620 research outputs found

### Spectral Oscillations, Periodic Orbits, and Scaling

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 Schrodinger 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.Comment: 19 pages, RevTeX4 with PicTeX. Minor improvements in content, new
references, typos correcte

### Mass Dependence of Vacuum Energy

The regularized vacuum energy (or energy density) of a quantum field
subjected to static external conditions is shown to satisfy a certain partial
differential equation with respect to two variables, the mass and the "time"
(ultraviolet cutoff parameter). The equation is solved to provide integral
expressions for the regularized energy (more precisely, the cylinder kernel) at
positive mass in terms of that for zero mass. Alternatively, for fixed positive
mass all coefficients in the short-time asymptotics of the regularized energy
can be obtained recursively from the first nontrivial coefficient, which is the
renormalized vacuum energy.Comment: 8 pages, RevTeX; v.2 has minor updates and format change

### Repulsive Casimir Pistons

Casimir pistons are models in which finite Casimir forces can be calculated
without any suspect renormalizations. It has been suggested that such forces
are always attractive. We present three scenarios in which that is not true.
Two of these depend on mixing two types of boundary conditions. The other,
however, is a simple type of quantum graph in which the sign of the force
depends upon the number of edges.Comment: 4 pages, 2 figures; RevTeX. Minor additions and correction

### Distributional Asymptotic Expansions of Spectral Functions and of the Associated Green Kernels

Asymptotic expansions of Green functions and spectral densities associated
with partial differential operators are widely applied in quantum field theory
and elsewhere. The mathematical properties of these expansions can be clarified
and more precisely determined by means of tools from distribution theory and
summability theory. (These are the same, insofar as recently the classic
Cesaro-Riesz theory of summability of series and integrals has been given a
distributional interpretation.) When applied to the spectral analysis of Green
functions (which are then to be expanded as series in a parameter, usually the
time), these methods show: (1) The "local" or "global" dependence of the
expansion coefficients on the background geometry, etc., is determined by the
regularity of the asymptotic expansion of the integrand at the origin (in
"frequency space"); this marks the difference between a heat kernel and a
Wightman two-point function, for instance. (2) The behavior of the integrand at
infinity determines whether the expansion of the Green function is genuinely
asymptotic in the literal, pointwise sense, or is merely valid in a
distributional (cesaro-averaged) sense; this is the difference between the heat
kernel and the Schrodinger kernel. (3) The high-frequency expansion of the
spectral density itself is local in a distributional sense (but not pointwise).
These observations make rigorous sense out of calculations in the physics
literature that are sometimes dismissed as merely formal.Comment: 34 pages, REVTeX; very minor correction

### The Dirichlet-to-Robin Transform

A simple transformation converts a solution of a partial differential
equation with a Dirichlet boundary condition to a function satisfying a Robin
(generalized Neumann) condition. In the simplest cases this observation enables
the exact construction of the Green functions for the wave, heat, and
Schrodinger problems with a Robin boundary condition. The resulting physical
picture is that the field can exchange energy with the boundary, and a delayed
reflection from the boundary results. In more general situations the method
allows at least approximate and local construction of the appropriate reflected
solutions, and hence a "classical path" analysis of the Green functions and the
associated spectral information. By this method we solve the wave equation on
an interval with one Robin and one Dirichlet endpoint, and thence derive
several variants of a Gutzwiller-type expansion for the density of eigenvalues.
The variants are consistent except for an interesting subtlety of
distributional convergence that affects only the neighborhood of zero in the
frequency variable.Comment: 31 pages, 5 figures; RevTe

- â€¦