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Paper presented at the AMS-IMS-SIAM Joint Summer Research Conference
on Quantum Graphs and Their Applications, Snowbird, June, 2005
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quantum graph, vacuum, spectral density, Robin
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% Fulling, Proceedings of JSRC on Quantum Graphs, 2005
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\begin{document}
\title[Local Densities near a Vertex]
{Local Spectral Density and Vacuum Energy \\
Near a Quantum Graph Vertex}
% Information for first author
\author[S. A. Fulling]{Stephen A. Fulling}
% Address of record for the research reported here
\address{Departments of Mathematics and Physics, Texas A\&M University,
College Station, TX, 77843-3368}
% Current address
%\curraddr{Department of Mathematics and Statistics,
%Case Western Reserve University, Cleveland, Ohio 43403}
\email{fulling@math.tamu.edu}
% \thanks will become a 1st page footnote.
%\thanks{The first author was supported in part by NSF Grant \#000000.}
% General info
\subjclass[2000]{Primary 81Q10; Secondary 34B45, 81V10}
\date{\today} %%%%%%%%%%%
%\date{July 28, 2005 and, in revised form, ?? ??, ????.}
%\dedicatory{This paper is dedicated to our advisors.}
\keywords{quantum graph, vacuum, spectral density, Robin}
\begin{abstract}
The delta interaction at a vertex generalizes the Robin
(generalized Neumann) boundary condition on an interval. Study
of a single vertex with $N$ infinite leads suffices to determine
the localized effects of such a vertex on densities of states,
etc. For all the standard initial-value problems, such as that
for the wave equation, the pertinent integral kernel (Green
function) on the graph can be easily constructed from the
corresponding elementary Green function on the real line. From the
results one obtains the spectral-projection kernel, local spectral
density, and local energy density. The energy density, which
refers to an interpretation of the graph as the domain of a
quantized scalar field, is a coefficient in the asymptotic
expansion of the Green function for an elliptic problem involving
the graph Hamiltonian; that expansion contains spectral/geometrical
information beyond that in the much-studied heat-kernel expansion.
\end{abstract}
\maketitle
\section*{Introduction}
A topic of perennial and renewed interest in quantum field theory
is the energy of the ``vacuum'' --- that is, of the ground state of
a field subjected to some nontrivial external condition
\cite{Casimir,BmC,Boyer,DC,Lamo,Milton,BMM}.
(The prototype is the electromagnetic field between two parallel
flat conductors.)
Although the only quantities indisputably open to experiment are
the derivatives of the total energy with respect to parameters
defining the configuration,
the energy itself and even its localization in space are of
theoretical interest, not least because energy \emph{density}
(along with associated quantities such as pressure) acts as the
source of the gravitational field in general relativity
\cite{Ford,DC}.
From a mathematical point of view, vacuum energy
is a probe of the spectral properties of a self-adjoint
differential operator, say $H$;
it contains ``nonlocal'' information not extractible from the
much-studied small-time expansion of the heat kernel
\cite{Kirsten,Gilkey}.
It reflects the oscillatory fine structure of the eigenvalue
distribution and, therefore, is directly related to the spectrum of
periodic orbits \cite{Gutz,BB3,Colin,Chazarain,DG,CPS,CRR,Uribe}
of the nonrelativistic
classical-mechanical
(or ray-optical) system associated with our operator $H$ as
quantum Hamiltonian (or wave operator).
The interplay among spectral theory, dynamics, and vacuum energy
is fascinating and rapidly developing, with each subject gaining
benefits from the others \cite{BmC,BD,SS,MSSvS,norman,JS}.
Inasmuch as quantum graphs provide instructive models of spectral
theory and semiclassical dynamics,
they should also be communicating with vacuum energy.
Here we show that some elementary techniques recently applied
\cite{BF} to certain traditional boundary-value problems of the
Robin type actually apply also to quantum graphs.
Indeed, they may be more valuable there, because of quantum graph
theory's supply of nontrivial problems susceptible to essentially
one-dimensional methods.
\section*{Infinite star graphs}
Here we will consider only the simplest type of quantum graph, one
consisting of a single vertex with $N$ infinite edges attached.
(Since many of the issues we will study are basically local, there
is a sense in which these graphs are the building blocks for all
others.)
The Hilbert space of the model thus consists of vector-valued
functions,
$u= \{u_j(x)\} \in L^2(0,\infty)^N$,
where $x$ as the argument of $u_j$ is the distance of the point in
question from the vertex along edge~$j$.
\[
\begin{picture}(150,100)(-50,-50)
\put(0,0){\circle*{5}}
\put(0,0){\line(1,0){100}}
\put(0,0){\line(2,1){100}}
\put(0,0){\line(1,-1){50}}
\put(0,0){\line(-2,3){30}}
\put(0,0){\line(-1,-1){50}}
\end{picture}
\]
The self-adjoint operator is
$H = -\,\frac{d^2}{dx^2}$ with certain boundary conditions.
We impose the usual continuity conditions,
\begin{equation} \label{continuity}
u_j(0) = u(0), \quad\forall j= 1,\ldots, N .
\end{equation}
The remaining condition will be one of these:
\begin{itemize}
\item the Dirichlet condition, $u(0)=0\,$;
\item the Kirchhoff, or generalized Neumann, condition,
\begin{equation} \label{kirchhoff}
\sum_{j=1}^N u_j'(0)= 0\,;
\end{equation}
\item our main concern, the Exner--\v{S}eba
or generalized Robin condition
\begin{equation} \label{exnerseba}
\sum_{j=1}^N u_j'(0)= \alpha u(0), \quad\alpha>0.
\end{equation}
\end{itemize}
% (\noindent deliberate)
Condition (\ref{exnerseba}) was apparently introduced in
\cite{ES}.
It is often \cite{Exner,Kuchment} called the $\delta$ condition
because it can be regarded as the effect of attaching a Dirac delta
potential at the vertex.
At a vertex with only one edge it reduces to the so-called Robin
(or convective cooling) condition, $u'(0)=\alpha u(0)$.
In passing we remark that the label ``Robin'' has almost no
historical justification \cite{GA}, but it is preferable to
``mixed'' because ``mixed boundary conditions'' has acquired other
meanings \cite{Kirsten,Gilkey}.
\smallskip
\textsc{Remark.} The case $\alpha <0$ can also be handled, but the
construction of the Bondurant transform (\ref{bondurant})
is then so different as to require a separate discussion.
With our sign convention, $\alpha\ge0$ is the more ``physical''
case, where heat flows from the hotter to the cooler and $H$ has no
negative eigenvalues.
\section*{The Bondurant transform}
In \cite{BF} we showed how to obtain solutions of the simplest
problems with Robin boundary conditions from solutions of the
corresponding Dirichlet problems.
The term ``Dirichlet-to-Robin transform'' has sometimes been
misunderstood as referring to an analogue of the
Dirichlet-to-Neumann map, hence the justification for immediately
naming the construction after my junior collaborator.
Here one is \emph{not} studying the relation between
nonhomogeneous Dirichlet data and nonhomogeneous Neumann data for a
fixed solution;
instead, one is constructing a \emph{new} solution to a different
problem, with homogeneous Robin (or Exner--\v{S}eba) data replacing
Dirichlet data.
The generalization of the key formula of \cite{BF} to an infinite
star graph with boundary conditions (\ref{continuity}) and
(\ref{exnerseba}) is
\begin{align} \label{bondurant}
u_j(x) &= (T^{-1}v)_j(x) \\
& \equiv
\frac1{\alpha}\left[ \frac1N \sum_{k=1}^N v_k(x) -v_j(x)\right]
-\frac1{N^2} \int_x^\infty e^{-\alpha(s-x)/N} \sum_{k=1}^N
v_k(s)\,ds. \nonumber
\end{align}
\medskip
\textsc{Theorem.} {\em
If $v(t,x)$ solves a Dirichlet problem for a constant-coefficient
partial differential equation [cf.\ (\ref{waveeq})--(\ref{cyleq})],
then $u(t,x)$ solves the corresponding
Exner-\v{S}eba problem (though with different initial data).
}\medskip
\textsc {Sketch of verification.}
(\ref{bondurant}) is obtained by solving the ordinary
differential equation $v=Tu$, where
\begin{equation}\label{bdop}
(Tu)_j (x) \equiv \sum_{k=1}^N u_k'(x) - \alpha u_j(x),
\end{equation}
with the condition of decay as $x\to\infty$.
The heuristics of finding the solution are less instructive than
the verification that it satisfies all the required conditions.
Since $v(0)=0$, one observes that
\begin{itemize}
\item $u_j(0)$ is independent of $j$ (condition
(\ref{continuity}));
\item $Tu(x)=v(x)$, so
$\displaystyle\sum_{j=1}^N u_j'(0)=\alpha
u(0)$ (condition (\ref{exnerseba}));
\item $T$ and $T^{-1}$ commute with the partial differential
operator, so $u$ is still a solution.
\hfill$\square$\hfilneg
\end{itemize}
\section*{Integral kernels}
Now consider any of the following initial-value problems.
\begin{alignat}{2}
\label{waveeq}&\mbox{Wave:}
&\quad &u_{tt} =u_{xx}\,, \quad u(0,x) = f(x), \ u_t(0,x) = 0 \\
\label{heateq}&\mbox{Heat:}
&\quad &u_t =u_{xx}\,, \quad u(0,x) = f(x) \\
\label{quanteq}&\mbox{Quantum:}
&\quad &iu_t =-u_{xx} =Hu, \quad u(0,x) = f(x) \\
\label{cyleq}&\mbox{Cylinder:}
&\quad &u_{tt} = Hu, \quad u(0,x) = f(x), \ u(+\infty,0)= 0
\end{alignat}
% (\noindent deliberate)
(Of course, there are many other problems involving the operator
$H$ on the graph that could be considered, but these form a natural
and highly useful quartet.)
We seek the integral kernel (Green function) that solves such a
problem via
\begin{equation}\label{kernel}
u_j(t,x) = \sum_{l=1}^N\int_0^\infty
G_{Sj}{}\!^l (t,x,y) f_l(y)\,dy.
\end{equation}
\medskip
\textsc{Corollary.} {\em Let $G_S(t,x,y) $ be the (matrix) Green
function for one of the initial-value problems
(\ref{waveeq})--(\ref{cyleq}) on the graph.
Let $G(t,\abs{x-y})$ be the corresponding (scalar) Green function on
the real line (also known as the ``free'' kernel).
Then
\begin{multline}\label{green}
G_{Sj}{}\!^l (t,x,y) = \delta_j{}\!^l G(t,\abs{x-y}) \\
+ \left(\frac2N -\delta_j{}\!^l\right) G(t,x+y)
-\frac{2\alpha}{N^2}\int_x^\infty e^{-\alpha(s-x)/N}
G(t,s+y)\,ds.
\end{multline}
}\medskip
\textsc{Sketch of derivation.}
The Dirichlet Green function consists of an incident term minus
an image term,
\begin{align*}
G_{Dj}{}\!^l(t,x,y) &= \delta_j{}\!^l [G(t,\abs{x-y}) -
G(t,x+y)] \\
&\equiv G_- -G_+\,.
\end{align*}
In operator language, we want $G_S = T^{-1} G_D T$.
(The final $T$ is needed to get the correct initial value,
$\delta_j{}\!^l \delta(x-y)$.)
In kernel language, therefore, we need
\[G_S(t,x,y) = T_x^{-1} T_y^\dagger G_D(t,x,y),\]
where $\dagger$ indicates the transpose (real adjoint).
But
\begin{align}\label{transposeD}
(T^\dagger u)_j &= -\sum_k u_k' -\alpha u_j \\
\label{transposeT} &= -(Tu)_j -2\alpha u_j \,.
\end{align}
From (\ref{transposeD}) and
$ -\partial_y G(t,\abs{x-y}) = + \partial_x G(t,\abs{x-y})$
one sees that the incident term passes through the similarity
transformation unchanged: $T^{-1}G_-T=G_-\,$.
From (\ref{transposeT}) and
$-\partial_y G(t,x+y) = - \partial_x G(t,x+y)$
one has $T_y G_+ = T_x G_+$ and hence
\begin{equation}\label{greenstep}
G_{Sj}{}\!^l(t,x,y) =\delta_j{}\!^l [G(t,\abs{x-y}) +G(t,x+y)]
+2\alpha T_{x,j}^{-1}[\delta_j{}\!^l G(t,x+y)] .
\end{equation}
(Note that the first two terms of (\ref{greenstep}) solve the true
Neumann (not Kirchhoff) problem, $u_j'(0)=0\ \forall j$.)
Working out the last term of (\ref{greenstep}) according to
(\ref{bondurant}), one obtains (\ref{green}).
\hfill$\square$\hfilneg\medskip
The factor $\left(\frac2N -\delta_j{}\!^l\right)$ in
(\ref{green})
will come as no
surprise to those who are familiar with the study of quantum graphs
by other methods (see \cite{KS}).
\smallskip
\textsc{Remark.} The kernel formula (\ref{green}) is correct for
$\alpha=0$ (the Kirchhoff condition),
although the intermediate steps are meaningless
(in particular, $T^{-1}$ doesn't exist in that case).
\smallskip
\textsc{Main results.}
As corollaries of the corollary, we derive formulas
(\ref{waveker})--(\ref{wavesol}), (\ref{specprojker}), (\ref{specdens}),
(\ref{intspec}), (\ref{cylkernel}), (\ref{graphendens})
for particular kernels and associated quantities.
\section*{The wave kernel}
In a one-dimensional system the simplest member of the quartet is
the wave problem (\ref{waveeq}), for which the free Green function
is (d'Alembert's solution)
\begin{equation}\label{dalembert}
G(t,z) = {\textstyle\frac12} [\delta(z-t) +\delta(z+t)].
\end{equation}
Applying (\ref{green}) and omitting terms that vanish for $t>0$
one gets
\begin{multline}\label{waveker}
G_{Sj}{}\!^l(t,x,y) =
{\textstyle\frac12}\delta_j{}\!^l [\delta(x-y-t) +\delta(x-y+t)]
\\
{} +\frac12 \left(\frac2N -\delta_j{}\!^l\right)\delta(x+y-t) -
\frac{\alpha}{N^2}\, e^{-\alpha(t-y-x)/N} \theta(t-y-x) ,
\end{multline}
where $\theta$ is the unit step function.
The meaning of (\ref{waveker}) becomes clearer when one applies
(\ref{kernel}) to get
\begin{multline}\label{wavesol}
u_j(t,x) = {\textstyle\frac12}[f_j(x-t)+f_j(x+t)]
-{\textstyle\frac12} f_j(t-x) \\
{}+ \frac1N \sum_{l=1}^N f_l(t-x)
-\frac{\alpha}{N^2} \theta(t-x) \int_0^{t-x}\!
e^{-\alpha\epsilon/N}
\sum_{l=1}^N f_l(t-x-\epsilon)\,d\epsilon.
\end{multline}
Here we see clearly the incident wave, the immediately reflected
and transmitted waves from a Kirchhoff vertex,
and some $\alpha$-dependent delayed transmission.
The Robin case ($N=1$) is \cite{BF}
\begin{equation}\label{robsol}
u(t,x) =
{\textstyle\frac12}[f(x-t)+f(x+t) + f(t-x)]
-\alpha \theta(t-x) \int_0^{t-x} e^{-\alpha\epsilon}
f(t-x-\epsilon)\,d\epsilon.
\end{equation}
What is the physical meaning of this time delay?
In the context of the wave equation, the Robin or Exner--\v{S}eba
boundary models an ideal spring, or elastic support, to which the
vibrating medium is attached. The spring absorbs energy from the
medium and leaks it back out.
It is a jolly exercise in integration by parts to show that for the
solution (\ref{robsol}) the total energy
\begin{equation}\label{robenergy0}
E= \frac12\int_0^\infty
\left[\left(\frac{\partial u}{\partial t}\right)^2 +
\left(\frac{\partial u}{\partial x}\right)^2\right] dx
+\frac{\alpha}2 u(t,0)^2
\end{equation}
is indeed conserved;
the field and boundary terms individually are not (unless
$\alpha=0$).
For later use we note that one integration by parts in
(\ref{robenergy0}) and use of $u'(t,0)=\alpha u(t,0)$ lead to an
alternative formula for the total energy,
\begin{equation}\label{robenergy1/4}
E= \frac12\int_0^\infty
\left[\left(\frac{\partial u}{\partial t}\right)^2 -
u\left(\frac{\partial^2 u}{\partial x^2}\right)\right] dx,
\end{equation}
in which the boundary term has been formally absorbed into the
field term.
\section*{The spectral projection kernel}
Let $P(\lambda)$ be the orthogonal projection operator onto the
interval $[0,\lambda]$ of the spectral resolution of~$H$.
Because $H$ on an infinite star graph has purely absolutely
continuous (and nonnegative) spectrum,
the integral kernel of $P(\lambda)$ may be written
\[
P(\lambda,x,y) = \int_{0}^{\sqrt{\lambda}}
\sigma(\omega,x,y)\,d\omega,
\]
where $\sigma$ is a well-defined matrix-valued
function (not just a distribution in $\omega$).
Alternatively, $\sigma$ can be defined as the
inverse Fourier cosine transform of the wave kernel.
(It is also the
inverse Laplace transform of the heat kernel, and so on for all
the standard ``spectral functions'' \cite{Kirsten}.)
That is, we write
\begin{equation}\label{fct}
G_{Sj}{}\!^l (t,x,y) =\int_0^\infty \cos (\omega t)
\sigma_{Sj}{}\!^l (\omega,x,y)\, d\omega
\end{equation}
and calculate
\begin{align}\label{specprojker}
\sigma_{Sj}{}\!^l (\omega,x,y) &= \frac2{\pi} \int_0^\infty \cos (\omega t)
G_{Sj}{}\!^l (t,x,y)\,dt \\ \begin{split}
&=\frac2{\pi} \, \delta_j{}\!^l \sin(\omega x) \sin(\omega y) \\
&\quad{} +\frac{2/\pi}{\alpha^2+N^2\omega^2} \{N\omega^2 \cos[\omega(x+y)]
+\alpha\omega \sin[\omega(x+y)]\}.
\end{split} \nonumber\end{align}
In a sense, (\ref{specprojker}) is the ultimate formula concerning
the operator~$H$, since all operator functions of~$H$ can be
calculated from it in principle and it contains all facts about
the spectral resolution in a rather explicit form.
Kottos and Smilansky \cite[Sec.~3B]{KS}
found the spectral resolution by treating the infinite star graph
as a scattering problem.
For each $\omega$ they provide the basis of incoming scattering
eigenfunctions
\begin{align}\label{scatbasis}
\psi_j{}\!^l(x) &\equiv \delta_j{}\!^l e^{-i\omega x}
+ \left[-\delta_j{}\!^l +
\frac1N \left(1+e^{-2i\tan^{-1}\frac{\alpha}{N\omega}}
\right)\right] e^{i\omega x} \\
&= -2i \delta_j{}\!^l \sin(\omega x) + 2\omega\,
\frac {N\omega -i\alpha}{ \alpha^2 +N^2\omega^2}\, e^{i\omega x} .
\nonumber \end{align}
These basis elements are orthonormal (up to the conventional factor
$\sqrt{2\pi}$)
and therefore
\begin{equation}\label{scatprojker}
\frac1{2\pi} \sum_{l=1}^N \psi_j{}\!^l(x)
\psi_{j'}{}\!^l(y)^*
= \sigma_{Sj}{}\!^{j'}(\omega,x,y).
\end{equation}
A calculation verifies that (\ref{specprojker}) and
(\ref{scatprojker}) agree.
\smallskip
\textsc{Remark:} Direct construction of eigenfunctions by applying
the Bondurant transform to orthonormal eigenfunctions of the
Dirichlet problem, while possible, is not recommended.
In the present problem the
immediate results are not orthogonal, much less normalized.
If $\alpha=0$ they are not even linearly independent
(because $T$ is not invertible), and one
basis element needs to be found by a separate argument.
\section*{The local spectral density}
\smallskip
Special interest attaches to the diagonal values of $\sigma$,
\begin{multline} \label{specdens}
\sigma_{Sj}{}\!^j(\omega,x,x) = \frac1{\pi} +
\frac1{\pi} \left(\frac2N -1\right) \cos(2\omega x) \\
{} +\frac{2\alpha/\pi}{\alpha^2+N^2\omega^2}
\left[\omega\sin(2\omega x)
-\frac{\alpha}N\, \cos(2\omega x)\right].
\end{multline}
Clearly here the $\frac1{\pi}$ is the universal Weyl term for a
one-dimensional system,
the next term is the spectral effect of a Kirchhoff vertex,
and the last term is the Exner--\v{S}eba correction.
In the limit $\alpha\to+\infty$ there is some cancellation between
the second and third terms, resulting in
\begin{equation}\label{specdensdir}
\frac1{\pi}- \frac1{\pi} \cos(2\omega x)
=\sigma_{Dj}{}\!^j(\omega,x,x) .
\end{equation}
as expected for a Dirichlet vertex.
Because the spectrum is continuous,
one can't integrate (\ref{specdens}) to get a density of states.
However, subtracting off the Weyl term and paying due attention to
distributional limits,
one can obtain a meaningful global spectral density:
\begin{align}\label{intspec}
\Delta\rho(\omega) &\equiv \int_0^\infty \sum_{j=1}^N
\left[ \sigma_{Sj}{}\!^j(\omega,x,x) -\frac1{\pi} \right] dx \\
& = \left(\frac12 -\frac N4\right)\delta(\omega)
+\left[ \frac{N\alpha/\pi}{\alpha^2 +N^2\omega^2}
-\frac12\, \delta(\omega)\right] .
\nonumber\end{align}
This expression approximates the incremental effect that such a
vertex would have in a problem with discrete spectrum.
The first term in (\ref{intspec}) is the Kirchhoff term;
it vanishes when $N=2$, because a Kirchhoff vertex with exactly
two edges is vacuous.
The other term is the Exner--\v{S}eba correction;
it vanishes when $\alpha=0$ because its first term
distributionally approaches $\frac12 \delta(\omega)$ in that limit.
Alternatively, (\ref{intspec}) can be simplified to
\begin{equation}\label{intspecdir}
\Delta\rho(\omega)=
-\,\frac N4\,\delta( \omega) + \frac{N\alpha/\pi}{\alpha^2
+N^2\omega^2}\,;
\end{equation}
here the first term is the correct formula for a Dirichlet vertex
and the remaining term is $O(\alpha^{-1})$ as $\alpha\to \infty$.
\smallskip
\textsc{Remark:} The meaning of a Dirac delta distribution in
formulas such as (\ref{intspec}) and (\ref{intspecdir}) is that
the spectral counting function $N(\omega)$ has a nonzero limit as
$\omega$ approaches $0$ from above
($N(\omega)$ being understood to be $0$ for negative~$\omega$).
For example, (\ref{intspecdir}) is simply the derivative of
the formula
\begin{equation}\label{staircase}
\Delta N(\omega) = \left[-\,\frac N4 + \frac1{\pi} \,\tan^{-1}
\frac{N\omega}{\alpha} \right]\theta(\omega)
\end{equation}
for the incremental effect of the vertex on the total number of
eigenvalues in the interval $0\le\lambda\le \omega^2$.
\smallskip
The Bondurant method cannot be applied directly to a finite
interval, because no transformation $T$ will work for both boundary
conditions simultaneously. However, the appropriate operators
$T^{-1}$ for the two boundaries can be applied alternately to
construct a solution as an infinite series (a sum over
closed classical paths, generalizing the classic method of images).
In \cite{BF} the wave kernel and hence the local and global
spectral densities were obtained in that way for a finite interval
with one Robin and one Dirichlet endpoint.
Numerical evaluation reveals the correct eigenvalues for the
problem emerging as spikes in the global density (the counterpart
of (\ref{intspec})).
See \cite{SPSUS} for a related study in two dimensions.
It should be straightforward to extend this analysis (and also the
study of vacuum energy, etc.)\ to an arbitrary \emph{finite} star
graph, and in principle to more complicated quantum graphs.
It is noteworthy that in these systems no semiclassical (or
stationary-phase) approximation is needed to obtain the
representation of the spectrum in terms of classical paths;
the only approximation involved is the truncation of the sum at
some maximum path length if and when one resorts to numerics.
\section*{Heat and quantum kernels}
The same machine (\ref{green}) can be used to treat the problems
(\ref{heateq}) and (\ref{quanteq}), for which the free kernels
are
\[
G(t,z) = (4\pi t)^{-1/2} e^{-z^2/4t}, \quad
G(t,z) = (4\pi i t)^{-1/2} e^{-z^2/4it},
\]
respectively.
(The results are qualitatively similar to (\ref{cylkernel}) below,
with the complementary error function appearing instead of the
exponential integral function.)
Studying the heat kernel is the traditional route to
equations like (\ref{staircase})
for partial differential operators.
In one-dimensional systems such as quantum graphs, however,
the wave kernel built from (\ref{dalembert}) appears to be easier to
calculate.
Using the heat kernel, the Robin case of (\ref{intspecdir})
was obtained in \cite[Secs.\ 3.3 and 5.5]{BF}.
That analysis extends to flat boundaries with constant $\alpha$ in
any dimension.
The results fit naturally with those of \cite{BB} in dimension $3$
and \cite{SPSUS} in dimension~$2$, where the leading orders in
boundary curvature are also included.
All these formulas are exact in $\alpha$; of course, when the
heat-kernel formulas are expanded in power series in $\alpha$ they
match and extend the relevant terms tabulated in such references
as \cite{Kirsten,Gilkey}.
(See also \cite{Dowker,BFSV}.)
Apart from a unifying point of view, it is not claimed that these
results are particularly new;
in fact, the Robin heat kernels in dimensions $1$ and $3$ were
found in 1891 by a closely related method \cite{Bryan,Bryan3}.
\section*{Vacuum energy density}
First, a paragraph which pure mathematicians are free to ignore:
From a \emph{physical} point of view,
vacuum energy involves a relativistic (usually massless and
bosonic) field.
(It is of no relevance, therefore, to quantum graphs if they are
regarded solely as models of nonrelativistic electrons in networks
of wires.)
Formally, the total energy corresponding to the wave operator~$H$
is
\begin{equation}\label{totenergy}
E = \frac12 \sum_{n=1}^\infty \omega_n
=\frac12\int_0^\infty \omega\, \rho(\omega)\, d\omega
\end{equation}
for an operator with purely point spectrum.
(To avoid irrelevant complications, let us also always assume that
$H$ has no negative spectrum.)
Equally formally, the local energy density is
\begin{equation}\label{energydens}
T_{00}(x)= \frac12 \int_0^\infty \omega\, \sigma(\omega,x,x)\,
d\omega
\end{equation}
(without the requirement of point spectrum).
The origin of these expressions is the ``second-quantized'' theory
of a field satisfying the wave equation (\ref{waveeq}), in which
each normal mode of the field (with frequency~$\omega$)
becomes a quantum harmonic oscillator (with ground state energy
$\frac12\omega$).
Then (\ref{energydens}) results from the integrand of
(\ref{robenergy1/4}), and (\ref{totenergy}) comes from
integrating (\ref{energydens}) over all space or just from adding
up the energies of all the modes.
Both integrals, (\ref{totenergy}) and (\ref{energydens}),
are divergent at the upper limit and are to be
defined by a renormalization procedure.
Our claim is that vacuum energy should be of mathematical interest
even in models whose physical relevance is questionable.
Therefore, we provide here a precise
\emph{mathematical} definition, which incorporates a particular
renormalization prescription (whose physical rationale need not
concern us):
{\em Consider the cylinder kernel
(the Green function of (\ref{cyleq}))
on diagonal ($y=x$, $l=j$),
find its asymptotic expansion as $t\to0$,
and extract the coefficient of the term
proportional to $t$, times~$-\frac12$;
this is the vacuum energy density, $T_{00}(x)$.
When appropriate, integrate over $x$
(and sum over $j$ in our graph problem),
before taking $t$ to $0$, to
define a total energy,~$E$.}
The intuition behind this definition is the following.
Let $G(t,x,y)$ be the cylinder kernel of the problem under study.
(For an infinite star graph it is the matrix~$G_S\,$.)
Then
\begin{equation}\label{cyldiag}
G(t,x,x) = \int_0^\infty e^{-\omega t} \sigma(\omega,x,x)\,
d\omega,
\end{equation}
and (when ``appropriate'') its trace is
\begin{equation}\label{cyltrace}
\mathop{\mathrm{Tr}}\nolimits G(t)
\equiv \int_0^\infty \sum_{j=1}^N G_j{}\!^j (t,x,x)\, dx
= \sum_{n=1}^\infty e^{-\omega_n t}.
\end{equation}
Now take $-\frac12 \,\frac{\partial}{\partial t}$ of
(\ref{cyldiag}) and (\ref{cyltrace}) and let $t$ approach~$0$,
formally obtaining (\ref{energydens}) and (\ref{totenergy}),
respectively.
Systematically throwing away the terms of negative order in the
Laurent expansions (and a possible logarithmic term), one arrives
at our definition.
To find the vacuum energy of an infinite star graph,
one can apply the Bondurant machine one
more time, to the free cylinder kernel
\begin{equation}\label{freecyl}
G(t,z) = \frac{t/\pi}{t^2+z^2}\,,
\end{equation}
obtaining
\begin{multline}\label{cylkernel}
G_{Sj}{}\!^l = \delta_j{}\!^l \, \frac{t/\pi}{t^2+(x-y)^2}
+\left(\frac 2N -\delta_j{}\!^l\right)\frac{t/\pi}{t^2+(x+y)^2}
\\
+ \frac{2\alpha}{\pi N^2} \, e^{\alpha(x+y)/N}
\mathop{\mathrm{Im}}\nolimits
\left[e^{-i\alpha t/N} \mathop{\mathrm{Ei}}\nolimits
\left( \frac{i\alpha t}{N}
-\frac{\alpha}N\,(x+y) \right)\right].
\end{multline}
It follows that
\begin{equation}\label{graphendens}
T_{00}(x) =
\left(1-\frac 2N\right) \frac1{8\pi x^2}
+ \frac{\alpha}{2\pi N^2 x} +\frac{\alpha^2}{\pi N^3}\,
e^{2\alpha x/N}
\mathop{\mathrm{Ei}}\nolimits
\left(-\,\frac{2\alpha x}{N}\right) .
\end{equation}
The most important parameter in this problem is the dimensionless
product $\alpha x$.
Therefore,
at short distance the Kirchhoff term dominates:
\begin{equation}\label{nearlim}
T_{00}(x) \sim \left(1-\frac 2N\right) \frac1{8\pi x^2}
+ \frac{\alpha}{2\pi N^2 x}
+\frac{\alpha^2}{\pi N^3} \,\ln|\alpha x| +O(\alpha^2x^2),
\end{equation}
whereas at large distance the energy density is almost pure Dirichlet:
\begin{equation}\label{farlim}
T_{00}(x) \sim \frac 1{8\pi x^2} +O(\alpha^{-1} x^{-3}).
\end{equation}
The nonintegrable $O(x^{-2})$ singularity in (\ref{nearlim})
would interfere with calculating a total energy by integration
over~$x$, even if the edges were finite. The renormalization
procedure implicit in our definition does not commute with the
spatial integration, however, and it leads to a finite total
energy \cite{BGH,systemat,norman}.
As agreed, we will not delve here into the physical issues thereby
raised (which are still somewhat controversial).
In a sense the calculation based on the cylinder kernel was
unnecessary, given the spectral formulas (\ref{specprojker}),
(\ref{specdens}), and (\ref{intspec}).
Indeed, (\ref{graphendens}) can be obtained from (\ref{cyldiag})
and (\ref{specdens}) and a Laurent expansion,
or even immediately from (\ref{energydens}) using the subtracted
spectral density appearing in the integrand of (\ref{intspec}).
(The subtraction of the Weyl term corresponds to the subtraction
of the leading Laurent term. In other problems, such as higher
dimensions, additional subtractions would be necessary.)
Similar methods were used in \cite{RS} to find the effect of a flat
Robin boundary in any dimension.
In general, however, it is easier to find the small-$t$ expansion
of a cylinder kernel than to find the detailed spectral resolution;
as for heat kernels, one expects useful calculations to run in the
opposite direction.
Romeo and Saharian \cite{RS} also gave complicated integral
formulas for the vacuum energy and energy density of a finite
interval with two Robin boundaries.
Our methods will instead give these quantities as infinite sums
over classical paths \cite{FL}.
\section*{A broader perspective}
Now let $H$ be a generic differential operator
(self-adjoint, elliptic, positive, second-order, with scalar
principal symbol)
in dimension~$d$.
Let $T$ be the cylinder kernel and $K$ be the heat kernel.
Then
\[
\begin{aligned}
\mathop{\mathrm{Tr}}\nolimits T &= \int_0^\infty e^{-t\omega} \,dN
=\int_0^\infty e^{-t\omega} \rho(\omega)\,d\omega, \\
\mathop{\mathrm{Tr}}\nolimits K &= \int_0^\infty e^{-t\lambda } \,dN
=\int_0^\infty e^{-t\omega^2 } \rho(\omega)\,d\omega,
\end{aligned}
\quad
\begin{aligned} T(t,x,x) &= \int_0^\infty e^{-t\omega}
\sigma(\omega,x,x)\,d\omega, \\
K(t,x,x) &= \int_0^\infty e^{-t\omega^2 }
\sigma(\omega,x,x)\,d\omega,
\end{aligned}
\]
where $N$ now is the number of eigenvalues less than or equal to
$\lambda=\omega^2$.
It is known that as $t\to0$ the global quantities have expansions
of the forms
\begin{equation}\label{cylseries}
\mathop{\mathrm{Tr}}\nolimits T \sim
\sum_{s=0}^\infty e_s t^{-d+s}
+\sum^\infty_{\genfrac{}{}{0pt}2{\scriptstyle s=d+1}{% \atop
\scriptstyle s-d \mbox{ \scriptsize odd}}} f_s t^{-d+s} \ln
t,
\end{equation}
\begin{equation}\label{heatseries}
\mathop{\mathrm{Tr}}\nolimits
K \sim \sum_{s=0}^\infty b_s t^{(-d+s)/2}.
\end{equation}
The local quantities, $T(t,x,x)$ and $K(t,x,x)$, have (nonuniform
in~$x$) expansions of precisely the same respective forms;
we do not introduce a separate notation for their coefficients.
{\em
If $d-s$ is even or positive,
\begin{equation}\label{evenrel}
e_s= \pi^{-1/2} 2^{d-s} \Gamma((d-s+1)/2) b_s\,.
\end{equation}
If $d-s$ is odd and negative, then
\begin{equation}\label{oddrel}
f_{s} = \frac{(-1)^{(s-d+1)/2}2^{d-s+1} }{
\sqrt{\pi}\, \Gamma((s-d+1)/2)} \, b_s\,,
\end{equation}
but, most strikingly, in that case
\begin{equation}\label{undetermined}
\mbox{$e_{s}$ is undetermined by the $b_{r}$}\,.
\end{equation}
}
I have expounded in detail elsewhere \cite{FG,systemat,norman}
how the connections (\ref{evenrel})--(\ref{undetermined}) come
about:
The $b_s$ are proportional to coefficients in the high-frequency
asymptotics of the Riesz means \cite{Hardy,Hor}
of $N$ (or of $\int\!\sigma$)
with respect to $\lambda$, while
the $e_s$ and $f_s$ are proportional to coefficients in the
asymptotics of the Riesz means with respect to $\omega$.
Alternatively, they are related through the poles of the zeta
functions of $H$ and $\sqrt{H}$ \cite{Gilkey4,GG,BM}.
The new half of the cylinder-kernel coefficients
(those in (\ref{undetermined}))
--- of which the first, $e_{d+1}\,$, is proportional to the vacuum
energy ---
are a new set of moments of the spectral distribution.
{\em What are they good for, mathematically?\/}
Unlike the old ones, they are \emph{nonlocal} in their dependence
on the geometry of the domain and the coefficient functions of~$H$.
They probe, in a comparatively crude, averaged way,
the oscillatory spectral structures that are correlated more
precisely with
periodic and closed classical paths.
Thus they embody, at least partially, the global dynamical structure
of the system;
they are a half-way house between the heat-kernel coefficients
and a full semiclassical closed-orbit analysis.
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\end{document}
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