Solvable models for the Schrodinger operators with $δ'$-like potentials

Details Solvable models for the Schrodinger operators with $δ'$-like potentials Solvable models for the Schrodinger operators with $δ'$-like potentials

2009-09-05

arxiv.org/abs/0909.1034v2

Ukrainian Mathematical Bulliten V.6, no 2 (2009), 169-203

Mathematics

Spectral Theory

The proof of Lemma 6.1 was corrected. The main results of the paper are unchanged. 30 pages, 1 figure

Scientific paper

We turn back to the well known problem of interpretation of the Schrodinger operator with the pseudopotential being the first derivative of the Dirac function. We show that the problem in its conventional formulation contains hidden parameters and the choice of the proper selfadjoint operator is ambiguously determined. We study the asymptotic behavior of spectra and eigenvectors of the Hamiltonians with increasing smooth potentials perturbed by short-range potentials. Appropriate solvable models are constructed and the corresponding approximation theorems are proved. We introduce the concepts of the resonance set and the coupling function, which are spectral characteristics of the shape of squeezed potentials. The selfadjoint operators in the solvable models are determined by means of the resonance set and the coupling function.

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