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On self-adjointness of 1-D Schrödinger operator with \(\delta\)-interaction matching conditions. (English) Zbl 1289.34238

The authors consider the operator \[ H_{X,\alpha}=-\frac{d^2}{dx^2}+\sum\limits_{n=1}^\infty \alpha_n\delta (x-x_n) \] on \(L^2(\mathbb R_+)\). Here, \(X=\{ x_n\}\) is a strictly increasing sequence of nonnegative numbers, \(\{ \alpha_n\}\) is a sequence of real numbers. The operator is defined by the differential expression \(-\frac{d^2}{dx^2}\) on the domain \[ D=\{ f\in W^{2,2}(\mathbb R_+\setminus X)\cap L_{\text{comp}}^2(\mathbb R_+):\;f'(0)=0,\,f'(x_n+0)-f'(x_n-0)=\alpha_n f(x_n)\}. \] Following A. S. Kostenko and M. M. Malamud [J. Differ. Equations 249, No. 2, 253–304 (2010; Zbl 1195.47031)], the authors find new conditions for selfadjointness (or, to the contrary, for the nontriviality of deficiency indices) of the operator \(H_{X,\alpha}\). The conditions known for the case where \(x_n-x_{n-1}=1/n\) are extended to wider classes of sequences \(\{ x_n\}\), like \(x_n=\frac{1}{n^\gamma \log^\eta n}\).

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47E05 General theory of ordinary differential operators
47B25 Linear symmetric and selfadjoint operators (unbounded)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis

Citations:

Zbl 1195.47031
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