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CLR estimate, weak type estimates, singular values
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\title[On the number of bound states for Schr\"odinger operators]{On the number of bound states for Schr\"odinger operators with operator-valued potentials}

\author{Dirk Hundertmark }

\thanks{Department of Mathematics 253--37, California Institute of Technology,    
Pasadena, CA 91125, U.S.A.; E-mail: dirkh@caltech.edu} 
\thanks{2000 Mathematics subject classification. 
Primary: 35P15, 47B10; Secondary: 81Q10, 47L20. 
Key words: CLR estimate, weak type estimates, singular values}
\thanks{\copyright 2000 by the author. Reproduction of this article, in its entirety, by any means is permitted for non-commercial purposes.}



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\noindent
\begin{abstract}
  \noindent 
Cwikel's bound is extended to  an operator-valued setting. One application of this result is a semi-classical bound for the number of negative bound states for Schr\"odinger operators with operator-valued potentials. 
We recover Cwikel's bound for the Lieb--Thirring constant $L_{0,3}$ which  is far worse than the best available by Lieb (for scalar potentials).  However, it leads to a uniform bound (in the dimension $d\ge 3$) for the quotient $L_{0,d}/ L^{\text{cl}}_{0,d}$, where $L^{\text{cl}}_{0,d}$ is the so-called classical constant. This gives some improvement in large dimensions.
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\label{sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The Lieb-Thirring inequalities bound certain moments of the negative eigenvalues of a one-particle Schr\"odinger operator by the corresponding classical phase space moment. More precisely, for ``nice enough" potentials one has 
\begin{equation}\label{eq:LT1}
\tr_{L^2(\rd)} (-\Delta +V)_-^\gamma 
\le
\frac{C_{\gamma,d}}{(2\pi)^d}
\iint\limits_{\rz^d \rz^d}d\xi dx\,(\xi^2+V(x))_-^\gamma .
\end{equation}
Here and in the following, $(x)_- =\tfrac{1}{2}(\vert x\vert-x)$ is the negative part of a real number or a self-adjoint operator. 
Doing the $\xi$ integration explicitly with the help of scaling the above inequality is equivalent to its more often used form
\begin{equation}\label{eq:LT2}
\tr_{L^2(\rd)} (-\Delta +V)_-^\gamma 
\le L_{\gamma,d}
\int_{\rz^d}dx\, V(x)_-^{\gamma+d/2} ,
\end{equation}
where the Lieb-Thirring constant $L_{\gamma,d}$ is given by
$L_{\gamma,d}= C_{\gamma,d} L_{\gamma,d}^{\text{cl}} $ with the classical 
Lieb-Thirring constant
\begin{equation}\label{eq:LTcl}
L_{\gamma,d}^{\text{cl}} =
\frac{1}{(2\pi)^d}\int_{\rz^d}dp (1-p^2)_+^\gamma .
\end{equation}
This integral is, of course, explicitly given by a quotient of Gamma functions, but we will have no need for this. The Lieb-Thirring inequalities are valid as soon as the potential $V$ is in $L^{\gamma+d/2}(\rd)$. 

These inequalities are important tools in the spectral theory of Schr\"odinger operators and they are known to hold if and only if $\gamma\ge \tfrac{1}{2}$ if $d=1$, $\gamma >0 $ if $d=2$, and $\gamma \ge 0$ if $d\ge 3$. The bound for the critical case $\gamma =0$, that is, the bound for the number of negative eigenvalues of a Schr\"odinger operator in three or more dimensions is the celebrated Cwikel-Lieb-Rozenblum bound \cite{Cwikel,Lieb,Rozenblum}. Later, different proofs for this were given by Conlon and Li and Yau \cite{Conlon, LY}. The remaining case $\gamma = \tfrac{1}{2}$ in $d=1$ was settled in \cite{Weidl}. The well-known Weyl asymptotic formula 
$$
\lim_{\lambda\to\infty}\tr (-\Delta+\lambda V)_-^\gamma
=
L_{\gamma,d}^{\text{cl}}\int dx\, V(x)_-^{\gamma+d/2}
$$ 
immediately gives the lower bound $C_{\gamma,d}\ge 1$. There are certain refined lower bounds \cite{LT2,HR} for small values of $\gamma$. In particular, one always has $C_{\gamma,d}>1$ for $\gamma<1$; see  \cite{HR}. In one dimension this even happens for $\gamma<3/2$, and in two dimensions, one always has $C_{1,2} > 1$ \cite{LT2}. 

Depending on the dimension there are certain conjectures for the optimal value of the constants in these inequalities \cite{LT,LT2}. 
One part of the conjectures on the Lieb-Thirring constants  is that, indeed, $C_{\gamma,d} =1$ for $d\ge 3$ and moments $\gamma \ge 1$. For the physically most important case $\gamma =1$, $d=3$ this would imply, via a duality argument, that the kinetic energy of fermions is bounded below by the Thomas-Fermi ansatz for the kinetic energy, which in turn has certain consequences for the energy of large quantum Coulomb systems \cite{Lieb,LT}.

Laptev and Weidl \cite{LW} realized that a, at first glance, purely technical extension of the Lieb-Thirring inequality from scalar to operator-valued potentials already suggested in \cite{Laptev} is a key in proving at least a part of the Lieb-Thirring conjecture. It allowed them to show that $C_{\gamma,d} = 1$ for all $d\in\nz$ as long as  $\gamma\ge 3/2$. To prove this they considered Schr\"odinger operators of the form $-\Delta\otimes \idG +V$ on the Hilbert space $L^2(\rd,\calG)$ where V now is an operator-valued potential with values $V(x)$ in the bounded self-adjoint operators on the auxiliary Hilbert space $\calG$. In this case the Lieb-Thirring inequalities (\ref{eq:LT1}) and (\ref{eq:LT2}) are modified to 
\begin{equation}\label{eq:LToper1}
\Tr (-\Delta\otimes\idG+V)_-^\gamma 
\le \frac{C_{\gamma,d}}{(2\pi)^d}
\iint\limits_{\rz^d \rz^d}dpdx\, \trG(p^2+V(x))_-^\gamma,
\end{equation}
or, again doing the $\xi$ integral explicitly with the help of the spectral theorem and scaling
\begin{equation}\label{eq:LToper2}
\Tr (-\Delta\otimes\idG+V)_-^\gamma 
\le L_{\gamma,d}
\int_{\rz^d}dx\, \trG(V(x)_-^{\gamma+d/2}).
\end{equation}
Here we abused the notation slightly in using the same symbol for the constants as in the scalar case. But in the following, we will only consider the operator-valued case anyway. Laptev and Weidl realized that this extension of the Lieb-Thirring inequality gives rise to the possibility of an inductive proof for $C_{3/2,d}=1$ as long as one has the a priori information $C_{3/2,1}=1$ for operator-valued potentials. 
This idea together with ideas in \cite{HLT} was then later used in \cite{HLW} to prove improved bounds on $C_{\gamma,d}$ in the range $1/2 \le \gamma \le 3/2$; in particular, it was shown that $C_{1,d}\le 2$ uniformly in $d\in\nz$. 

Unlike the scalar case, however, the range of parameters $\gamma$ and $d$ for which (\ref{eq:LToper1}) or equivalently (\ref{eq:LToper2}) holds is not known. The results in \cite{HLW} only show that these inequalities are true for $\gamma\ge 1/2$ and all $d\in \nz$. This shortcoming has to do with the way the Lieb-Thirring estimates are proven for operator-valued potentials: First, the estimate is shown to hold in one dimension. Then a suitable induction proof, using the one-dimensional result, is set up to prove the full result in all dimensions. This turns out to give good estimates for the coefficients $C_{\gamma,d}$ in the Lieb-Thirring inequality, for example, they are independend of the dimension. However, moments below $1/2$ cannot be addressed with this method, since the a priori estimate fails already for scalar potentials. 

This led Ari Laptev \cite{Laptevprivate}, see also \cite{LW2}, to ask the question whether, in particular, the Cwikel-Lieb-Rozenblum estimate holds for Schr\"odinger operators with operator-valued potentials. In this note we answer his question affirmatively, that is, the Lieb-Thirring inequalities for operator-valued potentials are shown to hold also for $\gamma=0$ as long as $d\ge 3$ and then, by a monotonicity argument also for all $\gamma\ge 0$. More precisely, we want to show that Cwikel's proof of the Cwikel-Lieb-Rozenblum bound can be adapted to the operator-valued setting.  However, the bound for $C_{0,d}$ is far from  being optimal since we use Cwikel's approach. But, nevertheless, reasoning similar to Laptev and Weidl, any a priori bound on $C_{0,3}$ implies the bound $C_{0,d}\le C_{0,3}$ for $d\ge 3$, thus giving a \emph{uniform} bound in the dimension, whereas the best available bound in the scalar case due to Lieb \cite{Lieb} grows like $\sqrt{\pi d}$, see \cite{LT2}.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Statement of the results}\label{sec:results}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $\calG$ be a (separable) Hilbert space with norm $\norm{.}_\calG$, scalar product $\langle.,.\rangle_\calG$, and let $\idG$ be the identity operator on $\calG$. We follow the convention that scalar products are linear in the second component. Furthermore, $\calB(\calG)$ is the Banach space of bounded operators equipped with the operator norm $\norm{.}_{\calB(\calG)}$ and $\calK(\calG)$ the (separable) ideal of the compact operators on $\calG$. For a compact operator $A\in\calK(\calG)$, the singular values $\mu_n(A)$, $n\in\nz$ are the eigenvalues of 
$\vert A\vert\defeq (A^*A)^{1/2}$ arranged in decreasing order counting multiplicity. $A^*$ is the adjoint of $A$. 
$\calS^q(\calG)$ denotes the ideal of compact operators $A\in\calK(\calG)$ whose singular values are $q$-summable, that is, 
$\sum_n \mu_n(A)^q <\infty$. In particular, $\calS^1(\calG)$ and $\calS^2(\calG)$ are the trace class and Hilbert-Schmidt operators on $\calG$. We will often write $\calB$, $\calK$, and $\calS^q$ if there is no ambiguity. 
Of course, $A\in\calS^q$ if and only if $\trG(\vert A\vert^q) = \trG((A^*A)^{q/2})<\infty$, where $\trG$ is the trace on $\calG$.

The Hilbert space $L^2(\rd,\calG)$ is the space of all measurable functions $\phi\!: \rd\to\calG$ such that
$$
\norm{\psi}_{L^2(\rd,\calG)}^2\defeq
\int_{\rd}dx\,\norm{\psi(x)}_{\calG}^2 <\infty
$$
and the Sobolev space $H^1(\rd,\calG)$ consists of all functions $\psi\in L^2(\rd,\calG)$ with finite norm
$$
\norm{\psi}_{H^1(\rd,\calG)}^2 \defeq
\sum_{l=1}^d \norm{\partial_l \psi}_{L^2(\rd,\calG)}^2
+
\norm{\psi}_{L^2(\rd,\calG)}^2 .
$$
As in the scalar case, the quadratic form 
$$
h_0(\psi,\psi) \defeq
\sum_{l=1}^d \norm{\partial_l \psi}_{L^2(\rd,\calG)}^2
$$
is closed in $L^2(\rd,\calG)$ on the domain $H^1(\rd,\calG)$. Naturally, this form corresponds to the Laplacian $-\Delta\otimes\idG$ on $L^2(\rd,\calG)$. 

$L^{q}(\rd,\calB(\calG))$ is the space of  operator-valued functions $f\!:\rd\to\calB(\calG)$ with finite norm
$$
\norm{f}_{q}^q
=
\norm{f}_{L^{q}(\rd,\calB(\calG))}^q
\defeq
\int_{\rd}dx\, \norm{f(x)}_{\calB(\calG)}^q
$$
and $ L^q(\rd,\calS^r(\calG))$ the space of operator-valued functions $f$ whose norm
$$
\norm{f}_{q,r}^q 
=
\norm{f}_{L^{q}(\rd,\calS^r(\calG))}^q
\defeq
\int_{\rd}dx\, \trG(\vert f(x)\vert^r)^{q/r} 
$$
is finite. 
A potential is a function $V\in L^{q}(\rd,\calB(\calG))$ such that $V(x)$ is a symmetric operator for almost every $x\in\rd$. If 
\begin{equation}\label{eq:formbound}
q\ge 1 \text{ for } d=1,\quad 
q>1 \text{ for } d=2\text{,} 
\quad\text{and } q\ge d/2 \text{ for } d\ge 3
\end{equation}
one sees, using Sobolev embedding theorems as in the scalar case, that the real-valued quadratic form
$$
v[\psi,\psi] \defeq \int_{\rd}dx\, 
\langle \psi(x), V(x)\psi(x)\rangle_{\calG}
$$
is infinitesimally form-bounded with respect to $h_0$. Hence the form sum
$$
h[\psi,\psi]\defeq
h_0[\psi,\psi] + v[\psi,\psi]
$$
is closed and semi-bounded from below on $H^1(\rd,\calG)$ and thus generates the self-adjoint operator 
$$
H= -\Delta\otimes\idG + V
$$
on $L^2(\rd,\calG)$ by the KLMN theorem \cite{RSII}. It is easy to see that any  potential $V\in L^{q}(\rd,\calB(\calG))$ satisfying (\ref{eq:formbound}) for which $V(x)\in\calK(\calG)$ for almost every $x\in\rd$ is relatively form compact with respect to $h_0$. Hence by Weyl's theorem for such potentials, the negative eigenvalues $E_0 \le E_1 \le E_3 \le \cdots\le 0$ are at most a countable set with accumulation point zero and their eigenspaces are 
finite-dimensional. In particular, this is the case for potentials 
$V\in L^{q}(\rd,\calS^r(\calG))$.


Our first result is a generalized version of a basic observation of Laptev and Weidl: The two versions (\ref{eq:LToper1}) and (\ref{eq:LToper2}) of the 
Lieb-Thirring inequality give rise to two different monotonicity 
properties of $C_{\gamma,d}$ in $d$.
\begin{theorem}[Sub-multiplicativity of $\mathbf{C_{\gamma,d}}$]
\label{thm:subm}
If, for dimensions $n$ and $d-n$, the Lieb-Thirring inequality holds for operator-valued potentials then it also holds in dimension $d$. Moreover, 
\begin{eqnarray}
C_{\gamma,d} &\le& C_{\gamma,n} C_{\gamma,d-n} \qquad\text{ and} \label{eq:subm1}\\
C_{\gamma,d} &\le& C_{\gamma,n} C_{\gamma+n/2,d-n} .
\label{eq:subm2}
\end{eqnarray}
\end{theorem}
\begin{remarks}\rm 
i) In the scalar case Aizenman and Lieb \cite{AL} showed that the map 
$\gamma \to C_{\gamma,d}= L_{\gamma,d}/L^{\text{cl}}_{\gamma,d}$ 
is decreasing. This monotonicity holds also in the general case, so, in fact, (\ref{eq:subm2}) implies (\ref{eq:subm1}). The monotonicity in $\gamma$ is most easily seen in the phase space picture: By scaling one has, for 
$\gamma > \gamma_0\ge 0$, 
$$
\int_0^\infty (s+t)_-^{\gamma_0}\, t^{\gamma-\gamma_0 -1} dt
=
(s)_-^\gamma\, B(\gamma - \gamma_0,\gamma_0+1) ,
$$
where 
$B(\alpha,\beta)= \int_0^1 t^{\alpha-1}(1-t)^{\beta-1} dt$ 
is the Beta function. In other words, for each choice of 
$\gamma\! >\! \gamma_0\!\ge\! 0$ there exists a positive measure $\mu$ on $\rz_+$ with
$(s)_-^\gamma = \int_{\rz_+}(s+t)_-^{\gamma_0}\, d\mu(t)$. Using this, the functional calculus, and the Fubini-Tonelli theorem, we immediately get
\begin{equation*}
\begin{split}
\Tr(\Delta +V)_-^\gamma 
& =
\int_0^\infty \Tr(\Delta +V +t)_-^{\gamma_0}\, d\mu(t) \\
&\le 
\frac{C_{\gamma_0,d}}{(2\pi)^d}
\int_0^\infty d\mu(t) 
\iint d\xi dx\, \trG (\xi^2 +V(x) +t)_-^{\gamma_0}  \\
&=
\frac{C_{\gamma_0,d}}{(2\pi)^d}
\iint d\xi dx 
\int_0^\infty d\mu(t) \, 
\trG (\xi^2 +V(x) +t)_-^{\gamma_0} \\
&=
\frac{C_{\gamma_0,d}}{(2\pi)^d}
\iint d\xi dx \,
\trG (\xi^2 +V(x))_-^{\gamma} .\\[0.3em]
\end{split}
\end{equation*} 
ii) Theorem \ref{thm:subm} is a slight extension of a very nice observation of Laptev and Weidl \cite{Laptev,LW}. They used it to show $C_{\gamma,d}=1$ as long as $\gamma\ge 3/2$. Basically this follows immediately by induction and the above monotonicity from (\ref{eq:subm1}) for $n=1$ once one knows that $C_{3/2,1}=1$. The beauty of this observation is that this bound is well-known in the scalar case \cite{LT2} and Laptev and Weidl gave a proof for it in the general case. See also \cite{BL} for an elegant alternative proof which avoids the proof of Buslaev-Fadeev-Zhakarov type sum rules for matrix-valued potentials.\\[0.3em]
iii) Using $C_{\gamma,d}=1$ for $\gamma\ge 3/2$ and (\ref{eq:subm2}), we get the bound 
$$
C_{\gamma,d} \le C_{\gamma, 3}
$$
in $d\ge 3$ for all $\gamma\ge 0$. In particular, this implies a uniform bound (in $d$) for the constant in the Cwikel-Lieb-Rozenblum bound as soon as such an estimate is established in dimension three for operator-valued potentials. Below we will recover Cwikel's bound $C_{0,3}\le 3^4=81$, see Corollary \ref{cor:boundstates}. 
It is, already for scalar potentials, known, that $C_{0,3}\ge 8/\sqrt{3} > 4.6188$, \cite{GGMT}  \cite[eq.\ (4.24)]{LT2} (see also the discussion in \cite[page 96--97]{Simon3}); in fact, it is conjectured to be the correct value \cite{GGMT,LT2,Simon2}. In the \emph{scalar case} Lieb's proof \cite{Lieb} of the CLR-bound gives by far the best estimate, 
$C_{0,3}^{\mathrm{scalar}}\le 6.87$. However, Lieb's estimate grows like $\sqrt{\pi d}$ for large dimensions \cite[eq. (5.5)]{LT2}. 
While we get a quite large bound on $C_{0,3}$ this at least furnishes the uniform bound $C_{0,d}\le 81$ for all $d\ge 3$. It would be nice to extend Lieb's or even Conlon's proof \cite{Conlon} of the CLR-bound to operator-valued potentials. 
\end{remarks} 
To state our second result, Cwikel's bound in the operator-valued case, we need some more notation: 
$L^q_w(\rd,\calB(\calG))$, the analog of the weak $L^q$-space $L^q_w(\rd)$, is given by all operator-valued functions $g\!:\rd\to\calB(\calG)$ for which 
$$
\norm{g}^*_{q,w} = 
\norm{g}^*_{L^q_w(\rd,\calB(\calG))}
\defeq
\sup_{t>0} \big(t \, 
\vert\{ \norm{g(\cdot)}_{\calB(\calG)}>t \}\vert^{1/q}\big)
<\infty.
$$
Here $\vert B\vert$ is the $d$-dimensional Lebesgue measure of a Borel set $B\subset\rd$. 
Note that $\norm{\cdot}^*_{q,w}$ is not a norm since it fails to obey the triangle inequality already for scalar $g$. But, as in the scalar case, one can give a norm on $L_w^q(\rd;\calB(\calG))$ which is equivalent to $\norm{\cdot}^*_{L^q(\rd;\calB(\calG))}$. However, we will not need this.

With $p$ we abbreviate the operator $-i\nabla$ and similarly to the scalar case we define the operator $f(x)g(p)$ to be
$$
\psi\to f(x)g(p)\psi (x)= 
f(x) \frac{1}{(2\pi)^{d/2}}
\int \e^{ix\zeta} g(\zeta) \hat{\psi}(\zeta)\,d\zeta,
$$
that is, $f(x)g(p)= M_{f}\calF^{-1}M_{g}\calF$ with $M_f$, $M_g$ the ``multiplication" operators by $f(x)$ and $g(\xi)$ and $\calF$ the Fourier transform. 
A priori, $f(x)g(p)$ is well-defined only for simple functions, but it will turn out to be a compact operator for rather general ``functions" $f$ and $g$. 
The extension of Cwikel's bound to the operator-valued case is 
\begin{theorem}[Cwikel's bound, operator-valued case]
\label{thm:Cwikel} 
Let $f$ and $g$ be operator-valued functions on an auxiliary Hilbert space $\calG$. Assume that $f\in L^q(\rz^d\!,\calS^q(\calG)\!)$ and $g\in L_w^q(\rz^d,\calB(\calG)\!)$ for some $q>2$. Then $f(x)g(p)$ is a compact operator on $L^2(\rd,\calG)$. In fact, it is in the weak operator ideal $\calS^q_w(L^2(\rd,\calG))$ and, moreover, 
\begin{equation}\label{eq:cwikelbound}
\norm{f(x)g(p)}^*_{q,w} := 
\sup_{n\ge 1}\, n^{1/q} \mu_n\big(f(x)g(p)\!\big) 
\le
K_q \norm{f}_{q,q}\norm{g}_{q,w}^*
\end{equation}
where the constant $K_q$ is given by
$$
K_q = (2\pi)^{-d/q} \frac{q}{2}  
\left(\frac{8}{q-2}\right)^{1-2/q} 
\left( 1+ \frac{2}{q-2}\right)^{1/q} .
$$
\end{theorem}

As in the scalar case Theorem \ref{thm:Cwikel} gives a bound for the number of negative eigenvalues of Schr\"odinger operators with operator-valued potentials.
\begin{corollary}\label{cor:boundstates}
Let $\calG$ be some auxiliary Hilbert space and $V$ a potential in $L^{d/2}(\rd,\calS^{d/2}(\calG))$. Then the operator 
$-\Delta\otimes\idG +V$ 
has a finite number $N$ of negative eigenvalues. 
Furthermore, we have the bound
$$
N\le L_{0,d} \int_\rd \trG (V(x)_-^{d/2})\, dx 
$$
with  
$$
L_{0,d} \le  (2\pi\, K_d)^d L^{\mathrm{cl}}_{0,d}, 
$$
that is, $C_{0,d}\le (2\pi\, K_d)^d$. 
\end{corollary}
\begin{proof} For completeness we explicitly derive the estimate for the number of negative eigenvalues of $-\Delta\otimes\idG +V$ from Theorem \ref{thm:Cwikel}. Replacing $V$ with $-(V)_-$ if necessary and using the min--max principle, we can assume $V$ to be non-positive. 
Let $N$ be the number of negative eigenvalues of $-\Delta\otimes +V$ and put 
$Y\defeq \vert V\vert^{1/2}\,(\vert p\vert^{-1}\!\otimes\!\idG)$. By the Birman-Schwinger principle \cite{Birman,Schwinger,BS,Simon,RSIV} one has 
$$
1 \le \mu_N(Y). 
$$
But $\xi\to \vert \xi\vert^{-1}\!\otimes\!\idG$ has weak $L^d(\rd,\calB(\calG))$-norm $\tau_d^{1/d}$, $\tau_d$ being the volume of the unit ball in $\rd$. With Theorem \ref{thm:Cwikel} we arrive at 
$$
1\le K_d \tau_d^{1/d} \Vert\vert V\vert^{1/2}\Vert_{d,d} N^{-1/d},
$$
that is,
$$
N\le K_d^d \tau_d \Vert\vert V\vert^{1/2}\Vert_{d,d}^d
= (2\pi K_d)^d L^{\text{cl}}_{0,d} \int \trG(\vert V(x)\vert^{d/2})\, dx,
$$
since $L^{\text{cl}}_{0,d} = \tau_d/(2\pi)^d$.
\end{proof}
\begin{remark}\rm
Corollary \ref{cor:boundstates} gives the a priori bound 
$C_{0,d}\le (2\pi\, K_d)^d$ for $d\ge 3$. Using Theorem \ref{thm:subm} and the fact that 
$C_{\gamma,d}=1$ if $\gamma\ge 3/2$, \cite{LW}, we know that
$C_{0,d}\le \min_{n=3,\ldots,d} C_{0,n}$. Since the a priori bound given in Corollary \ref{cor:boundstates} increases rather fast in the dimension, the best we can conclude is 
$C_{0,d}\le (2\pi\, K_3)^3=3^4=81$. 
\end{remark}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of the sub-multiplicativity of the Lieb-Thirring constants}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We proceed very similarly to \cite{LW},  but freeze the first $n<d$ variables. Let $x_< =(x_1,\ldots,x_n)$, 
$x_> =(x_{n+1},\ldots,x_d)$ and $\xi_<$, $\xi_>$ similarly defined. Put
$$
W(x_<)\defeq (-\Delta_> +V(x_<,.))_{-},
$$
where $\Delta_>$ is the Laplacian in the $x_>$ variables. Clearly, by assumption on $V$, $W$ is a non-negative compact operator on $L^2(\rz^{d-n},\calG)$ for almost all $x_<\in\rz^{n}$ and, moreover, 
\begin{equation}\label{eq:freeze}
\begin{split}
\Tr (-\Delta +V)_-^\gamma 
&\le\tr_{L^2(\rz^n,L^2(\rz^{d-n},\calG))} 
(-\Delta_< -W)_-^\gamma\\
&\le
\frac{C_{\gamma,n}}{(2\pi)^n}
\iint\limits_{\rz^n \rz^n}d\xi_<dx_<\, 
\tr_{L^2(\rz^{d-n},\calG))}(\xi_<^2-W(x_<))_-^\gamma .
\end{split}
\end{equation}
Since $(t-(s)_-)_- = (t+s)_-$ for $t\ge 0$, $s\in\rz$,  the spectral theorem gives
\begin{multline*}
\tr_{L^2(\rz^{d-n},\calG))} 
(\xi_<^2 -W(x_<))_-^\gamma 
=
\tr_{L^2(\rz^{d-n},\calG))} 
(\xi_<^2 -\Delta_>+V(x_<,.))_-^\gamma \\
\le
\frac{C_{\gamma,d-n}}{(2\pi)^{d-n}}
\iint\limits_{\rz^{d-n} \rz^{d-n}}d\xi_>dx_>\, 
\trG (\xi_<^2 +\xi_>^2+V(x_<,x_>))_-^\gamma.
\end{multline*}
This together with (\ref{eq:freeze}) and the Fubini-Tonelli theorem shows (\ref{eq:subm1}).\\
For the other inequality we use the more usual form (\ref{eq:LToper2}) of the Lieb-Thirring inequality. Again, freezing the first $n$ coordinates and proceeding as before, we immediately get
\begin{equation}\label{eq:subm3}
L_{\gamma,d}\le L_{\gamma,n} L_{\gamma+n/2,d-n},
\end{equation}
where $L_{\gamma+n/2,d-n}$ enters now because in the first application of the Lieb-Thirring inequality (\ref{eq:LToper2}) the exponent is raised from $\gamma$ to $\gamma+n/2$. Using the definition (\ref{eq:LTcl}) for the classical Lieb-Thirring constant together with the Fubini-Tonelli theorem and scaling, one easily sees 
\begin{equation*}
\begin{split}
L_{\gamma,d}^{\text{cl}} 
&= 
\int_{\rd}\! dp\, (|p|^2-1)_{-}^\gamma \\
&=
\int_{\rz^n}\! dp_{<}\, (\vert p_{<}\vert^2-1)_{-}^\gamma 
\int_{\rz^{(d-n)}}\! dp_{>}\, (\vert p_{>}\vert^2-1)_{-}^{\gamma+n/2} \\
&= 
L_{\gamma,n}^{\text{cl}} L_{\gamma+n/2,d-n}^{\text{cl}} .
\end{split}
\end{equation*}
This together with (\ref{eq:subm3}) proves (\ref{eq:subm2}) and thus Theorem \ref{thm:subm}.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of Cwikel's bound}\label{sec:cwikel}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The proof of Theorem \ref{thm:Cwikel} follows closely Cwikel's original proof. We first need a criterion for $f(x)g(p)$ to be a Hilbert-Schmidt operator. 
\begin{lemma}\label{lem:HS}
Let $f\in L^2(\rd,\calS^2(\calG))$ and assume $g$ obeys 
$\norm{g(.)}_{\calB(\calG)}\in L^2(\rd)$. Then
the operator $f(x)g(p)$ is Hilbert-Schmidt and we have the estimate
\begin{equation*}
\begin{split}
\norm{f(x)g(p)}_{HS}^2 &= 
(2\pi)^{-d} \iint \trG [g^*(\xi)f(x)^*f(x)g(\xi)]\, dx d\xi\\
&\le
(2\pi)^{-d} \int \trG [\vert f(x)\vert^2]\,dx \int \norm{g(\xi)}_{\calG}^2\, d\xi.
\end{split}
\end{equation*}
\end{lemma}
\begin{proof}
In the scalar case this is well-known and is usually shown by noting that in this case $f(x)g(p)$ is a convolution operator. Another proof is by changing the basis: 
Let $\calF$ be the Fourier transform on $L^2(\rd,\calG)$, that is, 
$$
\calF u (\xi) \defeq 
(2\pi)^{-d/2} \int_{\rd} \e^{-i\xi\cdot x} u(x)\,dx .
$$
Then the Hilbert-Schmidt norms of  $f(x)g(p)$ and 
$M_f\calF^{-1}M_g$ are equal. The operator $M_f\calF^{-1}M_g$ has ``kernel" $(2\pi)^{-d/2}\e^{ix\cdot\xi}f(x)g(\xi)$ and thus by \cite[Theorem VI.23]{RSI} or \cite[Section III.9]{GK} 
\begin{equation*}
\begin{split}
\norm{f(x)g(p)}_{HS}^2 &= \norm{f(x)\calF^{-1}g(\xi)}_{HS}^2 \\
&= (2\pi)^{-d}
\iint \trG(g(\xi)^*f(x)^*f(x)g(\xi))\,dx d\xi \\
&= (2\pi)^{-d}
\iint \trG(\vert f(x)\vert^2 \vert g(\xi)^*\vert^2)\,dx d\xi \\
&\le (2\pi)^{-d}
\iint \trG(\vert f(x)\vert^2) \norm{g(\xi)}_{\calG}^2)\,dx d\xi \\
&= (2\pi)^{-d}
\int \trG (\vert f(x)\vert^2)\,dx 
\int \norm{g(\xi)}_{\calG}^2)\,d\xi .
\end{split}
\end{equation*}
\end{proof}


The first step rests on splitting the operator $f(x)g(p)$ (which is a priori only defined on simple functions) into manageable pieces. Fix $t\!>\!0$, $r\!>\!1$ and assume that $f$ and $g$ are non-negative, in particular, self-adjoint, operator-valued functions. For a Borel subset $B$ of $\rz$ let $\chi_{B}(f(x))$ and $\chi_{B}(g(\xi))$ be the spectral projection operators of $f(x)$ and $g(\xi)$, respectively. By the functional calculus we have 
\begin{equation}\label{eq:decomposition}
\begin{split}
f(x)&= \sum_{l\in\gz} f(x)\chi_{t(r^{l-1}, r^l]}(f(x))
= \sum_{l\in\gz} f_l(x)\\
g(\xi)&= \sum_{l\in\gz} g(\xi)\chi_{(r^{l-1}, r^{l}]}(g(\xi))
= \sum_{l\in\gz} g_l(\xi) , 
\end{split}
\end{equation}
where $f_l$ (resp., $g_m$) are mutually orthogonal operators.  We use this decomposition of $f$ and $g$ to split the operator $f(x)g(p)$ into   
\begin{equation}
f(X)g(P) = B_t +H_t
\end{equation}
with $B_t\defeq \sum_{l+m \le 1}f_l(x)g_m(p)$, 
$H_t\defeq \sum_{l+m > 1}f_l(x)g_m(p)$. 
Note that this decomposition of $f(x)g(p)$ is slightly different from the one used by Cwikel. We have 

\begin{lemma} \label{lem:bounds} 
Let $f$ and $g$ be non-negative operator-valued functions. If $q>2$ and 
$f\!\in\! L^q(\rd,\calS^q(\calG))$, 
$g\!\in\! L^q_{w}(\rd,\calB(\calG))$ with $\norm{f}_{q,q}=1$ and $\norm{g}_{q,w}^*=1$ then  

\noindent
{\rm a)} $B_t$ is a bounded operator with operator norm bounded by
$$
\norm{B_t}_{L^2(\rz^d,\calG)} 
\le  t \frac{r}{1-r^{-1}}.
$$
{\rm b)} $H_t$ is a Hilbert-Schmidt operator with Hilbert-Schmidt norm bounded by
$$
\norm{H_t}_{\text{HS}}^2\le   
(2\pi)^{-d}t^{-(q-2)} \Big(1+\frac{2}{q-2} \Big).
$$
\end{lemma}
\begin{remarks}\rm i) Due to our choice of $B_t$, $H_t$ the bound in Lemma \ref{lem:bounds}.b) is \emph{independent} of $r$ and in a) it easy to see that the choice $r\!=\!2$ is optimal.\\[0.3em]
ii) This lemma also shows that $f(x)g(p)$ is a compact operator since it is the norm limit for $t\to 0$ of the Hilbert-Schmidt operators $H_t$.
\end{remarks}
\begin{proof}
Part a) follows completely Cwikel's original proof: Since the $f_l$ (resp., $g_m$) are orthogonal operators for different indices we get, for simple functions $\psi$ and $\phi$, say, 
\begin{equation*}
\begin{split}
\vert\langle\psi, & B_t \phi \rangle \vert
\le
\sum_{l+m\le 1} r^{l+m} 
\big\Vert r^{-l}f_l(x)\psi \big\Vert_2\, 
\big\Vert r^{-m}g_m(p)\phi \big\Vert_2 \\
&\le
\sum_{s\le 1} r^s
\sum_{m\in\nz} \big\Vert r^{-(s-m)}f_{s-m}(x)\psi\big\Vert_2
\big\Vert r^{-m}g_m(p)\phi \big\Vert_2 \\
&\le
\sum_{s\le 1} r^s
\left(\sum_{m\in\nz} 
\big\Vert r^{-(s-m)}f_{s-m}(x)\psi\big\Vert_2^2\right)^{1/2}
\left(\sum_{m\in\nz} 
\big\Vert r^{-m}g_m(p)\phi\big\Vert_2^2\right)^{1/2} \\
&= 
\sum_{s\le 1} r^s
\Big\Vert \sum_{m\in\nz}r^{-(s-m)}f_{s-m}(x)\psi\Big\Vert_2 
\Big\Vert \sum_{m\in\nz} r^{-m}g_m(p)\phi\Big\Vert_2 \\
&\le
r (1-r^{-1})^{-1} t \norm{\psi}\norm{\phi} ,
\end{split}
\end{equation*}
since $\sum_l r^{-l}f_l(x)\le t\idG$ and 
$\sum_{m}r^{-m}g_m(\xi) \le \idG$. Thus $B_t$ extends to a bounded operator on $L^2(\rd,\calG)$ with the given bound for its norm.\\[0.3em]
To prove part b) observe that by Lemma \ref{lem:HS} and the cyclicity of the trace, we have
$$
\norm{H_t}_{\text{HS}}^2 =
\sum_{l+m >1} \iint \tr_{\calG}[f_l(x)g_m(\xi)^2 f_l(x)]\, dx d\xi .
$$
Assume for $x,\xi\in\rd$ the operator inequality
\begin{equation}
\begin{split}\label{eq:operineq}
\sum_{l+m >1}f_l(x)g_m(\xi)^2 f_{l}(x) 
&\le 
\Big(
\norm{g(\xi)} f(x) \chi_{(t,\infty)}(\norm{g(\xi)}f(x)\!) 
\Big)^2
\\
&=: h(x,\xi)^2
\end{split}
\end{equation}
on the Hilbert space $\calG$. Note that the projection operator $\chi_{(t,\infty)}(\norm{g(\xi)}f(x))$ (on $\calG$) commutes with $f(x)$ for all $x,\xi\in\rd$. Let $\lambda_j(x)$ be the $j^{\text{th}}$ ordered eigenvalue of $f(x)$, and $E_{j}(\alpha)\defeq \{\norm{g(.)} \lambda_j(.)>\alpha \}$. Each $E_j$ has $2d$ dimensional Lebesgue measure
$$
\vert E_j(\alpha)\vert_{2d} 
=
\int \vert \{\norm{g(.)}>\alpha/\lambda_j(x) \}\vert_{d}\, dx
\le
\alpha^{-q} \int \lambda_j(x)^q \, dx , 
$$
since $\norm{g}_{q,w}^* =1$ by assumption. 
Thus we see 
\begin{equation*}
\begin{split}
\norm{H_t}_{HS}^2 &\le
(2\pi)^{-d}
\iint\trG[h(x,\xi)^2]\, dx d\xi \\
&= 
(2\pi)^{-d}\sum_{j} 
2 \int_0^\infty \vert E_j(\max(\alpha, t))\vert_{2d} \alpha\, d\alpha \\
& = (2\pi)^{-d}\left( \sum_j 2 \int_0^{t} 
\vert E_j(t)\vert_{2d} \alpha\, d\alpha
+\sum_j 2 \int_{t}^\infty \vert E_j(\alpha)\vert_{2d} 
\alpha\, d\alpha \right) \\
&\le
(2\pi)^{-d}
t^{-(q-2)}  \big(1+ 2/(q-2) \big)\sum_j \int \lambda_j(x)^q\, dx \\
&=
(2\pi)^{-d}
t^{-(q-2)}  \big(1+ 2/(q-2) \big),
\end{split}
\end{equation*}
since $\sum_j \int \lambda_j(x)^q\, dx = \norm{f}_{q,q}^q =1$ by assumption. It remains to prove (\ref{eq:operineq}): Again, let $s=l+m$ and note that the 
$g_m(\xi)= g(\xi)\chi_{(r^{m-1}, r^{m}]}(g(\xi)\!)$ 
are orthogonal operators for different indices. As operators on $\calG$, 
\begin{equation*}
\begin{split}
\sum_{l+m>1} & f_l(x)g_m(\xi)^2f_l(x)
=
\sum_{l\in\gz}\sum_{s\ge 2} f_l(x) g_{s-l}(\xi)^2 f_l(x) \\
& = 
\sum_{l\in \gz} f_l(x) \big(\sum_{s\ge 2} g_{s-l}(\xi)^2\big) f_l(x) 
=
\sum_{l\in\gz} f_l(x) g(\xi)^2 \chi_{(r^{1-l},\infty)}(g(\xi)\!) f_l(x) \\
&\le
\sum_{l\in\gz} 
f_l(x) 
\norm{g(\xi)}_{\calG}^2 \chi_{(r^{1-l},\infty)}(\|g(\xi)\|) 
f_l(x) \\
&= 
f(x)^2 \norm{g(\xi)}_\calG^2 
\sum_{l\in\gz}
\underbrace{\chi_{(r^{1-l},\infty)}(\|g(\xi)\|)
\chi_{t(r^{l-1},r^l]}(f(x)\!)
}_{\le \chi_{ (t,\infty)}( \norm{g(\xi)}f(x)) \chi_{t(r^{l-1},r^l]}(f(x)\!)} \\
&\le 
f(x)^2 \norm{g(\xi)}_\calG^2 
\chi_{ (t,\infty)}( \norm{g(\xi)}f(x))
\sum_{l\in\gz}
\chi_{t(r^{l-1},r^l]}(f(x)\!) \\
&= 
f(x)^2 \norm{g(\xi)}_\calG^2 \chi_{ (t,\infty)}( \norm{g(\xi)}f(x)),
\end{split}
\end{equation*}
which proves (\ref{eq:operineq}) and hence the lemma.
\end{proof}
Given the above bounds the proof of Theorem \ref{thm:Cwikel} is by now a standard interpolation argument. We give this argument for the sake of completeness:
\begin{proof}[Proof of Theorem \ref{thm:Cwikel}]
First, without loss of generality assume that $f$ and $g$ are non-negative operator-valued functions. Indeed, let $\calF$ be the Fourier transform and $M_f$ and $M_g$ the operators of ``multiplication" by $f$ and $g$ and note that $f(x)g(p)$ and $M_f\calF^{-1}M_g$ have the same singular values. With the polar decompositions $f(x)= U_1(x)\vert f(x)\vert$ and 
$g(\xi)= \vert g^*(\xi)\vert U_2^*(\xi)$ in the Hilbert space $\calG$ we have
$$
M_f\calF^{-1}M_g = 
U_1 M_{\vert f\vert}\calF^{-1}M_{\vert g^*\vert} U_2^* , 
$$
where $U_j$, $j\in \{1,2\}$ are fibered partial isometries in the space $L^2(\rd,\calG)$, for example, $(U_1\psi)(x)= U_1(x)\psi(x)$ . Hence the singular values of $f(x)g(p)$ are bounded by the singular values of 
$M_{\vert f\vert}\calF^{-1}M_{\vert g^*\vert}$ and 
$\norm{\vert g^*\vert}_{q,w}^* = \norm{g}_{q,w}^*$. 

By one of the consequences of Ky Fan's inequality \cite{GK} we have
\begin{equation*}
\begin{split}
\mu_n (f(x)g(p))  &= \mu_n(B_t+H_t) 
\le \mu_1 (B_t) + \mu_n(H_t) 
\le \norm{B_t} + \frac{1}{\sqrt{n}}\norm{H_t}_{HS} \\
&\le
t \frac{r}{1-r^{-1}} + 
(2\pi)^{-d/2}t^{-(q-2)/2} 
\big(1+\frac{2}{q-2} \big)^{1/2} \frac{1}{\sqrt{n}}
\end{split}
\end{equation*}
using Lemma \ref{lem:bounds}. Choosing $t$ and $r$ ($=2$) optimal gives 
\begin{equation*}
\mu_n (f(x)g(p)) \le 
(2\pi)^{-d/q} \frac{q}{2}  
\left(\frac{8}{q-2}\right)^{1-2/q} 
\left( 1+ \frac{2}{q-2}\right)^{1/q} 
n^{-1/q} 
\end{equation*}
which proves Theorem \ref{thm:Cwikel}.
\end{proof}


\noindent{\textbf{ Acknowledgment:}}
It is a pleasure to thank Ari Laptev and Timo Weidl for stimulating discussions 
and Michael Solomyak for comments and discussions on an earlier version of the paper.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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