Content-Type: multipart/mixed; boundary="-------------0012151052433"
This is a multi-part message in MIME format.
---------------0012151052433
Content-Type: text/plain; name="00-501.comments"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="00-501.comments"
also available at quant-ph/0012070
---------------0012151052433
Content-Type: text/plain; name="00-501.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="00-501.keywords"
scaling, semiclassical, Gutzwiller trace formula, period, action
---------------0012151052433
Content-Type: application/x-tex; name="oscscal.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="oscscal.tex"
\documentclass[pra,preprint,tightenlines,eqsecnum]{revtex4}
%(18 pages, 12pt, optimal page breaks)
%\documentclass[pra,preprint,showpacs,eqsecnum]{revtex4}
%(23 pages, double-spaced)
%\documentclass[pra,twocolumn,showpacs,eqsecnum]{revtex4}
%(10 pages, 10 pt, some formulas cramped)
\DeclareFixedFont{\fiverm}{OT1}{cmr}{m}{n}{5pt}
\input{prepictex}
\input{pictex}
\input{postpictex}
%\endinput
% \usepackage{graphicx}%
\begin{document}
\title[Spectrum, Orbits, and Scaling]{Spectral Oscillations,
Periodic Orbits, and Scaling}
\author{S. A. Fulling}
\email{fulling@math.tamu.edu}
\homepage{http://www.math.tamu.edu/~fulling}
\affiliation{Mathematics Dept., Texas A\&M University,%
College Station, TX, 77843-3368 USA}
%\date{\today}
\date{December 13, 2000}
\begin{abstract}
The eigenvalue density of a quantum-mechanical system
exhibits oscillations, determined by
the closed orbits of the corresponding classical system;
this relationship is simple and strong for waves in billiards or on
manifolds, but becomes slightly muddy for a Schr\"odinger equation
with a potential,
where the orbits depend on the energy.
We discuss several variants of a way to restore the simplicity
by rescaling the coupling constant or the size of the orbit or
both.
In each case the relation between the oscillation frequency and
the period of the orbit is inspected critically;
in many cases it is observed that a characteristic length of the
orbit is a better indicator.
When these matters are properly understood, the periodic-orbit
theory for generic quantum systems recovers the clarity and
simplicity that it always had for the wave equation in a cavity.
Finally, we comment on the alleged ``paradox'' that semiclassical
periodic-orbit theory is more effective in calculating low energy
levels than high ones.
\end{abstract}
\pacs{03.65.Sq}
\maketitle
%%%%% \tableofcontents
\section{Introduction}\label{sec:intro}
The basic principle relating classical periodic orbits to quantum
spectra has been succinctly stated by Delos and Du~\cite{DD}:
\begin{quotation}
The average density of states as a function of energy [is] equal to
a smooth monotonic function, related to the volume occupied by the
energy-shell in phase-space, plus a sum of sinusoidal oscillations.
The wavelength and amplitude of each oscillation are respectively
correlated with the period and the stability of a periodic orbit
of the system.
For given energy resolution $\Delta E$, only those periodic orbits
are significant for which the period is less than
$2\pi \hbar/\Delta E$.
\end{quotation} \noindent
An example of a precise theorem along these lines can be found
in~\cite{CRR}.
(Other expository works include \cite{FW,Berry,Bolte,Rob}.)
This principle was simultaneously discovered thirty years ago in
three classic research works dealing with three distinct contexts:
\begin{enumerate}
\item Balian and Bloch \cite{BB3} considered a bounded region
in space.
The differential operator
(corresponding to a quantum Hamiltonian~$H$)
is the Laplacian with standard
(Dirichlet or Neumann) boundary conditions.
The relevant classical paths are straight lines with specular
reflection at the boundary.
\item Gutzwiller \cite{Gutz3,Gutz4} studied a Schr\"odinger
equation in infinite space, with a potential that is responsible
for confining particles to compact regions and creating a discrete
energy spectrum. The periodic orbits involved are those of the full
classical dynamics governed by that potential function.
(Balian and Bloch also studied this situation in a later
paper~\cite{BB5}.)
\item Duistermaat and Guillemin \cite{DG}, building on earlier
papers in the French mathematical literature
\cite{Chaz,ColV,Cots},
worked on a compact Riemannian manifold without boundary.
The special case of their problem most closely analogous to the
other two is that where $H$ is the
Laplace--Beltrami operator for that manifold, plus
a possible potential function that is treated as a
{\em perturbation}.
That is, the manifold itself is responsible for confining the orbits
and making the spectrum discrete, and the periodic orbits that
appear in the theorem are the closed geodesics of the Riemannian
metric, independent of the potential.
\end{enumerate}\noindent
Although ultimately the most important thing about these three
bodies of work is their similarity,
there are some characteristic differences among them.
First,
Gutzwiller works with the Schr\"odinger equation,
\begin{equation}
i\, \frac{\partial\psi}{\partial t} = H \psi \qquad(\hbar=1),
\label{1.1}\end{equation}
the others with the
wave equation,
\begin{equation}
-\, \frac{\partial^2\psi}{\partial t^2} = H\psi
\label{1.2}\end{equation}
or its first-order pseudodifferential form
\begin{equation}
i\,\frac{\partial\psi}{\partial t} = \sqrt H\, \psi.
\label{1.3}\end{equation}
(Indeed, the authors of \cite{DG} go so far as to say that
``No statement like [our theorem] holds if we replace the first
order operator [$\sqrt H$] by a higher order operator.'')
Second, Gutzwiller and the many physicists who have followed him
usually speak of periodicities in the {\em energy\/} spectrum,
determined (reciprocally) by the {\em times\/} (periods) of the
classical orbits.
But the conclusions of the other works are formulated in terms of
periodicity in wave frequency, $\omega$
(essentially the {\em square root\/}
of energy, $E$),
determined by the {\em lengths\/} of the orbits.
A reader who has not devoted years to the comparative study of
these papers might form the impression that the distinction between
energy periodicity and frequency periodicity arises from the
respective concentration on the Schr\"odinger equation or the wave
equation;
in other words, that the square root relating $E$ to $\omega$ is
merely the square root relating the quantum Hamiltonian to the
pseudodifferential operator $\sqrt H$ appearing in the wave
equation~(\ref{1.3}).
That would be wrong.
Indeed, in hindsight, it is obviously wrong:
The conclusions of the theory are objective statements about the
spectrum of the {\em same\/}
elliptic second-order differential operator~$H$.
These facts cannot depend upon which time-dependent equation,
(\ref{1.1}) or (\ref{1.2}),
was used as a technical tool in discovering them.
A third difference is that the orbits studied in \cite{BB3}
and \cite{DG}
are energy-independent objects, but the closed orbits in a
potential vary with the energy.
The latter fact means that the parametrization of spectral
oscillations by orbits
can be meaningful only over rather short energy
intervals, in general.
(An exception is systems with homogeneous potentials,
where the orbits at different energies are related merely by
dilation in space. See Sec.~\ref{sec:hom} and
\cite[text surrounding
(1.40)]{Bolte}.)
Friedrich and Wintgen \cite{WF,FW}
pointed out that this problem can be at least partially avoided by
broadening the point of view to allow variation of a coupling
constant with energy.
The primary aim of the present paper is to generalize their
observation,
showing that it has several variants, one that is applicable
to any system, and others that may be more appropriate for
systems with certain scaling symmetries.
In each case the natural variable of the spectral oscillations is
a certain power of the energy.
It is argued tht the identification of the frequencies of these
oscillations with the {\em periods\/} of the orbits is rather
forced;
some other quantity, often a {\em length}, is more pertinent.
These matters occupy most of the remaining sections of this paper.
Sec.~\ref{sec:peri}
is a digression to establish a (known) relation among
action, energy, and period when the coupling constant
is {\em not\/} scaled; in the context of scaling, this relation no
longer holds, but neither is it necessary.
In Sec.~\ref{sec:wave}
we elucidate the relationship between the wave
(\ref{1.2})--(\ref{1.3})
and Schr\"odinger (\ref{1.1}) approaches (and their correlates in
classical phase space);
this point is understood by workers in the field but is
seldom spelled out.
%^& footnote?
(To keep the treatments of the two approaches as parallel as
possible, we discuss the wave equation in the terminology of
relativistic quantum theory;
this should not be allowed to obscure the applicability of
periodic-orbit analysis in other, more classical, contexts, such
as optics and acoustics, where quantization is just a metaphor.)
Finally, Sec.~\ref{sec:reso}
addresses the frequently remarked-upon counterintuitive fact
that the ``semiclassical'' periodic-orbit method is more effective
in reproducing low-lying eigenvalues than eigenvalues in the regime
of large quantum numbers.
\section{Wave and Schr\"odinger dynamics}\label{sec:wave}
A Hamiltonian system is defined by a function $H(\mathbf x,\mathbf p)$ of
$\mathbf x\in\Omega$, a $d$-dimensional region or manifold, and
$\mathbf p\in\mathbf R^d$.
Its classical equations of motion are
\begin{equation}
\frac{d \mathbf x}{dt} = \nabla_{\mathbf p} H, \qquad
\frac{d \mathbf p}{dt} = - \nabla_{\mathbf x} H.
\label{2.1}\end{equation}
Energy is conserved: Each trajectory of the system (\ref{2.1})
remains on a phase-space submanifold
\begin{equation}
H(\mathbf x,\mathbf p) = E \label{2.2}\end{equation}
of dimension $2d-1$.
Suppose that $H(\mathbf x,\mathbf p)$ is quadratic in~$\mathbf p$ and
nonnegative, and consider a new Hamiltonian function
\begin{equation}
h(\mathbf x,\mathbf p) \equiv H(\mathbf x,\mathbf p)^{1/2}.
\label{2.3}\end{equation}
Calling the new time parameter~$\tau$, calculate the resulting
equations of motion:
\begin{equation}
\frac{d\mathbf x}{d\tau} = \nabla_{\mathbf p} h =
{\textstyle\frac12}H^{-1/2} \frac{d\mathbf x}{dt}\,, \qquad
\frac{d\mathbf p}{d\tau} =
{\textstyle\frac12}H^{-1/2} \frac{d\mathbf p}{dt}\,.
\label{2.4}\end{equation}
It follows that the trajectories of the new system in phase space
are exactly the same as those of the old one, but parametrized
differently.
At a fixed energy the two time scales are simply related by
\begin{equation}
\frac{d\tau}{dt} = 2\sqrt E.
\label{2.5}\end{equation}
The relation between $\mathbf p$ and the velocity is changed by
the same factor.
Consider the very special case $H= \mathbf p^2/2m$.
Then $\mathbf p$ is a constant vector with magnitude
$p=\sqrt{2mE}$,
and the velocity in system (\ref{2.1}) satisfies
\begin{equation}
\mathbf v \equiv \frac{d\mathbf x}{dt} =
{\mathbf p\over m}\,, \qquad v^2 = {2E\over m}\,,
\label{2.6}\end{equation}
while the velocity in system (\ref{2.4}) satisfies
\begin{equation}
\mathbf u \equiv \frac{d\mathbf x}{d\tau} =
\frac1{2\sqrt E}\,{\mathbf p\over m}\,,
\qquad u^2 = \frac1{2m} = \text{constant}.
\label{2.7}\end{equation}
The trajectories of $H$~dynamics in space-time are, of course,
those of a nonrelativistic particle: straight lines
of arbitrary slope depending on energy.
The trajectories of $h$~dynamics in space-time are
{\em independent of energy}, as appropriate for a
{\em relativistic massless particle}.
(Clearly, in this context $m$ does not have the physical
significance of a mass;
rather, it parametrizes the wave speed as $c=(2m)^{-1/2}$.)
Passing to a quantum system by the prescription
\begin{equation}
\mathbf p \mapsto -i\nabla_{\mathbf x}\,,\qquad
E \mapsto i \,\frac{\partial}{\partial\tau}
\label{2.8}\end{equation}
converts the relativistic energy relation
$h(\mathbf x,\mathbf p)=E$
to a special case of (\ref{1.3}),
which is equivalent to the wave equation
\begin{equation}
\frac{\partial^2\psi}{\partial t^2} =\frac1{2m} \nabla^2\psi
\label{2.9}\end{equation}
supplemented by a condition of positive frequency.
The nonrelativistic energy relation (\ref{2.2})
of course becomes the
free Schr\"odinger equation, a case of (\ref{1.1}).
The energy independence of relativistic trajectories
(which extends to the very general systems studied in~\cite{DG})
leads to a
very direct relationship between spectral oscillations and the
{\em singularities\/}
of the fundamental solution of the wave equation
(\ref{1.3}).
This is the meaning of the remark of Duistermaat and Guillemin
quoted in Sec.~\ref{sec:intro}.
An up-to-date exposition of that methodology is given
in~\cite{SV}.
Now turn to a Hamiltonian including a potential energy function,
\begin{equation}
H(\mathbf x,\mathbf p) =
{\mathbf p^2\over 2m} + V(\mathbf x),
\label{2.10}\end{equation}
where $V(\mathbf x)$ is nonnegative.
The momentum is no longer constant, and the velocities satisfy
\begin{equation}
v^2 = \frac2m [E-V(\mathbf x)]
\label{2.11}\end{equation}
(the usual nonrelativistic kinetic-energy relation) and
\begin{eqnarray}
u^2 &=& {p^2 \over 2m(p^2 + 2mV(\mathbf x))}
\nonumber\\
&=& \frac1{2m}\left( 1-\frac{V(\mathbf x)}E\right) .
\label{2.12} \end{eqnarray}
The trajectories in space-time are now curves, whose slopes at
each point are greater than they would have been for a free
particle.
In particular, in the relativistic case the trajectories stay
always inside the local light cones,
as appropriate for a relativistic
particle with mass (induced here by an $\mathbf x$-dependent
interaction).
The main point is that the traces of these orbits on configuration
space are the same as in the nonrelativistic case.
In passing, note that if $V$ were allowed to be negative, the
classical relativistic trajectories would be
``tachyonic'' according to~(\ref{2.12}).
Nevertheless, the wave equation~(\ref{1.2}) still obeys hyperbolic
causality.
This situation was studied by Schroer and others
in the context of quantum field theory~\cite{Sch}.
We shall not consider this case further, because it interferes
with the definition of the square roots in
(\ref{2.3}) and (\ref{1.3}).
In what follows we restrict attention to Hamiltonians of the form
(\ref{2.10}).
Then the quantization prescription (\ref{2.8})
leads unambiguously to a
second-order elliptic partial differential operator
\begin{equation}
H = -\,\frac1{2m}\nabla^2 + V(\mathbf x).
\label{2.13}\end{equation}
(In more general cases some variant of Weyl quantization must be
chosen to resolve operator-ordering ambiguities~\cite{Full}.)
The complete specification of the problems
(\ref{1.1})--(\ref{1.3}) may also
require boundary conditions, which we take to be of the standard
Dirichlet, Neumann, or Robin type.
\section{Scaling a billiard}\label{sec:bill}
Henceforth we adopt the nonrelativistic, or
Schr\"odinger-equation, point of view, and we restore the
conventional constant $\hbar$.
It is assumed that the spectrum of the operator~$H$ is
discrete, at least below some threshold.
The essence of the Gutzwiller trace theory
\cite{DD,FW,Berry,Bolte}
is that the density of states, $\rho(E)$, contains a term
\begin{equation}\rho_\gamma(E) =
a_\gamma \sin\left[{S_\gamma\over \hbar} + \eta_\gamma\right]
\label{3.1}\end{equation}
for each classical periodic orbit $\gamma$ of energy~$E$;
here $S_\gamma(E)$ is the {\em action\/}
$\oint \mathbf p\cdot d\mathbf x$ of the orbit~$\gamma$,
and the details of the amplitude $a_\gamma(E)$ and the phase
$\eta_\gamma$ do not concern us now.
(Also, this simplified formulation does not do justice to the
complications that arise when the orbits are not isolated.
See \cite[text surrounding (1.35)]{Bolte} for a summary, and
\cite{DG} for a technical treatment.)
Let us examine what (\ref{3.1}) asserts
for a system of the type studied
by Balian and Bloch~\cite{BB3}.
The classical paths are straight lines in a region
$\Omega\subset \mathbf R^d$
with specular reflections at the boundary of~$\Omega$.
(The distinction among different boundary conditions shows up in
the phases~$\eta_\gamma\,$.)
In particular, these paths are the {\em same\/} for all~$E$.
The speed is fixed at $v = \sqrt{2E/m} $
(see (\ref{2.6})).
If the length of such a closed polygonal path, $\gamma$, is
$L_\gamma\,$,
then its period is
\begin{equation}
T_\gamma(E) = {L_\gamma \over v} = \sqrt{m\over 2E} \, L_\gamma\,.
\label{3.2}\end{equation}
The momentum is parallel to the path with fixed magnitude
$p=\sqrt{2mE}$, so the action is
\begin{equation}
S_\gamma(E) \equiv \oint_\gamma \mathbf p\cdot d\mathbf x =
\sqrt{2mE} \,L_\gamma = 2ET_\gamma\,. \label{3.3}\end{equation}
Thus (\ref{3.1}) becomes
\begin{equation}
\rho_\gamma(E) = a_\gamma \sin\left[{2T_\gamma E\over \hbar} +
\eta_\gamma\right],
\label{3.4}\end{equation}
which is superficially a sinusoidal function of $E$ with period
$\pi\hbar/T_\gamma\,$.
But because $T_\gamma$ is itself a function of~$E$,
whereas $L_\gamma$ is not,
a more accurate picture is obtained by reexpressing (\ref{3.4}) as
\begin{equation}
\rho_\gamma(E) = a_\gamma \sin\left[{\sqrt{2m E}\over \hbar}\,L_\gamma
+ \eta_\gamma\right].
\label{3.5}\end{equation}
Thus $\rho_\gamma$ is exactly
(to the extent that $a_\gamma$ and $\eta_\gamma$ can be regarded as
constant)
a sinusoidal function of $\omega \equiv \sqrt E$
with period
\begin{equation}
P_\omega =\sqrt{2\over m} {\pi\hbar \over L_\gamma }\,,
\label{3.6}\end{equation}
or $P_\omega={2\pi/L_\gamma}$ if ${\hbar^2\over 2m} =1$,
the most natural
normalization when dealing with the wave equation.
This is the result obtained by Balian and Bloch~\cite{BB3}
working from
the relativistic, or wave-equation, point of view:
The eigenvalue density oscillates with frequency $L_\gamma$ as a
function of~$\omega$.
On the other hand, noting again that $L_\gamma\,$, not $T_\gamma\,$,
is independent of~$E$, we have
\begin{equation}
\frac{d S_\gamma}{dE} = \sqrt{m\over 2E}\, L_\gamma = T_\gamma
= {S_\gamma\over 2E}\,.
\label{3.7}\end{equation}
Approximating $S_\gamma(E)$ locally by a linear
function~\cite{DD},
one gets from (\ref{3.1}) and (\ref{3.7})
\begin{equation}
\rho_\gamma(E) \approx a_\gamma
\sin\left[{T_\gamma\over\hbar}(E-E_0)
+{S_\gamma(E_0)\over\hbar} + \eta_\gamma\right].
\label{3.8}\end{equation}
Thus $\rho_\gamma$ oscillates {\em locally\/} as a function of $E$
with frequency $T_\gamma(E_0)/\hbar$, or period
\begin{equation}
P_E = {2\pi\hbar\over T_\gamma}\,.
\label{3.9}\end{equation}
(Note the factor of $2$ discrepancy with our initial naive attempt.)
This is the sort of statement about spectral oscillations that is
most common in the physics literature surrounding the Gutzwiller
trace formula (see the next section).
For completeness we record the time-dependent action
\begin{equation}
R_\gamma(T) = S_\gamma(E) - ET_\gamma = ET_\gamma =
{mL_\gamma{}\!^2 \over 2T} \,,
\label{3.10}\end{equation}
where
\begin{equation}
E = {\textstyle \frac12}mv^2 = {mL^2\over T^2}
\label{3.11}\end{equation}
has been used.
It satisfies
\begin{equation}
\frac{d R_\gamma}{dT} = -E,
\label{3.12}\end{equation}
which is dual to (\ref{3.7}) when (\ref{3.10})
is interpreted as a Legendre transformation.
It can also be obtained as
\begin{equation}
R_\gamma \equiv \oint_\gamma \mathcal L\, dt = ET,
\label{3.13}\end{equation}
where $\mathcal L$ is the Lagrangian,
here equal to the kinetic energy.
\section{Interlude: The action--period relation}
\label{sec:peri}
In converting (\ref{3.1}) to (\ref{3.8}), only the relation
\begin{equation}
\frac{d S_\gamma}{dE} = T_\gamma
\label{4.1}\end{equation}
was used.
This identity is frequently cited as
``a well known theorem of classical mechanics''\negthinspace,
but a proof is hard to find in the literature.
Indeed, at second glance it is not even clear what (\ref{4.1})
{\em means\/} for a generic Hamiltonian (\ref{2.10}),
since $S_\gamma$ was defined for a single closed orbit
$\gamma$ existing
at a particular energy, say $E_0\,$.
The discussion in Sec.~\ref{sec:bill} shows that
for a billiard the orbit
indeed persists unchanged (in configuration space) as $E$ varies;
but when the dynamics involves a potential function the classical
path defined by an initial point and direction on $\gamma$ will
generally cease to be a periodic orbit when $E$ deviates
from~$E_0\,$.
It is reasonable to expect that generically $\gamma$ will smoothly
evolve with $E$ into a family of nearby periodic orbits $\gamma(E)$
(in general, disjoint from $\gamma(E_0)$).
But this picture obviously breaks down at a maximum of the
potential in dimension~1 (more generally, at a separatrix),
where two families of orbits merge into one.
In higher dimensions, moreover, it is not obvious that the
embedding of $\gamma(E_0)$ into a family is unique.
The main point of the remainder of this paper is that these
existence and uniqueness questions can be avoided by adopting a
broader point of view, in which other quantities are varied in
addition to the energy.
Here, however, we provide a simple derivation of (\ref{4.1}) in
circumstances where it makes sense.
{\em Assume\/} the existence of a family $\gamma(E)$ of classical
closed orbits of (\ref{2.1}) depending smoothly on the energy,~$E$
(as $E$ varies over some interval, possibly small).
Assume also that the period $T_\gamma$ depends smoothly and
monotonically on~$E$, so that $\gamma$ can be alternatively parametrized
by~$T$.
We shall prove
\begin{equation}
\frac{d R_\gamma}{dT} = - E,
\label{4.2}\end{equation}
which is equivalent to (\ref{4.1}) under the Legendre transformation
\begin{equation}
S = R+ET \qquad \bigl(S= S(E), \quad R= R(T)\bigr).
\label{4.3}\end{equation}
For a classical trajectory connecting two points, $\mathbf x$ and
$\mathbf x'$, and depending on the elapsed time $T$ as a parameter,
(\ref{4.2}) is the usual Hamilton--Jacobi equation for
$R$ as a function
of one endpoint (with $E = H(\mathbf x,\nabla R)$);
but for periodic orbits a different argument is necessary.
By definition,
\begin{equation}
R = \oint_\gamma \mathbf p\cdot d\mathbf x - ET =
\int_0^T \mathcal L \, dt,
\label{4.4}\end{equation}
\begin{equation}
\mathcal L = {\textstyle\frac12} m\dot{\mathbf x}(t)^2
- V(\mathbf x(t)).
\label{4.5}\end{equation}
In the standard way, consider a variation
\begin{equation}
\delta R = \int_0^T {\delta \mathcal L\over \delta\mathbf x}\,
\delta\mathbf x \, dt
+ \bigl.\mathcal L\bigr|_T\, \delta t,
\label{4.6}\end{equation}
where
\begin{equation}
\delta\mathcal L =
m \dot\mathbf x\cdot \delta\dot\mathbf x
-\nabla V\cdot \delta\mathbf x
= \frac{d}{dt}( m\dot\mathbf x\cdot \delta\mathbf x ) -
(m\ddot\mathbf x + \nabla V)\cdot \delta \mathbf x .
\label{4.7}\end{equation}
The last term in (\ref{4.7}) vanishes by the equation of motion,
and integrating the other term yields
\begin{equation}
m\dot\mathbf x(T)\cdot\delta\mathbf x(T)
- m\dot\mathbf x(0)\cdot\delta\mathbf x(0)
= -m \dot\mathbf x(T)^2 \, \delta T,
\label{4.8}\end{equation}
because
\begin{equation}
\mathbf x(T)= \mathbf x(0), \quad
\dot\mathbf x(T)= \dot\mathbf x(0), \quad
(\mathbf x+\delta\mathbf x)(T+\delta T) =
(\mathbf x+\delta\mathbf x)(0),
\label{4.9}\end{equation}
so that
\begin{eqnarray}
\delta\mathbf x(T) &=& (\mathbf x+\delta\mathbf x)(T) -
\mathbf x(T) \nonumber\\
&=& (\mathbf x+\delta\mathbf x)(T+\delta T)
- (\dot\mathbf x+\delta\dot\mathbf x)(T) - \mathbf x(0)
\nonumber\\
&=& \delta\mathbf x(0) - \dot\mathbf x(T)\,\delta T
+ O\bigl((\delta T)^2\bigr) .
\label{4.10}\end{eqnarray}
Putting all the ingredients into (\ref{4.6}) yields
\begin{eqnarray*}
{\delta R\over \delta T} &=&
-m\dot\mathbf x(T)^2 + {\textstyle\frac12}
m\dot\mathbf x(T)^2 - V(\mathbf x(t)) \\
&=& - {\textstyle\frac12} m\dot\mathbf x(T)^2 - V(\mathbf x(t))
\\
&=& -E, \end{eqnarray*}
as desired.
From (\ref{4.2}) follows (\ref{4.1}), and then (\ref{3.1})
leads as
before~\cite{DD} to (\ref{3.8})
and (\ref{3.9}), under the additional assumption that $T_\gamma$ is
sufficiently slowly varying for the concept of a local frequency of
oscillation of $\rho_\gamma$ to make sense.
An interesting example is the one-dimensional harmonic oscillator,
\begin{equation}
V(x) = {\textstyle\frac12} m\varpi^2 x^2.
\label{4.11}\end{equation}
The general solution of its equation of motion is
\begin{equation}
x(t) = x_0 \cos {\varpi t} + \frac{p_0}m \,\sin{\varpi t}.
\label{4.12}\end{equation}
One calculates
\begin{equation}
T = {2\pi\over \varpi}\,,
\label{4.13}\end{equation}
\begin{equation}
E = {p_0{}\!^2\over 2m} + {\textstyle\frac12} m\varpi^2 x_0{}\!^2,
\label{4.14}\end{equation}
\begin{equation}
S = ET = {2\pi E\over \varpi}\,,
\label{4.15}\end{equation}
\begin{equation}
R = 0.
\label{4.16}\end{equation}
Thus
\begin{equation}
\frac{dS}{dE} = T = \frac SE
\label{4.17}\end{equation}
in agreement with (\ref{4.1}), but in contrast to (\ref{3.7}).
Ironically, our derivation of (\ref{4.1})
does not apply to this case,
because $T$ is constant instead of parametrizing the orbits.
In view of (\ref{4.13}) and (\ref{4.14}),
(\ref{4.2}) holds only in the formal sense
of $\frac 00$.
\section{Scaling a coupling constant} \label{sec:coup}
Consider a Hamiltonian
\begin{equation}
H(\mathbf x,\mathbf p) =
{\mathbf p^2 \over 2m} + \lambda V(\mathbf x)
\label{5.1}\end{equation}
with a generic potential $V$ and a coupling constant~$\lambda$.
The equations of motion are
\begin{equation}
\frac{d \mathbf x} {dt} = \frac {\mathbf p}m \,,\qquad
\frac{d \mathbf p} {dt} = - \lambda\nabla V(\mathbf x),
\label{5.2}\end{equation}
and any solution curve lies on an energy surface
\begin{equation}
{\mathbf p^2 \over 2m} + \lambda V(\mathbf x) = E.
\label{5.3}\end{equation}
Let $\bigl( \mathbf x_0(t), \mathbf p_0(t) \bigr)$ be a solution of
(\ref{5.2}) and (\ref{5.3}) with $\lambda=\lambda_0$ and $E=E_0\,$.
For any positive real number~$\alpha$,
consider
\begin{equation}
\mathbf x(t) \equiv \mathbf x_0(\alpha t), \qquad
\mathbf p(t) \equiv \alpha \mathbf p_0(\alpha t).
\label{5.4}\end{equation}
A short calculation shows that
$\bigl( \mathbf x(t), \mathbf p(t) \bigr)$ satisfies
(\ref{5.2}) and (\ref{5.3}) with
\begin{equation}
\lambda=\alpha^2 \lambda_0\,, \qquad E=\alpha^2 E_0\,.
\label{5.5}\end{equation}
In particular, the path $\mathbf x(t)$ in configuration space is
independent of the parameter $\alpha^2 = E/E_0\,$.
In this sense the same closed orbit exists for all values of the
energy.
When $V=0$ this is the familiar billiard orbit reviewed in
Sec.~\ref{sec:bill}.
When the potential is not zero, however, one must pay the price of
varying $\lambda$ --- i.e., considering {\em different\/} physical
systems at different energies ---
to gain the convenience of fixed orbits.
For the orbit (\ref{5.4}) it is clear that
\begin{equation}
T = \frac {T_0}{\alpha} \,,
\label{5.6}\end{equation}
and one calculates
\begin{eqnarray}
S = \oint_\gamma \mathbf p \cdot d\mathbf x &=&
\alpha \int_0^T \mathbf p_0 (\alpha t) \cdot
[\alpha \dot\mathbf x_0(\alpha t)] \,dt \nonumber\\
&=& \alpha\int_0^{T_0} \mathbf p_0(\tau) \cdot
\dot\mathbf x_0(\tau) \, d\tau =
\alpha S_0\,. \label{5.7} \end{eqnarray}
Thus (\ref{3.1}) becomes (by use of (\ref{5.5}))
\begin{eqnarray}
\rho_\gamma(E) &=& a_\gamma \sin\left[ {S_0\over \hbar} \sqrt{E\over
E_0} + \eta_\gamma\right] \nonumber\\
&=& a_\gamma \sin\left[ {S_0\over \hbar\omega_0}\,\omega
+ \eta_\gamma\right].
\label{5.8}\end{eqnarray}
That is, just as for a billiard, we have a globally sinusoidal
factor of $\omega\equiv\sqrt{|E|}$
with frequency $S_0/\hbar \omega_0\,$, or period
\begin{equation}
P_\omega = {2\pi \hbar \omega_0 \over S_0}\,.
\label{5.9}\end{equation}
Thus the analogue of the $L_\gamma$ of a billiard is
\begin{equation}
{S_0 \over \sqrt{2m |E_0|}} \,.
\label{5.10}\end{equation}
For the harmonic oscillator orbit (\ref{4.12}),
this characteristic length is
$\pi$ times the maximum value of $x(t)$
(not 4 times, which would be the actual length of the orbit).
It should be noted that (\ref{5.9}) or (\ref{5.10})
depends only on the
orbit~$\gamma$, not on the reference scale arbitrarily chosen to
correspond to $\alpha=1$.
One can also look at the local frequency with respect to~$E$,
following (\ref{3.7})--(\ref{3.9}).
From (\ref{5.7}) and (\ref{5.5}) one has
\begin{equation}
\frac{d S}{dE} = {S \over 2E}\,.
\label{5.11}\end{equation}
But this quantity is no longer equal to~$T$,
because of the variation of the coupling constant with~$E$.
For the harmonic oscillator it equals $\frac12 T$, by (\ref{4.17}).
More generally, since the momentum is parallel to the velocity for
a system of type (\ref{5.1}), one can compute
\begin{equation}
S = \oint_\gamma \mathbf p\cdot \dot\mathbf x \, dt
= 2 \int_0^T [E- \lambda V(\mathbf x(t))]\, dt,
\label{5.12}\end{equation}
and hence
\begin{equation}
{S\over 2E} = T - \frac{\lambda}E \int_0^T V(\mathbf x(t))\, dt.
\label{5.13}\end{equation}
Note also that $S/2E$ scales as $1/\alpha$ and therefore cannot be
regarded as a characteristic time of the orbit~$\gamma$
(in contrast to the length (\ref{5.10})).
Finally, because nearby nonperiodic orbits also obey the scaling
law, the amplitude $a_\gamma$ in (\ref{5.8})
(determined by the monodromy of
the Poincar\'e map) is independent of~$E$, whereas the same cannot
be said for the $a_\gamma$ in (\ref{3.8}).
The phase shift $\eta_\gamma$ also is constant for Dirichlet and
Neumann boundary conditions (not Robin).
\section{Scaling a homogeneous potential} \label{sec:hom}
When the potential in (\ref{5.1}) is homogeneous of degree~$\nu$,
\begin{equation}
V(\beta \mathbf x) = \beta^\nu V(\mathbf x)
\quad(\text{for $\beta>0$}),
\label{6.1}\end{equation}
an alternative scaling procedure exists that avoids changing the
coupling constant.
Assume that the configuration space~$\Omega$,
if not all of~$\mathbf R^d$,
is invariant under dilations
(e.g., a half-space or a cone)
with boundary conditions that are either Dirichlet or pure
Neumann (not Robin).
(If $\Omega$ does not satisfy this condition, then it must be dilated
along with the orbits, and again the system has been replaced by a
one-parameter family.)
In place of (\ref{5.4}) consider
\begin{equation}
\mathbf x(t) \equiv\alpha^2 \mathbf x_0(\alpha^{\nu-2} t), \qquad
\mathbf p(t) \equiv \alpha^\nu \mathbf p_0(\alpha^{\nu-2} t).
\label{6.2}\end{equation}
This is a solution of (\ref{5.2}) and (\ref{5.3}) with
\begin{equation}
\lambda= \lambda_0\,, \qquad E=\alpha^{2\nu} E_0\,.
\label{6.3}\end{equation}
The orbits for different values of $\alpha$ are not the same,
but they are geometrically similar;
one can say that they are fixed if the unit of length is scaled
by~$\alpha^2$.
We now have (parallel to (\ref{5.6})--(\ref{5.8}))
\begin{equation}
\alpha = \left( \frac E{E_0} \right) ^{\frac1{2\nu}},
\label{6.4}\end{equation}
\begin{equation}
T = \alpha^{2-\nu} T_0\,, \qquad S = \alpha^{\nu+2} S_0\,,
\label{6.5}\end{equation}
\begin{equation}
\rho_\gamma(E) = a_\gamma \sin\left[{S_0\over \hbar}
\left( \frac E{E_0}\right) ^{\nu+2\over 2\nu} + \eta_\gamma \right].
\label{6.6}\end{equation}
Thus the variable with respect to which the spectral oscillations
take place is a peculiar power of~$E$, and the analogue of the
characteristic length (\ref{5.10}) is a rather inconvenient
function of the fiducial action and energy.
Note that the exponent of $|E|$ in (\ref{6.6}) equals~$1$ for the
harmonic oscillator;
that it acquires the value $\frac12$ (familiar for billiards)
only as $\nu\to +\infty$;
that it is negative for $-2<\nu<0$; and
that the cases $\nu=0$ and $\nu=-2$ are singular.
The analogue of (\ref{5.11}) is
\begin{equation}
\frac{d S}{dE} = {\nu+2\over 2\nu} \, \frac SE\,,
\label{6.7}\end{equation}
(\ref{5.12}) remains true, and the analogue of (\ref{5.13}) is
\begin{equation}
{\nu+2\over 2\nu}\, \frac SE =
{\nu+2\over\nu} \left[ T - \frac{\lambda}E
\int_0^T V(\mathbf x(t))\,dt\right].
\label{6.8}\end{equation}
On the other hand, the theorem of Sec.~\ref{sec:peri} applies to the orbit
family (\ref{6.2}), and so
\begin{equation}
\frac{d S}{dE} = T.
\label{6.9}\end{equation}
Equating (\ref{6.8}) and (\ref{6.9}) yields the identity
\begin{equation}
\int_0^T \lambda V(\mathbf x(t))\, dt = {2ET \over \nu + 2}
\label{6.10}\end{equation}
{\em for a periodic orbit of a homogeneous potential of
degree~$\nu$}, which is a rewriting of the virial
theorem~\cite{Gold}.
The Coulomb potential,
\begin{equation}
V(\mathbf x) = - \,{e^2\over r} \qquad
(r^2 \equiv x^2 + y^2 + z^2),
\label{6.11}\end{equation}
has
\begin{equation}
\nu = -1, \qquad
{\nu+2 \over 2\nu} = -\,\frac12\,.
\label{6.12}\end{equation}
Furthermore, in the regime of discrete spectrum and periodic
orbits of the atom, $E$~is negative.
Thus (\ref{6.7}) becomes
\begin{equation}
\frac{d S}{dE} = + \, {S\over 2|E|}\,,
\label{6.13}\end{equation}
which displays a superficial coincidence with the corresponding
relation for a billiard, (\ref{3.7}).
\section{Scaling of mixed type} \label{sec:mix}
A mixture of the strategies of Secs.\ \ref{sec:coup}
and~\ref{sec:hom} is appropriate in
some situations.
Suppose that the potential is a sum of two terms, each of which is
homogeneous:
\begin{equation}
V_1(\beta \mathbf x) = \beta^{\nu_1} V_1(\mathbf x), \qquad
V_2(\beta \mathbf x) = \beta^{\nu_2} V_2(\mathbf x).
\label{7.1}\end{equation}
The equations of motion and the energy equation are
\begin{equation}
\frac{d \mathbf x} {dt} = \frac {\mathbf p}m\,,\qquad
\frac{d \mathbf p}{dt} = - \lambda_1 \nabla V_1(\mathbf x)
- \lambda_2 \nabla V_2(\mathbf x),
\label{7.2}\end{equation}
\begin{equation}
{\mathbf p^2 \over 2m} + \lambda_1 V_1(\mathbf x)
+ \lambda_2 V_2(\mathbf x) = E.
\label{7.3}\end{equation}
(This is {\em not\/} a perturbative situation:
Neither potential is assumed to be small compared to the other.)
Following the usual pattern,
suppose that we have a solution with coupling constants
$\lambda_{10}$ and $\lambda_{20}$ and energy $E_0\,$.
Perform the scaling (\ref{6.2}) appropriate to $V_1\,$:
\begin{equation}
\mathbf x(t) \equiv\alpha^2 \mathbf x_0(\alpha^{\nu_1-2} t), \qquad
\mathbf p(t) \equiv \alpha^{\nu_1} \mathbf p_0(\alpha^{\nu_1-2} t).
\label{7.4}\end{equation}
Then (\ref{7.4}) solves (\ref{7.2}) and (\ref{7.3}) with
\begin{equation}
\lambda_1 = \lambda_{10}\,, \qquad
\lambda_2 = \alpha^{2(\nu_1-\nu_2)} \lambda_{20}\,,
\qquad E = \alpha^{2\nu_1} E_0\,.
\label{7.5}\end{equation}
Equations (\ref{6.4})--(\ref{6.6}) continue to apply,
with $\nu=\nu_1\,$.
The point of this transformation is that a fixed orbit can be
associated with a one-parameter family of classical situations, in
which one coupling constant is held fixed but the other varies with
energy.
The prototype is the Hamiltonian studied by Friedrich and
Wintgen~\cite{WF,FW},
\begin{equation}
H(\mathbf x,\mathbf p) = {\mathbf p^2 \over 2m} - {e^2\over r} +
{\textstyle\frac12} m\varpi^2 (x^2+ y^2).
\label{7.6}\end{equation}
It describes, in a rotating coordinate system, a hydrogen atom in
a constant magnetic field
$B = (2mc/e)\varpi$ along the $z$~axis.
This is
``a real physical system that can be and has been studied in the
laboratory''\negthinspace.
The magnetic field is a continuous variable that is under the
experimenter's control;
the charge of the proton is~not!
Therefore, one applies (\ref{7.1}), (\ref{7.4}), (\ref{7.5})
with $\nu_1= -1$, $\nu_2= 2$.
Thus
\begin{equation}
\lambda_2 = \alpha^{-6} \lambda_{20}\,, \qquad
E = \alpha^{-2} E_0\,,
\label{7.7}\end{equation}
and the bottom-line equation (\ref{6.6}) becomes
\begin{equation}
\rho_\gamma(E) = a_\gamma \sin\left[{S_0\over \hbar}
\left( \frac E{E_0}\right) ^{-1/2} + \eta_\gamma \right].
\label{7.8}\end{equation}
In the notation of Friedrich and Wintgen,
\begin{equation}
\Gamma \equiv {\varpi \over \varpi_0} = \alpha^{-3}\,, \qquad
\frac E{E_0} = \Gamma^{2/3},
\label{7.9}\end{equation}
and the contribution $\rho_\gamma$ to the spectral density
is a sinusoidal function of $\Gamma^{-1/3}$.
(We write $\Gamma$ and $\varpi$ for the $\gamma$ and $\omega$
of \cite{FW} to avoid notational collisions.)
For the beautiful fruits of this approach as applied to (\ref{7.6})
we refer to the original papers~\cite{WF,FW}.
From $S= \alpha^{\nu_1+2}S_0$ and $E= \alpha^{2\nu_1} E_0$
we still get
\begin{equation}
\frac{d S}{dE} = {\nu_1+ 2 \over 2\nu_1}\, \frac SE\,,
\label{7.10}\end{equation}
but this is not equal to an orbital period in these incompletely
homogeneous systems.
Mixed scaling could be applied to a potential with arbitrarily many
terms, even an infinite series, but it would be necessary to
rescale all but one of the coupling constants.
The practical advantage over the procedure of Sec.~\ref{sec:coup}
is thereby reduced.
\section{Implications} \label{sec:imp}
A central feature of the relation between classical and quantum
mechanics is that quantization introduces into each problem a new
fundamental scale, set by the quantum of action,~$\hbar$.
That is, one-parameter families of situations that are equivalent
classically become distinct quantum-mechanically.
Taking the semiclassical limit refers to motion along one of these
families in a certain direction, the opposite direction leading to
``deep quantum'' behavior.
Moving from one family to another, on the other hand, corresponds
to various purely classical distinctions, such as integrable versus
chaotic, or different values of angular momentum.
When the potential is homogeneous (Sec.~\ref{sec:hom})
or absent (Sec.~\ref{sec:bill}), a
``situation'' simply means a point
in phase space.
In the more general contexts of Secs.\ \ref{sec:coup}
and~\ref{sec:mix}
the space of situations should be enlarged by one dimension,
representing the coupling constant $\lambda$ or $\lambda_2\,$.
In either case the points are grouped into equivalence classes,
the classical trajectories.
Finally, the
trajectories (including, in particular, the closed orbits)
fall into families, (\ref{5.4}) or (\ref{6.2}),
related by a geometrical
similarity and therefore mathematically equivalent.
The classical trajectories of a family are exactly the same
at all energies, except for a trivial rescaling. But the
quantum states at various energies are quite different.
In the study of spectra the most convenient choice of parameter
along each family is the energy.
What determines whether an energy is large or small?
To compare it with $\hbar$ one must construct a quantity with
dimensions of action.
The dimensions of $\sqrt{2m|E|}$ are action divided by length,
so a suitable measure is the product of $\sqrt{2m|E|}$
with some length characteristic of the entire system.
The statement ``$\hbar$ is small'' is meaningless unless translated
into such a criterion.
In the end one can always choose units in which $\hbar=1$
and $2m=1$,
and then the only independent physical dimension is length.
%^& footnote?
(Once the quantum of action and the characteristic length of the
system are fixed, rescaling $m$ amounts to changing the unit of
time.
Since $m$ has been totally inert throughout all our
considerations, it could have been eliminated
at the very beginning, but so that the classical-mechanical
equations would look familiar, we did not do so.)
The criterion now is whether the length
$|E|^{-1/2}\equiv \omega^{-1}$
is large or small relative to the length scale of the system.
This restatement makes sense in classical wave theories, such as
optics or acoustics, as originally studied by Balian and
Bloch~\cite{BB1,BB3}.
The length scale can be set by the geometry of the boundary,
if there is one.
Otherwise it must be a characteristic of the potential.
At first sight this is a simple matter:
Quantities such as $\nabla^2V/V$ (evaluated at a minimum of the
potential, say)
are independent of the coupling
constant and characterize the spatial scale of the potential, as
distinct from its strength.
However, if $V$ is homogeneous, (\ref{6.1}) shows that the
distinction between the spatial scale and the coupling constant is
a mirage;
and Sec.~\ref{sec:mix} suggests that this ambiguity can be
imported into more general potentials as well.
For a homogeneous potential there is an alternative way to set the
scale of the system, exploiting (\ref{6.1}) to ``transmute'' the
coupling constant into a length, $x_0 \equiv \beta^{-1}$.
For example, if $d=1$ and the potential energy is $\lambda x^\nu$,
then $\lambda$ has dimensions $\text{[energy][length]}^{-\nu}$
and can be written
\begin{equation}
\lambda \equiv {\hbar^2\over 2m} \, x_0{}\!^{-(\nu+2)},
\label{8.1}\end{equation}
so that
\begin{equation}
\lambda x^\nu = {\hbar^2\over 2m x_0{}\!^2}
\left ({x\over x_0}\right) ^\nu.
\label{8.2}\end{equation}
It is instructive to look at everyone's favorite homogeneous
potentials, the harmonic oscillator (\ref{4.11})
and the hydrogen atom (\ref{6.11}),
in the light of the foregoing remarks.
If we treat these systems according to Sec.~\ref{sec:coup},
we are taking a fixed ``classical situation'' to be a fixed ratio
$E/\lambda$, with $\lambda\propto \varpi^2$ or $e^2$ respectively.
We expect to encounter semiclassical behavior as $|E| \to \infty$
along one of the diagonal lines in Figs.\
\ref{scalharm} and~\ref{scalcoul},
\begin{figure}%[bp]
$$\beginpicture
\setcoordinatesystem units <.3truein, .3truein>
\setplotarea x from 0 to 9, y from 0 to 9
\putrule from 0 0 to 0 9
\putrule from 0 0 to 9 0
\put{$E$} [lt] <1pt,0pt> at 0 9
\put{$\lambda$} [rt] <0pt,-1pt> at 9 0
\put{$\vdots$} [b] at 3.5 7
\put{$n=0$} [lb] <2pt,1pt> at 6.5 1.4
\put{$n=1$} [lb] <2pt,1pt> at 6.5 3.5
\put{$n=2$} [lb] <2pt,0pt> at 7 6.5
\put{$n=3$} [lt] <2pt,0pt> at 6.5 9
\setquadratic
\vshade
0 0 0
.25 0 .25
1 0 .5
4 0 1
9 0 1.5 /
\plot
0 0
.0625 .125
.25 .25
.49 .35
.74 .43
1.44 .6
2.25 .75
5.76 1.2
9 1.5 /
\plot
0 0
.0625 .375
.25 .75
.49 1.05
.74 1.29
1.44 1.8
2.25 2.25
5.76 3.6
9 4.5 /
\plot
0 0
.0625 .625
.25 1.25
.49 1.75
.74 2.15
1.44 3
2.25 3.75
5.76 6
9 7.5 /
\plot
0 0
.0625 .875
.25 1.75
.49 2.45
.74 3.01
1.44 4.2
2.25 5.25
4 7
6.63 9 /
\setlinear
\setdashes
\plot
0 0
9 9 /
\plot
0 0
9 6 /
\plot
0 0
6 9 /
\linethickness=1pt
\setdots
\putrule from 2.5 0 to 2.5 9
\putrule from 4.5 0 to 4.5 9
\putrule from 6.5 0 to 6.5 9
\endpicture$$
\caption{Solid curves:
Energy levels of the harmonic oscillator,
$E_n = \bigl(n+\frac12\bigr)\hbar\varpi$,
as functions of coupling
constant, $\lambda\propto \varpi^2$.
No quantum states exist in the shaded area.
Dashed lines: Loci of typical classical orbits fixed in space
(Sec.~\ref{sec:coup}).
Dotted lines: Loci of typical homogeneously scaled orbits
(Sec.~\ref{sec:hom}).
The direction of increasing $E$ along either set of lines leads
into the semiclassical region of large quantum numbers.}
\label{scalharm}
\end{figure}
\begin{figure}%[bp]
$$\beginpicture
\setcoordinatesystem units <.3truein, .3truein>
\setplotarea x from 0 to 9, y from -9 to 0
\putrule from 0 0 to 0 -9
\putrule from 0 0 to 9 0
\put{$E$} [rb] <-1pt,0pt> at 0 -9
\put{$\lambda$} [rt] <0pt,-1pt> at 9 0
\put{$\vdots$} [b] at 5.5 -1.5
\put{$n=4$} [lb] <2pt,0pt> at 6.5 -2.6
\put{$n=3$} [lb] <2pt,1pt> at 7 -5.5
\put{$n=2$} [lb] <3pt,1pt> at 5 -7
\put{$n=1$} [lb] <2pt,1pt> at 3 -9
\setquadratic
\hshade
-9 0 3
-4 0 2
-1 0 1
-.64 0 .8
-.25 0 .5
-.0625 0 .25
0 0 0 /
\plot
0 0
.25 -.0625
.5 -.25
.7 -.49
.8 -.64
.9 -.81
1 -1
1.414 -2
1.732 -3
2 -4
2.236 -5
2.449 -6
2.646 -7
2.828 -8
3 -9
/
\plot
0 0
.5 -.0625
1 -.25
1.4 -.49
1.6 -.64
1.8 -.81
2 -1
2.828 -2
3.464 -3
4 -4
4.472 -5
4.899 -6
5.292 -7
5.657 -8
6 -9
/
\plot
0 0
.75 -.0625
1.5 -.25
2.1 -.49
2.4 -.64
2.7 -.81
3 -1
4.243 -2
5.196 -3
6 -4
6.708 -5
7.348 -6
7.937 -7
8.485 -8
9 -9
/
\plot
0 0
1 -.0625
2 -.25
2.8 -.49
3.2 -.64
3.6 -.81
4 -1
5.657 -2
6.928 -3
8 -4
8.944 -5
/
\setlinear
\setdashes
\plot
0 0
9 -9 /
\plot
0 0
9 -6 /
\plot
0 0
6 -9 /
\linethickness=1pt
\setdots
\putrule from 2.5 0 to 2.5 -9
\putrule from 4.5 0 to 4.5 -9
\putrule from 6.5 0 to 6.5 -9
\endpicture$$
\caption{Solid curves:
Energy levels of the hydrogen atom,
$E_n \propto - e^4/n^2$, as functions of coupling
constant, $\lambda\propto e^2$.
No quantum states exist in the shaded area.
Dashed lines: Loci of typical classical orbits fixed in space
(Sec.~\ref{sec:coup}).
Dotted lines: Loci of typical homogeneously scaled orbits
(Sec.~\ref{sec:hom}).
The semiclassical region of large quantum numbers
is reached by moving
along the dashed lines in the direction of
increasing $|E|$ ($E\to -\infty$),
or
along the dotted lines in the direction of
decreasing $|E|$ ($E\to 0$).}
\label{scalcoul}
\end{figure}
and indeed this leads into the region of large quantum numbers
in each case.
There is no unique way to associate a natural length with either of
these potentials, but lengths independent of~$\lambda$ can be
built out of derivatives of~$V$ at an arbitrarily chosen point.
If we treat the systems according to Sec.~\ref{sec:hom},
we keep $\lambda$ constant and therefore move along one of the
vertical lines in the figures.
From
\begin{equation}
x_0 = \lambda^{-1/(\nu+2)} \qquad (\hbar^2= 2m),
\label{8.3}\end{equation}
we have
\begin{equation}
x_0{}\!^2 = \frac1\varpi\,,
\qquad E_nx_0{}\!^2 = n+\frac12
\label{8.4}\end{equation}
for the energy levels of the oscillator, and
\begin{equation}
x_0{}\!^2 =e^{-4}, \qquad
E_nx_0{}\!^2 \propto - \,\frac1{n^2}
\label{8.5}\end{equation}
for those of the atom.
The semiclassical regime of large quantum number~$n$ is reached by
going vertically upward in either of the figures.
For the oscillator this is again the limit of large~$E$, albeit in
a different direction in the $E$--$\lambda$ plane.
But for the Coulomb potential this limit corresponds to $E \to 0$,
not $|E|\to \infty$.
(In view of (\ref{6.13}) this is still the direction of
increasing~$S$.)
This result perhaps should not be surprising in view of the
negative exponent in (\ref{6.6}), (\ref{6.13}).
But it indicates that the identification of ``semiclassical'' with
``large energy'' can be ambiguous, even within the theory of the
same physical system.
\goodbreak
Let us summarize the main points:
\begin{enumerate}
\item
The concept of ``spectral oscillations'' (associated
with the basic formula (\ref{3.1}), which we took as given)
has meaning only if the orbit structure is at least locally
independent of energy.
\item {\em The orbits can be ``nailed down'' by scaling the
coupling constant along with~$E$.}
\item For a homogeneous or partially homogeneous potential
there are other options for fixing the orbit structure, involving
dilations.
\item The wave frequency $\omega = E^{1/2}$ is a better variable
than $E$ for describing the oscillations globally.
(In the homogeneous case the exponent $\frac12$ is replaced by
$(\nu+1)/ 2\nu$.)
Correspondingly, the best descriptor of the oscillations
caused by a particular orbit
is a characteristic length (\ref{5.10}) of the orbit,
not the orbital period.
\item Even the local frequency of spectral oscillation with
respect to energy,
$dS/dE$, is not equal to the orbital period when the coupling
constant is varied.
(In passing we supplied a proof of the equality when $\lambda$ is
constant.)
\end{enumerate}
\section{A final comment} \label{sec:reso}
When periodic-orbit theory was first applied numerically to the
spectra of concrete systems, early authors expressed surprise that
it gave accurate results ``even'' for the lowest eigenvalues.
As a semiclassical method, the technique had been expected to be
applicable primarily in the regime of ``large quantum numbers'' and
hence high energy.
Later it became clear that the method was generally practical
{\em only\/} in the low-energy regime, because the number of
periodic orbits increases roughly exponentially with period,
and as energy increases it becomes necessary to consider
increasingly long orbits in order to resolve individual
eigenvalues~\cite{Berry,Bolte}.
Long after this fact has been accepted, it is still often regarded
as ``paradoxical''\negthinspace.
We suggest that, in hindsight, this phenomenon is merely an
instance of the familiar principle that when a calculation is very
stable and involves some kind of smoothing or averaging, the
inverse calculation is likely to be unstable (highly dependent on
the details of the input) and hence difficult.
There is a loose analogy with the solution of an initial-value
problem for the heat equation.
For large positive time the problem is very easy to solve and very
insensitive to the details of the initial data;
consequently, reconstructing the data from the final solution is
hopeless.
For very small time an approximate solution to the backwards
problem, adequate for some practical purpose, may be feasible;
of course, the forward problem is less trivial in that case.
In spectral asymptotics we are interested in deducing the spectrum
of a differential operator from its geometry,
or vice versa.
Here ``geometry'' is meant in an extended sense, including not
only the literal geometry of the region $\Omega$ where the wave
functions are defined, but also the potential function
(or other coefficient functions in the operator~$H$).
The classical periodic orbits of the Hamiltonian
$H(\mathbf x,\mathbf p)$
are also aspects of the geometry of the system.
In the ``old'' spectral asymptotics, associated with the names
Weyl, Thomas--Fermi, Minakshisundaram, Schwinger--DeWitt, etc.,
one associated the (high-$E$) asymptotic behavior of the density of
states with the global geometry of the operator: the volume of
$\Omega$, the integrated curvature of its boundary, the integral of
the Ricci curvature of $\Omega$ if it is a manifold,
the integral of the potential $V$ over $\Omega$, and so on.
Via the asymptotics of the heat kernel, the passage from the
spectrum to the geometry is rigorously asymptotic.
In the inverse direction, however, the geometry does not determine
a genuine asymptotic expansion of the eigenvalue density in powers
of~$E^{-1}$, precisely because of the presence of the oscillatory
terms that are the subject of the periodic-orbit theory.
The formal expansion of the eigenvalue density becomes literally
asymptotic only when some kind of averaging is performed, such as
Lorentzian smoothing~\cite{BB1} or Riesz means~\cite{Hor}.
A complementary situation has always existed at the low end of the
spectrum.
If we know $\Omega$ and $H$ exactly, then it is relatively easy to
construct the lowest-lying eigenvalues and eigenfunctions, by
variational methods, numerical methods, etc.
Larger eigenvalues are harder.
Furthermore, merely from a knowledge of the lowest energies one
would not expect to be able to reconstruct $\Omega$ or~$H$.
(Complete knowledge of a single eigen{\em function\/} is a
different matter, however.)
The ``new'' spectral asymptotics of periodic orbits extends this
picture.
Decreasing the width over which the
eigenvalue distribution is averaged,
one supplements the power-law asymptotics
with the longest-wavelength oscillatory components (\ref{3.1}),
characterized by the lengths or periods of the shortest classical
orbits. This is still an asymptotic (high-energy) matter.
It is best thought of as a prediction of classical behavior
(well-defined orbits for wave packets) in the high-energy,
large-action regime.
It still involves an averaging of the spectrum, albeit on a
smaller scale.
The stable direction of prediction is {\em from\/} the spectrum
{\em to\/} the classical, geometrical description.
The counterpart of this at low energy is that from the classical
orbits
(and their associated amplitudes and phases, cf.\ (\ref{3.1}))
one can predict the formation of discrete eigenvalues
(resonant frequencies), with
greatest precision near the bottom of the spectrum.
In some sense this involves an averaging over geometrical
information.
More precise geometrical information in principle allows precise
construction of longer-period orbits and hence more complete
reconstruction of the spectrum (both improved resolution
and extension to higher energies).
The stable direction of prediction is {\em from\/} the
geometry {\em to\/} the spectrum.
It is misleading to think of the regime of low energy
(or small quantum numbers)
as the ``deep quantum regime''\negthinspace,
as if the duality between the spectrum of quantum
eigenvalues and the spectrum of classical
periodic orbits is irrelevant there.
A better phrase is ``resonant regime'':
There, resonant behavior emerges on the spectral side, out of the
geometry and classical mechanics,
much as, in the opposite ``classical regime''\negthinspace,
classical-mechanical behavior emerges on the geometrical side, out
of the quantum substrate.
\begin{acknowledgments}
The calculation in Sec.~\ref{sec:peri}
developed in conversations with David Garfinkle
at the Parkerfest (Oakland University, 2000).
\end{acknowledgments}
\goodbreak
\begin{thebibliography}{}
\frenchspacing
\bibitem{DD} J. B. Delos and M.-L. Du,
{\em IEEE J. Quantum Electron. \bf 24}, 1445 (1988).
\bibitem{CRR} M. Combescure, J. Ralston, and D. Robert,
{\em Commun. Math. Phys. \bf 202}, 463 (1999).
\bibitem{FW} H. Friedrich and D. Wintgen,
{\em Phys. Reports\/ \bf 183}, 37 (1989).
\bibitem{Berry} M. Berry, in {\em Chaos et Physique Quantique /
Chaos and Quantum Physics}, Les Houches Session LII, 1989,
ed. by M.-J. Giannoni, A. Voros, and J. Zinn-Justin,
Elsevier, 1991, pp. 251--303.
\bibitem{Bolte} J. Bolte, {\em Semiclassical Trace Formulae and
Eigenvalue Statistics in Quantum Chaos}, Lectures at the Third
International Summer School/Conference at the University of
Maribor, Slovenia, 1996 ({\tt chao-dyn/9702003}).
\bibitem{Rob} D. Robert, {\em Helv. Phys. Acta \bf 71}, 44 (1998).
\bibitem{BB3} R. Balian and C. Bloch,
{\em Ann. Phys. (NY)\/ \bf 69}, 76 (1972).
\bibitem{Gutz3} M. C. Gutzwiller,
{\em J. Math. Phys. \bf11}, 1791 (1970).
\bibitem{Gutz4} M. C. Gutzwiller,
{\em J. Math. Phys. \bf 12}, 343 (1971).
\bibitem{BB5} R. Balian and C. Bloch,
{\em Ann. Phys. (NY)\/ \bf 85}, 514 (1974).
\bibitem{DG} J. J. Duistermaat and V. W. Guillemin,
{\em Invent. Math. \bf 29}, 39 (1975).
\bibitem{Chaz} J. Chazarain, {\em Invent. Math. \bf 24}, 65 (1974).
\bibitem{ColV} Y. Colin de Verdi\'ere,
{\em Comp. Math. \bf 27}, 159 (1973).
\bibitem{Cots} M. Cotsaftis,
{\em Compt. Rend. Acad. Sci. Paris\/ \bf 274}, A911 (1973).
\bibitem{WF} D. Wintgen and H. Friedrich,
{\em Phys. Rev. A \bf36}, 131 (1987).
\bibitem{SV} Yu. Safarov and D. Vassiliev,
{\em The Asymptotic Distribution of Eigenvalues of Partial
Differential Operators}, Transl. Math. Monog. 155,
American Mathematical Society, 1997.
\bibitem{Sch} B. Schroer, {\em Phys. Rev. D \bf3}, 1764 (1971).
\bibitem{Full} S. A. Fulling, in
{\em Relativity, Particle Physics and Cosmology},
Proceedings of the Richard Arnowitt Fest,
ed. by R. E. Allen, World Scientific, 1999, pp. 329--342.
\bibitem{Gold} H. Goldstein, {\em Classical Mechanics},
Addison--Wesley, 1950, Sec. 3.4.
\bibitem{BB1} R. Balian and C. Bloch,
{\em Ann. Phys. (NY)\/ \bf60}, 401 (1970).
\bibitem{Hor} L. H\"ormander, {\em Acta Math. \bf 121}, 193 (1968).
\end{thebibliography}
\end{document}
---------------0012151052433--