149 research outputs found
Elementary linear algebra for advanced spectral problems
We discuss the general method of Grushin problems, closely related to Shur
complements, Feshbach projections and effective Hamiltonians, and describe
various appearances in spectral theory, pdes, mathematical physics and
numerical problems.Comment: 2 figure
Viscosity Limits for Zeroth-Order Pseudodifferential Operators
Motivated by the work of Colin de Verdière and Saint-Raymond on spectral theory for zeroth-order pseudodifferential operators on tori, we consider viscosity limits in which zeroth-order operators, P, are replaced by P + iν Δ, ν > 0. By adapting the Helffer–Sjöstrand theory of scattering resonances, we show that, in a complex neighbourhood of the continuous spectrum, eigenvalues of P + iν Δ have limits as the viscosity ν goes to 0. In the simplified setting of tori, this justifies claims made in the physics literature. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC
Quantum ergodicity for restrictions to hypersurfaces
Quantum ergodicity theorem states that for quantum systems with ergodic
classical flows, eigenstates are, in average, uniformly distributed on energy
surfaces. We show that if N is a hypersurface in the position space satisfying
a simple dynamical condition, the restrictions of eigenstates to N are also
quantum ergodic.Comment: 22 pages, 1 figure; revised according to referee's comments. To
appear in Nonlinearit
Fractal Weyl laws for chaotic open systems
We present a result relating the density of quantum resonances for an open
chaotic system to the fractal dimension of the associated classical repeller.
The result is supported by numerical computation of the resonances of the
system of n disks on a plane. The result generalizes the Weyl law for the
density of states of a closed system to chaotic open systems.Comment: revtex4, 4 pages, 3 figure
Probabilistic Weyl laws for quantized tori
For the Toeplitz quantization of complex-valued functions on a
-dimensional torus we prove that the expected number of eigenvalues of
small random perturbations of a quantized observable satisfies a natural Weyl
law. In numerical experiments the same Weyl law also holds for ``false''
eigenvalues created by pseudospectral effects.Comment: 33 pages, 3 figures, v2 corrected listed titl
Symmetry of bound and antibound states in the semiclassical limit
We consider one dimensional scattering and show how the presence of a mild
positive barrier separating the interaction region from infinity implies that
the bound and antibound states are symmetric modulo exponentially small errors
in 1/h. This simple result was inspired by a numerical experiment and we
describe the numerical scheme for an efficient computation of resonances in one
dimension
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