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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:420

Article

关键词: Schrodinger operator;    Perturbations by measures;    Eigenvalues;    Bound states;   

DOI  :  10.1016/j.jmaa.2014.06.053

来源: Elsevier

【 摘 要 】

We consider a self-adjoint two-dimensional Schrodinger operator H-alpha mu, which corresponds to the formal differential expression -Delta-alpha mu, where mu is a finite compactly supported positive Radon measure on R-2 from the generalized Kato class and alpha > 0 is the coupling constant. It was proven earlier that sigma(ess)(H-alpha mu) = [0, +infinity). We show that for sufficiently small a the condition #sigma(d)(H-alpha mu) = 1 holds and that the corresponding unique eigenvalue has the asymptotic expansion lambda(alpha) = (C-mu + o(1)) exp(- 4 pi/alpha mu(R-2)), alpha -> 0+, with a certain constant C-mu > 0. We also obtain a formula for the computation of C-mu. 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. (C) 2014 The Authors. Published by Elsevier Inc.

【 授权许可】

Free