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Spectral asymptotics of a broken delta interaction
This paper is concerned with the spectral analysis of a Hamiltonian with a
-interaction supported along a broken line with angle . The
bound states with energy slightly below the threshold of the essential spectrum
are estimated in the semiclassical regime .Comment: 20 page
Spectral asymptotics for the Schr\"odinger operator on the line with spreading and oscillating potentials
This study is devoted to the asymptotic spectral analysis of multiscale
Schr\"odinger operators with oscillating and decaying electric potentials.
Different regimes, related to scaling considerations, are distinguished. By
means of a normal form filtrating the oscillations, a reduction to a
non-oscillating effective Hamiltonian is performed
Tunneling for the Robin Laplacian in smooth planar domains
We study the low-lying eigenvalues of the semiclassical Robin Laplacian in a
smooth planar domain symmetric with respect to an axis. In the case when the
curvature of the boundary of the domain attains its maximum at exactly two
points away from the axis of symmetry, we establish an explicit asymptotic
formula for the splitting of the first two eigenvalues. This is a rigorous
derivation of the semiclassical tunneling effect induced by the domain's
geometry. Our approach is close to the Born-Oppenheimer one and yields, as a
byproduct, a Weyl formula of independent interest
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