{d/2} dx\\
&\le & e^{\frac{d-2}{2}} K^d_{d,2^\ast} \int_{{\bf R}^d} V_-(x)^{d/2} dx\ .
\end{eqnarray*}
\item{(iv)} Finally we exhaust ${\bf R}^d$ by bounded subdomains
$\Omega_n \to {\bf R}^d$ and use the fact that
the negative eigenvalues of $-\Delta_D - q(x)$ converges
towards the negative eigenvalues of $-\Delta - q(x)$. $\Box$
\end{description}
\smallskip
There are several extensions of these results:
In [BSR87] the generalised
Sobolev inequalities were used to derive the following estimate on the number
of bound states:
Let $\gamma \ge d/2$ with $d \ge 3$. Then for any $x_0 \in {\bf R}^d$
\begin{equation}
N_0 (V) \le e^{\gamma -1} K^{2\gamma}_{d,p} \int_{{\bf R}^d} V_-(x)^\gamma
|x-x_0|^{2\gamma -d} dx
\end{equation}
where $p = \frac{2\gamma}{\gamma -1}$ and $K_{d,p}$ is the Sobolev constant
given in (2.17).
In [St90] potentials with so called critical decay were considered. Suppose,
for example that
\begin{equation}
\lim_{|x|\to\infty} V_-(x) |x|^2 = c
\end{equation}
with $0 \le c < (\frac{d-2}{2})^2$. Then
\begin{eqnarray}
N_0 (V) &\le & e^{\frac{d-2}{2}} K^d_{d,2^\ast} (1 - \frac{4c}{(d-2)^2}
)^{\frac{d-1}{2}}\\
& & \int_{{\bf R}^d} (V(x) + \frac{c}{|x|^2} )_-^{d/2} dx \ . \nonumber
\end{eqnarray}
Consequently, a potential may have only a finite number of bound states
although its classical phase space volume is infinite.
\subsection{Improved bounds on $R(\gamma,d)$}
We briefly describe a slightly more sophisticated method to obtain better
numerical constants on $R(\gamma,d)$ for $\gamma > 0$. Basically it relies
on the use of the one parameter family of generalised Birman-Schwinger operators
$K_{\alpha E}, \alpha \in [0,1)$ given in (2.7). This method was introduced
by Lieb and Thirring [LT75,76] when they derived the celebrated bound on the sum
of the eigenvalues of a Schr\"odinger operator, i.e.~$R (1,d)$, without
having a bound on $R(0,d)$!
In order to illustrate the basic idea we sketch the original
proof of Lieb and Thirring for $R(1,3)$.
By Theorem 2.2 and (2.6) we have for all $\alpha \in [0,1)$
\begin{eqnarray}
N_E (V) &\le& tr K_{\alpha E}^\ast K_{\alpha E}^{} \\
&=& \frac{1}{(4\pi)^2} \int_{{\bf R}^3} \int_{{\bf R}^3}
|V_\alpha (x) | |V_\alpha (y)| |x-y|^{-2} \exp (-2\sqrt{\alpha E}
|x-y|) dx dy \nonumber
\end{eqnarray}
where $V_\alpha (x) = | V(x) - (1-\alpha) E|_-$.
We estimate the integral by Young's inequality (2.13) using exponents
$p=r=2$ and $q=1$. This yields the bound
\begin{equation}
N_E (V) \le \frac{1}{8\pi} |\alpha E|^{-\frac{1}{2}} \int_{{\bf R}^3}
|V_\alpha (x) |^2 dx\ .
\end{equation}
\bigskip
We integrate (5.36) with respect to $E$. Using the change of variables
$|E| \to \frac{|V(x)|_-}{1-\alpha} \beta$ we surprisingly obtain the correct
phase space term $\int_{{\bf R}^3} |V(x)|_-^{5/2} dx$. Optimizing
with respect to $\alpha$ (take $\alpha = \frac{1}{2}$) we find
\begin{equation}
S_1 (V) \le \frac{1}{4\pi} \frac{\Gamma (3) \Gamma (\frac{1}{2})}{\Gamma (
\frac{3}{2})} \int_{{\bf R}^3} |V(x)|_-^{5/2} dx
\end{equation}
or equivalently
\begin{equation}
R (1,3) \le 4 \pi\ .
\end{equation}
\bigskip
In the following we use the same trick in Lieb's trace formula (Theorem 5.2).
We obtain for $f$ convex and satisfying the conditions of Theorem 5.2
\begin{equation}
N_E (V) \le \frac{1}{F(1)} (4\pi)^{-d/2} \int^\infty_0 dt\ t^{-1-d/2}
e^{-\alpha|E|t} \int_{{\bf R}^d} dx f (t V_\alpha (x))
\end{equation}
\noindent
where $F$ and $f$ are related by the transformation $F(z) = \int^\infty_0
\frac{ds}{s} f(s z) e^{-s}$.
Integrating (5.39) with respect to the energy and doing the following
changes of variables
\begin{equation}
|E| \to \frac{|V(x)|_-}{1-\alpha} \beta\ \mbox{and} \ t \to \frac{1-\alpha}{\alpha}
\frac{1}{\beta|V(x)|_-} s
\end{equation}
we find
\begin{equation}
R(1,d) \le \frac{\Gamma (2+\frac{d}{2})}{1-\alpha} \left(
\frac{\alpha}{1-\alpha}\right)^{1/2} \frac{\int^\infty_0 ds s^{-1-d/2} e^{-s}
\int^\infty_0 d\beta\ \beta^{d/2} f(\frac{1-\alpha}{\alpha}
\frac{1-\beta}{\beta}
s)}{\int^\infty_0 ds\ s^{-1} e^{-s} f(s)}\ .
\end{equation}
Choosing $f$ as a piecewise linear function as in Section 5.3 we can do the
$\beta$-integration.
Finally, defining a new parameter $y$ by $y = \frac{\alpha}{1-\alpha} x$ where
$x > 0$ we obtain
\begin{equation}
R(1,d) \le F_d (x,y) \qquad \mbox{for all}\ x, y \ge 0
\end{equation}
where
\begin{equation}
F_d (x,y) = \Gamma \left(\frac{d}{2} \right) \left(\frac{y}{x} \right)^{d/2}
\left( \frac{1}{x} + \frac{1}{y} \right)
\frac{\Gamma(\frac{2-d}{2}, y)}{\Gamma (-1,x)}
\end{equation}
and $\Gamma (d,x) = \int^\infty_x dt t^{d-1} e^{-t}$ denotes the incomplete
$\Gamma-$function.
The limit $y \to 0$ corresponds to $\alpha \to 0$ and the limit $y \to \infty$
corresponds to $\alpha \to 1$.
We have
\begin{eqnarray}
\lim_{y\to 0} F_d (x,y) &=& \frac{\Gamma (\frac{d-2}{2})}{x^{d/2} \Gamma (-1,x)}
\qquad \mbox{for}\ d > 2 \\
and & & \nonumber\\
\lim_{y\to\infty} F_d (x,y) &=& \frac{\Gamma (d/2)}{x^{d/2 +1} \Gamma (-1,x)},
\end{eqnarray}
which corresponds to the bounds (5.16) and (5.15) respectively.
\bigskip
In the following table we list the minimum value of $F_d (x,y), \mbox{Min}\
F_d (x,y)$ and the position $(x,y)$ where it is attained.
\begin{table}
%\caption{..}
\[
\begin{array}{l | l | l | l}
\mbox{Dimension}\ d & \mbox{Min}\ F_d (x,y) & x & y \\
\hline
& & & \\
1 & 5.81029 & 0.037191 & 0.060372\\
2 & 5.17690 & 0.23019 & 0.207166\\
3 & 5.21803 & 0.533354 & 0.356744\\
4 & 5.39902 & 0.896431 & 0.491124\\
5 & 5.62201 & 1.29481 & 0.608821\\
6 & 5.85781 & 1.71579 & 0.711530
\end{array}
\]
\end{table}
\medskip
Generalizing to arbitrary $\gamma$ we obtain
\begin{eqnarray}
\lefteqn{R(\gamma, d) \le \frac{\Gamma (\gamma + \frac{d}{2} - 1)}{\Gamma
(-1,x)} \cdot (x+y)^\gamma} \nonumber \\
& & \cdot\ x^{-\gamma - \frac{d}{2}} \cdot U (\gamma, 2 - \frac{d}{2}, y)
\end{eqnarray}
for all $x, y \ge 0$ where $U(\gamma, 2 - \frac{d}{2}, y) = \frac{1}{\Gamma(\gamma)}
\int^\infty_0 e^{-yt} t^{\gamma-1} (1+t)^{1-\gamma-d/2}$ is the Kummer
function.
(5.46) yields some improvements for small $d$ and large $\gamma$, for example
we find $R(20,2) \le 4.0099$ or $R(100,2) \le 3.9601$.
\subsection{Bounds for one-dimensional Schr\"odinger operators}
The only case where it can be shown that $R(\gamma, d) = 1$ is for $d=1$ and $\gamma \ge
3/2$ by a technique completely different from those considered before.
This was first done in [LT76].
Let $V \in C^\infty_0$ and consider the initial value problem for the
Korteweg-de Vries equation
\begin{eqnarray}
W_t + W_{xxx} - 6WW_x &=& 0\\
W(x,0) &=& V(x)\ . \nonumber
\end{eqnarray}
>From the theory of this equation the following results are well known:
\begin{description}
\item{1.} The eigenvalues of $-\frac{d^2}{dx^2} + W(x,t)$ remain
time independent.
\bigskip
\item{2.} $\int_{\bf R} W^2 (x,t) dx = \int_{{\bf R}} V^2 (x)\ dx\ .$
\bigskip
\item{3.} As $t \to \infty, W(x,t)$ evolves into a train of solitons of the form
$f_a (\xi) =$\\ $= - 2 a^2 \cosh^{-2} (a\xi)$ where
\[
\xi = x - ct\ , \qquad c = 4 a^2
\]
plus a dispersive part going to zero in the sup-norm. These solitons are separated
as $t\to\infty$ since they have different velocities. Any soliton potential
has exactly one bound state. Its energy is $-a^2$.
\end{description}
\bigskip
\noindent
As a consequence we have
\begin{equation}
S_{3/2} (V) = \sum |E_j |^{3/2} = \sum_{\mbox{solitons}} a^3
\end{equation}
and
\begin{equation}
\int V^2 (x) \ge \sum_{\mbox{solitons}} \int f_a^2 (x)\ dx = \frac{16}{3}
(\sum_{\mbox{solitons}} a^3)
\end{equation}
which implies
\bigskip
\bigskip
\noindent
\underline{\bf Theorem 5.4.}\enspace
\begin{equation}
R (\frac{3}{2}, 1) = 1\ .
\end{equation}
\bigskip
As we shall see in the next Section general properties of $R(\gamma, d)$
will then imply
\begin{equation}
R(\gamma, 1) = 1
\end{equation}
for all $\gamma \ge 3/2$. (see Theorem 5.5.)
\subsection{General Properties of $R(\gamma, d)$ and some Conjectures}
In this Section we derive some properties of $R(\gamma, d)$. The first
result is due to Aizenman and Lieb [AL78] stating that for $d$ fixed
$R(\gamma, d)$ is a non-increasing function in $\gamma$.
\bigskip
\bigskip
\noindent
\underline{{\bf Theorem 5.5.}} \enspace For any $\epsilon \ge 0$
\begin{equation}
R (\gamma + \epsilon, d) \le R(\gamma, d)\ .
\end{equation}
\bigskip
\bigskip
\noindent
\underline{{\bf Proof:}}\enspace We fix the potential $V$. Let $\epsilon > 0$.
Then
\begin{equation}
S_{\gamma + \epsilon} (V) = \frac{\Gamma (\epsilon + \gamma +1)}{\Gamma(\epsilon)
\Gamma(\gamma +1)} \int^\infty_0 da a^{\epsilon-1} \sum_j
|E_j(V) + a|^\gamma_- \ .
\end{equation}
However, $(E_j(V) + a)$ are the eigenvalues of the potential $V(x) + a$\ .
Hence
\begin{eqnarray*}
S_{\gamma + \epsilon} (V) &\le& \frac{\Gamma (\epsilon)
\Gamma(\gamma + 1)}{\Gamma (\epsilon + \gamma + 1)} \\
&\cdot & R(\gamma, d) L^{class}_{\gamma + \epsilon, d} \int^\infty_0 da
a^{\epsilon -1}
\int_{{\bf R}^d} dx |V(x)+ a|^\gamma_-\\
&=& R(\gamma, d) L^{class}_{\gamma, d} \int_{{\bf R}^d} |V(x)|^{\gamma
+ \epsilon} dx
\end{eqnarray*}
which implies the desired result. $\Box$
\bigskip
\bigskip
\noindent
\underline{{\bf Remark:}}\enspace As stressed in the preceding Section Theorem
5.5 implies that $R(\gamma, 1) = 1$ for all $\gamma \ge 3/2$.
\bigskip
The second result is due to Martin [M90] and relates different dimensions $d$.
\bigskip
\bigskip
\noindent
\underline{{\bf Theorem 5.6.}}
\begin{equation}
R(\gamma + \frac{1}{2}, d) \le R(\gamma, d+1)\ .
\end{equation}
\bigskip
\bigskip
\noindent
\underline{{\bf Proof:}} \enspace By the definition of $R(\gamma + \frac{1}{2}, d)$
for any $\epsilon > 0$, there is a potential $V$, $V \le 0$, such that
\begin{equation}
\frac{\sum | E_j (V) |^{\gamma + \frac{1}{2}}}{L^{class}_{\gamma +
\frac{1}{2}, d} \int_{{\bf R}^d} |V|^{\gamma + \frac{1}{2} +d} dx}
\ge R(\gamma, d) - \epsilon\ .
\end{equation}
In the following we take $\epsilon$ and $V$ fixed, satisfying condition
(5.57).
We consider the following Schr\"odinger operator on ${\bf R}^{d+1}$ which is
a direct sum of the operator $H = - \Delta + V(x)$ acting on $L^2
({\bf R}^d)$
and an one dimensional infinite square well (or Dirichlet Laplacian on an
interval)
\begin{equation}
{\tilde H} = - \Delta - \partial^2_t + V(x) + W_R (t)
\end{equation}
with
\begin{eqnarray*}
W_R(t) = \left\{ \begin{array}{ll}
0 & |t| \le R\\
\infty & |t| \ge R
\end{array} \right.
\end{eqnarray*}
Its eigenvalues are given by
\begin{equation}
{\tilde E}_{j,n} = E_j (V) + \frac{\pi^2}{4R^2} n^2\ .
\end{equation}
Since
\begin{eqnarray*}
\lefteqn{ \int_{{\bf R}^{d+1}} \int |V(x) + W(t)|^{\gamma + \frac{d+1}{2}} dx\ dt } \\
& & = 2 R \int_{{\bf R}^d} |V(x)|^{\gamma + \frac{1}{2} + d/2} dx
\end{eqnarray*}
and
\begin{eqnarray*}
\lefteqn{ \lim_{R\to\infty} \frac{1}{2R} \sum_j \sum_n |E_j (V) +
\frac{\pi^2}{(2R)^2} n^2 |^\gamma_- } \\
& & = \sum_{E_j < 0} \int^\infty_0 |E_j (V) + \pi^2 \gamma^2 |^\gamma_- ds\\
& & = \frac{1}{\pi} (\sum_{E_j < 0} |E_j (V)|^{\gamma + \frac{1}{2}} ) \int^1_0
(1-\sigma^2)^\gamma d\sigma\\
& & = (4\pi)^{-\frac{1}{2}} \frac{\Gamma(\gamma + 1)}{\Gamma (\gamma + 3/2)}
\sum_{E_j < 0} |E_j (v)|^{\gamma + \frac{1}{2}}
\end{eqnarray*}
we have
\begin{eqnarray*}
R(\gamma, d+1) &\ge& \lim_{R\to\infty} \frac{\sum_j \sum_{n_s} |E_j (V) +
\frac{\pi^2}{(2R)^2} n^2|^\gamma_-}{L^{class}_{\gamma , d+1}
\int_{{\bf R}^{d+2}}\int |V(x) + W(t) |^{\gamma + \frac{d+1}{2}} dx\ dt}_-\\
&=& (4\pi)^{-\frac{1}{2}} \frac{\Gamma(\gamma +1)}{\Gamma(\gamma + \frac{3}{2})}
\frac{\sum_j |E_j (V)|^{\gamma + \frac{1}{2}}}{L^{class}_{\gamma , d+1}
\int_{{\bf R}^d} |V(x)|^{\gamma + \frac{1}{2}+ \frac{d}{2}}_- dx}\\
&=& \frac{\sum |E_j (V)|^{\gamma+\frac{1}{2}}}{L^{class}_{\gamma +
\frac{1}{2},d} \int_{{\bf R}^d} |V(x)|^{\gamma + \frac{1}{2}+d} dx}\\
&\ge& R(\gamma + \frac{1}{2}, d) - \epsilon
\end{eqnarray*}
which proves the Theorem.
\bigskip
The following facts about $R(\gamma ,d)$ are known. Comparing the zero or
one bound state constant $K^d_{d,2^\ast}$ with $L^{class}_{0 , d}$
one sees that $R(0,d) > 1$ if $d<7$. A more sophisticated argument by
Glaser, Grosse and Martin [GGM78] yields $R(0,d) > 1$ for $d\ge 7$.
\bigskip
It is conjectured that for $3 \le d \le 6$,
\[
R(0 ,d) = \frac{K_{d,2^\ast}^d}{L^{class}_{0,d}}\ .
\]
$R(\gamma ,d)$ is finite if $\gamma + d/2 > 1$. The two boundary points are
$(\gamma ,d) = \{ (\frac{1}{2}, 1), (0,2)\}$.
\bigskip
It was claimed in [LT76] that $R(\frac{1}{2}, 1) < \infty$.
This was recently proved by Weidel [Weid95]. On the other hand
$R(\gamma ,1)$ is infinite if $0 \le \gamma < \frac{1}{2}$. This can be
seen by taking a potential well $V_R (x) = - \frac{1}{R} \chi (0,R)$.
As $R \to 0\ , V_R (x)$ converges towards $-\delta(x)$ which has a
finite energy bound state while $\lim_{R\to 0} \int |V_R(x)|^{\frac{1}{2} +
\gamma} = 0$.
>From Theorem 4.2 we know that $R(0,2)$ is infinite.
>From Theorem 3.3 together with (3.26) we deduce that if the operator
$-\frac{d^2}{dx^2} + V$ has a bound state then its eigenvalue satisfies
\[
|E|^\gamma < M^r_p \int_{{\bf R}} |V(x)|^{\frac{1}{2} + \gamma}\ dx
\]
with $M_p$ given in (2.20).
As pointed out in Section 3.2 this condition is optimal. Comparing $M^\sigma_p$
with $L^{class}_{\gamma, 1}$ we see that $R(\gamma ,1) > 1$ for
$1/2 \le \gamma < 3/2$.
\bigskip
These considerations lead to conjecture that for each $d\le 7$ there is a
$\gamma_d > 0$ such that $R(\gamma ,d) > 1$ if $\gamma < \gamma_d$ and
$\gamma_d$ is defined by the point where the one bound state constant and the
classical constant coincide. This conjecture was originally formulated
by Lieb and Thirring for any $d$, but the argument by Glaser et al.~[GGM]
mentioned above shows that this cannot hold for $d\ge 8$. The conjecture
is true for $d=1$.
An important conjecture, related to the nonrelativistic stability of matter
is that $R(1,d) = 1$ for $d \ge 3$.
Glaser et al.~conjectured also that
\[
\lim_{d\to\infty} R(0,d) = 1 \ .
\]
\subsection{Schr\"odinger operators with magnetic fields}
Recently phase space bounds have been generalized to Schr\"odinger
operators with magnetic fields [LSY94,LLS95]. As in [LSY94] we only consider
the case of a constant magnetic field. Some generalizations to
nonconstant fields have been obtained in [Er95]. The two dimensional
case has already been presented in [LSY95].
We consider the Schr\"odinger operator for a spin $\frac{1}{2}$ particle
\begin{eqnarray}
H &=& ( p + {\bf A} (x) )^2 + {\bf \sigma} \cdot {\bf B} + V(x) \nonumber\\
&=& H_{\bf A} + V(x)
\end{eqnarray}
where ${\bf B} = (0,0,B), {\bf A} = \frac{1}{2} {\bf B} \wedge x$ and $\sigma$ denotes
the Pauli matrices. We suppose $V_- \in L^{3/2} ({\bf R}^3)
\cap L^{5/2} ({\bf R}^2)$.
The spectrum of $H_{\bf A}$ is described by the Landau bounds
\begin{equation}
E_{p,\nu} = 2 \nu B + p^2
\end{equation}
where $\nu = 0, 1, \ldots$ and $p \in {\bf R}$ is the momentum along the
field $B$.
The spin-dependent term ${\bf \sigma} \cdot {\bf B}$ makes the problem interesting.
Without the ${\bf \sigma} \cdot {\bf B}$ term, by the diamagnetic inequality [Si79] we
can apply all phase space bounds for the $B = 0$ case.
With the spin-term, however, the bottom of the essential spectrum is shifted
by $-B$ down to zero so that one estimates all the spectrum below $\sigma_{ess}$.
A further complication comes from the observation that $H$ may have
infinitely many eigenvalues even when $V$ has compact support [AHS78].
However, the phase space bounds obtained in [LSY94] show that these
infinitely many eigenvalues are summable.
In fact there is the following bound on $S_1 (V,B) = \sum_{E_j < 0} |E_j (V,B)|$
\begin{equation}
S_1 (V,B) \le L_1 B \int_{{\bf R}^3} V_- (x)^{5/2} dx + L_2 \int_{{\bf R}^3}
V_-^{3/2} (x) dx
\end{equation}
with $L_1 = \frac{4}{3\pi}$ and $L_2 = \frac{8\sqrt{6}}{5\pi}$.
(5.62) is proved by using the generalized Birman Schwinger operators
$K_{\alpha E}, \alpha \in [0,1)$ corresponding to (2.7) and a splitting
of the operators into a projection onto the lowest Landau bound and onto
higher bounds.
The bound (5.62) does not correspond to the classical phase space expression
which is given by
\begin{equation}
S_1 (V,B) \simeq \frac{B}{3\pi^2} \int |V_- (x) |^{3/2} + 2 \sum^\infty_{v=1}
|V(x) + 2 \nu B |_-^{3/2} dx \ .
\end{equation}
However, the two terms in the bound (5.62) correspond, respectively, to the
$B \to \infty$ (first term) and the $B \to 0$ (second term) asymptotics of
(5.63).
To obtain a bound in terms of the r.h.s. of (5.63) we use the inequality
\begin{equation}
\frac{|V(x)|^{5/2}}{5B} \equiv \int^\infty_0 | V(x) + 2 B\sigma
|_-^{3/2} d\sigma \le \sum^\infty_{v=0} |V(x) + 2 B\nu|_-^{3/2} \ .
\end{equation}
\section{Applications}
The most important applications of the phase space bounds presented
in this review are related to the problem of the stability of matter and to
the theory of dynamical systems. We outline them in the sequel.
\subsection{A Sobolev inequality for Fermions}
The bound on the sum of the bound state energies of a Schr\"odinger
operator with potential $V \le 0$ given by
\begin{equation}
\sum_{E_j \le 0} |E_j| \le L_{1,d} \int_{{\bf R}^d} |V|^{d/2 + 1} dx
\end{equation}
will imply a Sobolev-type inequality for the kinetic energy of $N$ Fermions.
The constant $L_{1,d}$ is related to $R (1,d)$ by $L_{1,d} = R(1,d) \quad
L_{1,d}^{class}$ with $L^{class}_{1,d} = (4\pi)^{-d/2} \frac{1}{\Gamma (2 + d/2)}$.
For simplicity we do not take into account the spin of the particles.
\bigskip
\bigskip
\noindent
\underline{{\bf Theorem 6.1.[LT75,76]:}} \enspace Let $\psi : R^{dN} \to {\bf C}$,
$\psi = \psi
(x_1, \ldots, x_N),\ x_j \in {\bf R}^d,$\linebreak $j = 1, \ldots, N$
be an antisymmetric
$L^2$-wave function and let $\rho_\psi (x) =$\linebreak $= N \int_{{\bf R}^{d(N-1)}}
|\psi (x_1, x_2, \ldots, x_N)|^2 dx_2 \ldots dx_N$ the corresponding
one-particle density. Then
\begin{eqnarray}
\sum^N_{i=1} & \int_{{\bf R}^{dN}} |\nabla_{x_i} \psi (x_1, \ldots, x_N)|^2
dx_1, \ldots, dx_N \\
& \ge \frac{d}{d+2} K_d \int_{{\bf R}^d} \rho_\psi (x)^{1 + 2/d} dx
\nonumber
\end{eqnarray}
with
\begin{eqnarray}
K_d &=& \left( \frac{d+2}{2} L_{1,d} \right)^{-2/d} \nonumber \\
&=& 4\pi \left[ \frac{\Gamma (\frac{d+2}{2})}{R(1,d)} \right]^{2/d}\ .
\end{eqnarray}
\bigskip
\bigskip
\noindent
\underline{{\bf Proof:}} \enspace Consider the N-particle Hamiltonian
\be
H_N = \sum^N_{L=1} - \Delta_{x_i} + V(x_i) \quad , \qquad V \le 0
\ee
By (6.1) we have for any antisymmetric N-particle wave function
\begin{eqnarray}
\lefteqn{< H_N \psi, \psi >_{L^2(R^{dN})}} \\
& & = \sum^N_{L=1} \int_{{\bf R}^{dN}} |\nabla_i \psi (x_1, \ldots, x_N)|^2
dx_1
\ldots dx_N + \int_{{\bf R}^d} V(x) \rho (x) dx \nonumber \\
& & \ge -L_{1,d} \int_{{\bf R}^d} |V|^{1 + d/2} dx \nonumber \ .
\end{eqnarray}
By standard Sobolev's inequality one shows $\rho \in L^{1 + 2/d} ({\bf R}^d)$
if the kinetic energy of $\psi$ is finite. Hence $\rho \in L^1 \cap L^{1 +2/d}$.
Choosing $V = - \lambda \rho^{2/d} (x)$ we have $V \in L^{d/2} ({\bf R}^d)
\cap L^{1 + d/2} ({\bf R}^d)$ and consequently $V$ has only a finite number
of bound states and the sum over the bound state energies is finite. Optimizing
with respect to $\lambda$ yields the desired inequality. $\Box$
The corresponding classical value (or Thomas-Fermi value) $K_d^{cl}$ in the
kinetic energy inequality (6.2) is given by
\be
K^{cl}_d = 4\pi (\Gamma (1 + d/2))^{2/d} \ .
\ee
The case $d=3$ is relevant in the proof of the stability of matter.
Using our estimates in Section 5 we have the following numerical values
\begin{eqnarray}
K^{cl}_3 &=& (6\pi^2)^{2/3} = 15.1927 \\
K_3 \ge (5.21)^{-2/3} K^{cl}_3 &=& 0.3324 K^{cl}_3 = 5.0501\ .
\end{eqnarray}
The Lieb-Thirring bound on the ground state of N identical fermions with
charge $-1$ and mass $m$ having $q$ different spin states and $k$ point
nuclei with charges $Z_1, \ldots, Z_k$ has the form
\be
E(N,q) > - 4.42 (K_3)^{-1} q^{-1} N \left[ 1 + \sqrt{\frac{\sum^k_{j=1}
Z_j^{7/3}}{N}} \right] m \alpha
\ee
where $\alpha$ denotes the fine structure constant. For N electrons
$(q=2)$ we obtain the constant 2.316 which is only a minor improvement of
the constant 2.52 of [M90]. The best possible value is 0.770.
In view of recent interests in two dimensional quantum systems we give the
values for $d=2$.
\begin{eqnarray}
K_2^{cl} &=& 4\pi\\
K_2 &\ge & (5.1769)^{-1} \quad K^{cl}_2 = 2.4273 \ .
\end{eqnarray}
The problem of stability of matter in magnetic fields was recently solved
in [LLS95].
The second application is related to the Navier-Stokes equations on a bounded
domain $D$ in ${\bf R}^d$. As in [L84] we consider the Hamiltonian
$H = \nu \Delta - W(x)$ on $L^2(D)$ with Dirichlet boundary conditions; $\nu$ is
the kinematic viscosity and the nonnegative potential $W$ is related
to the average rate of energy dissipation per unit mass in the flow governed
by the Navier Stokes equations. The number $N^+$ defined by
\be
N^+ = \ \mbox{smallest}\ N \ \mbox{such that} \quad \sum^N_{j=1} E_j (V) \ge 0,
\ee
is related to the number of nonnegative characteristic exponents of the
Navier Stokes equations. In the following we sketch a simplified derivation
of a bound on $N^+$ yielding improved constants which is due to [L84].
However, we improve this derivation and obtain a better estimate (see (6.17)
and (6.18)).
As in [L84] we use a result by Li and Yau [LY83] on the energy of $N$ fermions
in a bounded domain with impenetrable walls
\be
\sum^N_{i=1} \int_{D^N} |\mbox{grad} \psi (x_1, x_2, \ldots, x_N)|^2 dx_1 \ldots dx_N
> \frac{d}{d+2} K^{cl}_d (N/|D|)^{2/d} N
\ee
where $|D|$ denotes the volume of $D$.
Let $(\phi_1, \ldots, \phi_N)$ be a set of eigenfunctions corresponding to
the eigenvalues $E_1, \ldots, E_N$. Then
\be
\sum^N_{j=1} E_j \ge \nu \sum^N_{i=1} \int_D |\mbox{grad} \phi_i|^2 dx - \int_D
W\rho dx
\ee
where $\rho (x) = \sum^N_{i=1} |\phi_i (x) |^2$ is the corresponding one
particle density.
H\"older's inequality yields
\be
\sum^N_{j=1} E_j \ge \nu \sum^N_{i=1} \int_D |\mbox{grad}\ \phi_i |^2 dx - \|
W\|_{1+d/2} \|\rho \|_{1+2/d}\ .
\ee
The following trick improves the result of [L84].
Writing $\sum^N_{i=1} \int_D |\mbox{grad} \phi_i |^2 dx$ as
$\left( \sum^N_{i=1} \int_D |\mbox{grad}\ \phi_i |^2 dx \right)^\frac{d}{d+2}$
\linebreak $\left( \sum^N_{i=1} \int_D |\mbox{grad}\ \phi_i |^2 dx \right)^\frac{d}{d+2}$
and applying (3.1) and (3.7), respectively, we obtain
\be
\sum^N_{j=1} E_j \ge \left[ \nu \frac{d}{d+2} (K_d) \frac{d}{d+2}
(K^{cl}) \frac{2}{d+2} N^{\frac{2}{d}} |D|^{\frac{-4}{d(d+2)}} - \| W\|_{1+d/2}
\right] \| \rho \|_{1+2/d} \ .
\ee
Therefore $\sum^N_{j=1} E_j \ge 0$ if the square bracket in the above
inequality is nonnegative. Hence $N^+$ is bounded above by
\be
N^+ \le \nu^{-d/2} A_d |D| \left[ \frac{\int_D W(x)^{1+d/2} dx}{|D|}
\right]^{\frac{d}{d+2}}
\ee
where the constant $A_d$ is given by
\begin{eqnarray}
(A_d)^{2/d} &=& \frac{d+2}{d} (K^{cl}_d)^{-2/(d+2)} (K_d)^{-d/(d+2)} \nonumber\\
&=& \left( \frac{d}{d+2} K^{cl}_d\right)^{-1} R(1,d)^{\frac{2}
{d+2}}\ .
\end{eqnarray}
Notice that (6.18) improves eq. (41) in [L84]. In dimensions $d=2$ and
$d=3$ we have the following bounds on $A_d$
\be
0.15915 \le A_2 \le ~0.3621
\ee
\be
0.03633 \le A_3 \le 0.09791
\ee
where the left hand side is the corresponding classical value.
\subsection{No binding results and lower bounds for N-body Hamiltonians}
Our first example deals with muonic atoms and is taken from [GGMT76].
We consider an atom with a $\mu^-$ and $N$ electrons and a nucleus of charge
$Z$ as described by the Schr\"odinger equation.
We write the Hamiltonian as
\be
H = H_{N,Z,m} + h_{Z\mu}
\ee
where $H_N$ denotes the $N$-electron Hamiltonians given by
\be
H_{N,Z,m} = \sum^N_{i=1} \frac{p^2}{2m} - \frac{Z}{|x_i|} +
\sum \sum_{i \rightarrow E(N,Z-1) <\psi,\psi> $.
Then we find
\be
E \le - \frac{Z^2\mu}{2} + E (N, Z-1)\ .
\ee
The upper bound corresponds to the energy one obtains from the naive assumption
that the electrons just feel an effective charge $Z-1$.
To obtain lower bounds we employ the projection method
\begin{eqnarray}
H &\ge& \frac{p^2}{2\mu} - \frac{Z}{|x|} + H_{N,Z,m} \cr
& & + \sum^N_{i=1} P (P |x-x_i| P)^{-1} P \nonumber
\end{eqnarray}
where $P$ is the projector onto the ground state of $\frac{p^2}{2\mu}
- \frac{Z}{|x|}$. Then
\be
P(P|x-x_i|P)^{-1}P = \frac{1}{|x_i|} - \frac{1}{|x_i|} U (Z\mu|x_i|)
\frac{1}{|x_i|} - \frac{1}{|x_i|} U (Z\mu|x_i|)
\ee
where
\be
U(r) = \left[ 1 + \frac{r^2}{1-e^{-2r}(1 + \frac{r}{2})} \right]^{-1}\ .
\ee
For any $m_1 > m$ we have the operator inequality
\begin{eqnarray}
H &\ge& \left[ - \frac{Z^2\mu}{2} + H_{N, Z-1, m^\prime}\right] P \nonumber\cr
& & + (1-P) \left[- \frac{Z^2\mu}{8} + E(N,Z)\right] \cr
& & + P \left\{ \sum^N_{i=1} \frac{P^2_i}{2m_1} - \frac{1}{|x_i|}
U(Z\mu|x_i|)\right\} \nonumber
\end{eqnarray}
with $m^\prime = \frac{m m_1}{m_1 - m}$.
According to Theorem 3.1 the bracket $\{ \}$ is nonnegative if
\begin{eqnarray}
\frac{2m_1}{Z\mu} &\le& \frac{r}{r -1} \left[ \frac{(r-1)\Gamma^2 (r)}
{\Gamma(2r)} \right]^{1/r}\cr
& & \cdot \left[ \int^\infty_0 U^r (t) t^{r -1} dt \right]^{-1/r}
\nonumber\cr
\end{eqnarray}
for some $r \ge 1$.
The numerical result of [GGMT] for the best choice of $r$ is $r = 1.8242$
giving
\be
\frac{2m_1}{Z\mu} \le 1.2706\ .
\ee
Using the scaling properties of Coulomb Hamiltonians we find
\begin{eqnarray}
E &\ge& \left\{ (1 - \frac{1.574 m}{Z\mu} )^{-1} E(N,Z-1) - \frac{Z^2\mu}{2}
\right. \cr
& & \left. - \frac{Z^2\mu}{8} + E(N,Z)\right\}
\end{eqnarray}
provided $\frac{Z\mu}{m} \ge 1.574$.
Again from scaling we infer that
\be
E(N,Z) > (\frac{Z}{Z-1})^2 \cdot E(N,Z-1)\ .
\ee
Therefore
\be
E \ge - \frac{Z^2\mu}{2} + (1 - \frac{1.574 m}{Z\mu})^{-1} E(N,Z-1)
\ee
if
\be
|E(N,Z-1)| \left[ \left( \frac{Z}{Z-1}\right)^2 - \left(1 -
\frac{1.574m}{Z\mu} \right)^{-1}\right] < \frac{3}{8} Z^2 \mu
\ee
which is satisfied for all $\mu \ge m$.
This argument also shows that a system composed of a proton, an electron
and a negative particle of mass $\mu \ge 1.574 m$ is not bounded.
Our second example concerns the N-body Hamiltonian with a general
pair interaction $V$
\begin{equation}
H_N = \sum^N_{i=1} \frac{1}{2m} p_i^2 + \sum \sum_{i