4,778 research outputs found
WKB Approximation to the Power Wall
We present a semiclassical analysis of the quantum propagator of a particle
confined on one side by a steeply, monotonically rising potential. The models
studied in detail have potentials proportional to for ; the
limit would reproduce a perfectly reflecting boundary, but at
present we concentrate on the cases and 2, for which exact
solutions in terms of well known functions are available for comparison. We
classify the classical paths in this system by their qualitative nature and
calculate the contributions of the various classes to the leading-order
semiclassical approximation: For each classical path we find the action ,
the amplitude function and the Laplacian of . (The Laplacian is of
interest because it gives an estimate of the error in the approximation and is
needed for computing higher-order approximations.) The resulting semiclassical
propagator can be used to rewrite the exact problem as a Volterra integral
equation, whose formal solution by iteration (Neumann series) is a
semiclassical, not perturbative, expansion. We thereby test, in the context of
a concrete problem, the validity of the two technical hypotheses in a previous
proof of the convergence of such a Neumann series in the more abstract setting
of an arbitrary smooth potential. Not surprisingly, we find that the hypotheses
are violated when caustics develop in the classical dynamics; this opens up the
interesting future project of extending the methods to momentum space.Comment: 30 pages, 8 figures. Minor corrections in v.
On cscK resolutions of conically singular cscK varieties
In this note we discuss the problem of resolving conically singular cscK
varieties to construct smooth cscK manifolds, showing a glueing result for
(some) crepant resolutions of cscK varieties with discrete automorphism groups
Bounding the Heat Trace of a Calabi-Yau Manifold
The SCHOK bound states that the number of marginal deformations of certain
two-dimensional conformal field theories is bounded linearly from above by the
number of relevant operators. In conformal field theories defined via sigma
models into Calabi-Yau manifolds, relevant operators can be estimated, in the
point-particle approximation, by the low-lying spectrum of the scalar Laplacian
on the manifold. In the strict large volume limit, the standard asymptotic
expansion of Weyl and Minakshisundaram-Pleijel diverges with the higher-order
curvature invariants. We propose that it would be sufficient to find an a
priori uniform bound on the trace of the heat kernel for large but finite
volume. As a first step in this direction, we then study the heat trace
asymptotics, as well as the actual spectrum of the scalar Laplacian, in the
vicinity of a conifold singularity. The eigenfunctions can be written in terms
of confluent Heun functions, the analysis of which gives evidence that regions
of large curvature will not prevent the existence of a bound of this type. This
is also in line with general mathematical expectations about spectral
continuity for manifolds with conical singularities. A sharper version of our
results could, in combination with the SCHOK bound, provide a basis for a
global restriction on the dimension of the moduli space of Calabi-Yau
manifolds.Comment: 32 pages, 3 figure
Diffractive Theorems for the Wave Equation with Inverse Square Potential
We first establish the presence of a diffractive front in the fundamental
solution of the wave operator with a diract delta intial condition in two
dimensional euclidean space caused by the potentials perturbation on the
spherical laplacian. This motivates a result which restricts the propagation of
singularities for the wave operator with a more general potential to precisely
these diffractive fronts higher dimensional euclidean spaces. This is proven
using microlocal energy estimates.Comment: 41 pages, 6 figure
Quantum ergodicity and quantum limits for sub-Riemannian Laplacians
This paper is a proceedings version of \cite{CHT-I}, in which we state a
Quantum Ergodicity (QE) theorem on a 3D contact manifold, and in which we
establish some properties of the Quantum Limits (QL). We consider a
sub-Riemannian (sR) metric on a compact 3D manifold with an oriented contact
distribution. There exists a privileged choice of the contact form, with an
associated Reeb vector field and a canonical volume form that coincides with
the Popp measure. We state a QE theorem for the eigenfunctions of any
associated sR Laplacian, under the assumption that the Reeb flow is ergodic.
The limit measure is given by the normalized canonical contact measure. To our
knowledge, this is the first extension of the usual Schnirelman theorem to a
hypoelliptic operator. We provide as well a decomposition result of QL's, which
is valid without any ergodicity assumption. We explain the main steps of the
proof, and we discuss possible extensions to other sR geometries.Comment: Appears in S{\'e}minaire Laurent Schwartz - EDP et applications,
2015, Palaiseau, France. 2015. arXiv admin note: text overlap with
arXiv:1504.0711
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