170 research outputs found
Asymptotic Inverse Problem for Almost-Periodically Perturbed Quantum Harmonic Oscillator
Consider quantum harmonic oscillator, perturbed by an even almost-periodic
complex-valued potential with bounded derivative and primitive. Suppose that we
know the first correction to the spectral asymptotics
(, where
and is the spectrum of the unperturbed and the perturbed
operators, respectively). We obtain the formula that recovers the frequencies
and the Fourier coefficients of the perturbation.Comment: 6 page
Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions
We consider the Laplacian in a domain squeezed between two parallel
hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet
boundary conditions on one of the hypersurfaces and Neumann boundary conditions
on the other. We derive two-term asymptotics for eigenvalues in the limit when
the distance between the hypersurfaces tends to zero. The asymptotics are
uniform and local in the sense that the coefficients depend only on the
extremal points where the ratio of the area of the Neumann boundary to the
Dirichlet one is locally the biggest.Comment: 9 pages, 1 figure; written for proceedings of Equadiff 2013, to
appear in Mathematica Bohemic
PT Symmetric Schr\"odinger Operators: Reality of the Perturbed Eigenvalues
We prove the reality of the perturbed eigenvalues of some PT symmetric
Hamiltonians of physical interest by means of stability methods. In particular
we study 2-dimensional generalized harmonic oscillators with polynomial
perturbation and the one-dimensional for
Zero Energy Scattering for One-Dimensional Schr\"odinger Operators and Applications to Dispersive Estimates
We show that for a one-dimensional Schr\"odinger operator with a potential
whose (j+1)'th moment is integrable the j'th derivative of the scattering
matrix is in the Wiener algebra of functions with integrable Fourier
transforms. We use this result to improve the known dispersive estimates with
integrable time decay for the one-dimensional Schr\"odinger equation in the
resonant case.Comment: 9 page
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