1-D Schrödinger operators with local point interactions on a discrete set

Abstract

MSC: 34L05 34L40 47E05 47B25 47B36 81Q10 Keywords: Schrödinger operator Local point interaction Self-adjointness Lower semiboundedness Discreteness Spectral properties of 1-D Schrödinger operators H X,αOur paper is devoted to the case d * = 0. We consider H X,α in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding Weyl functions. We show that the spectral properties of H X,α like self-adjointness, discreteness, and lower semiboundedness correlate with the corresponding spectral properties of certain classes of Jacobi matrices. Based on this connection, we obtain necessary and sufficient conditions for the operators H X,α to be self-adjoint, lower semibounded, and discrete in the case d * = 0. The operators with δ -type interactions are investigated too. The obtained results demonstrate that in the case d * = 0, as distinguished from the case d * > 0, the spectral properties of the operators with δ-and δ -type interactions are substantially different.

Spectral asymptotics for the Sturm—Liouville operator with point interaction

“…This result was proved in [5] and can also be established by a method in [10], where a similar condition was obtained for a usual (continuous) potential.…”

“…First, we consider the case of α k < 0. Under certain conditions, the spectrum of A proves to be discrete (see [5]). We give asymptotic estimates of the spectrum.…”

Schrödinger operators with δ′-interactions and the Krein-Stieltjes string

“…The operator H X, α is known as the Hamiltonian with δ interactions of strengths α n at the centers x n [2]. The analogs of Lemma 1 and Theorem 3 hold true for the operator H X, α (see [11,Sect. 5]) and the role of the matrix (15) is played by the Jacobi matrix…”

“…By (16) and (26), the structure of matrices (15) and (25) are completely different in the case d * = 0. There fore, the spectral properties of Hamiltonians with δ' and δ interactions on X are essentially different in this case (see [11]). …”

One-dimensional Schrödinger operators with singular potentials: A Schwartz distributional formulation

“…We shall denote them generically by L. In the one-dimensional case they form a four-parameter family of operators, each of which is characterized by two boundary conditions at x = 0. These objects yield exactly solvable models [2,4,5,6,8,11,12,26,38,45] and have been widely used in applications in quantum mechanics (e.g. in models of low-energy scattering [3,13,14,35] and quantum systems with boundaries [22,23,24,27,32]), condensed matter physics [10,17,25] and, more recently, on the approximation of thin quantum waveguides by quantum graphs [1,15,16,20].…”