No abstract

Schrödinger operators with nonlocal point interactions are considered as new solvable models with point interactions. Examples in one and three dimensions are discussed. Corresponding direct and inverse scattering problems in one dimension are also discussed.

A one-dimensional Schrödinger operator with point interactions on Sobolev spaces is studied on the basis of the extension theory of nondensely defined operators.Key words: Schrödinger operator, point interaction, Sobolev space, self-adjoint extension, nondensely defined operator.Self-adjoint Schrödinger operators are usually studied on the spaces L 2 (R d ). However, there is a number of special problems where one has to consider Schrödinger operators on the Sobolev spaces W m 2 (R d ). The idea of studying the operators on spaces different from L 2 (R d ) is not new. Various problems in analysis, differential equations, and mathematical physics contributed to the creation and study of operators both on abstract scales of Hilbert spaces and on scales of Sobolev spaces [1][2][3][4][5]. However, the spectral theory of operators on Sobolev spaces is practically not developed despite its significance and possible applications [6].1. The Sturm-Liouville problem with higher derivatives in the boundary conditions leads to selfadjoint operators on Sobolev spaces. For example, the problem −y = λy on the interval 0 < x < π with the boundary conditions y (0) − y(0) = 0 and y (π) − y(π) = 0 is self-adjoint on the Sobolev space W 1 2 (0, π) with the eigenfunctions u n = sin nx, n = 1, 2, . . . , and u ± = exp(±x). These functions form a complete orthogonal system in the space W 1 2 (0, π). If the boundary conditions y (0) + y(0) = 0 and y (π) + y(π) = 0 are taken instead, then the eigenvalue problem is self-adjoint with respect to the indefinite inner product [u, v] The free Schrödinger operator −∆ is self-adjoint both on L 2 (R d ) and on the Sobolev spaces W m 2 (R d ). One way to define a Schrödinger operator with a point interaction at a point x 0 is to consider self-adjoint extensions of the minimal symmetric operator defined on C ∞ 0 (R d \{x 0 }) by the formula L min ϕ = −∆ϕ [7][8][9]. The operator L min is also a symmetric operator on the Sobolev space W m 2 (R d ) with equal deficiency indices i(m, d). If m 0 is an integer, then i(m, d) is equal to the number of all integer multi-indices α = (α 1 , . . . , α d ) such that |α| = α 1 +· · ·+α d < 2+m− 1 2 d [10]. For the space L 2 (R d ), i.e., for m = 0, this leads to the well-known fact that the deficiency indices are nonzero only in dimensions d = 1, 2, 3 and are equal to i(0, d) = 2, 1, 1, respectively.For the important dimensions d = 1, 2, 3, the deficiency indices of L min on W m 2 (R d ) are given by i(m, 1) = m + 2, i(m, 2) = 1 2 (m + 1)(m + 2), and i(m, 3) = 1 6 (m + 1)(m + 2)(m + 3). Although L min is not densely defined on W m 2 (R d ) (unless m = 0 or d 2 + 2m), one can constructively describe all of its self-adjoint extensions. This leads to a great variety of self-adjoint Schrödinger operators with point interactions on Sobolev spaces W m 2 (R d ), in particular, the ones that describe supersingular spherically nonsymmetric point interactions [10].3. To study one-dimensional Schrödinger operators on Sobolev spaces, we need the following auxiliary facts. ...

A norm resolvent convergence result is proved for approximations of general Schr6dinger operators with zero-range potentials. An approximation of the 6'-interaction by nonlocal non-Hermitian potentials (without a renormalization of the coupling constant) is also constructed.