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J Math Anal Appl. 2014 Dec 15;420(2):1416-1438.

Weakly coupled bound state of 2-D Schrödinger operator with potential-measure.

Abstract

We consider a self-adjoint two-dimensional Schrödinger operator [Formula: see text], which corresponds to the formal differential expression[Formula: see text] where μ is a finite compactly supported positive Radon measure on [Formula: see text] from the generalized Kato class and [Formula: see text] is the coupling constant. It was proven earlier that [Formula: see text]. We show that for sufficiently small α the condition [Formula: see text] holds and that the corresponding unique eigenvalue has the asymptotic expansion[Formula: see text] with a certain constant [Formula: see text]. We also obtain a formula for the computation of [Formula: see text]. The asymptotic expansion of the corresponding eigenfunction is provided. The statements of this paper extend the results of Simon [41] to the case of potentials-measures. Also for regular potentials our results are partially new.

KEYWORDS:

Bound states; Eigenvalues; Perturbations by measures; Schrödinger operator

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