Radial positive definite functions and spectral theory of the Schrödinger operators with point interactions

Abstract

Key words Schrödinger operator, point interactions, self-adjoint extension, spectrum, positive definite function MSC (2000) 47A10, 47B25 Dedicated to the 75th anniversary of Eduard Tsekanovskii.We complete the classical Schoenberg representation theorem for radial positive definite functions. We apply this result to study spectral properties of self-adjoint realizations of two-and three-dimensional Schrödinger operators with point interactions on a finite set. In particular, we prove that any realization has purely absolutely continuous non-negative spectrum.

Schoenberg Matrices of Radial Positive Definite Functions and Riesz Sequences of Translates in $$L^2({\mathbb R}^n)$$ L 2 ( R n )

“…It was proved in [27] (see also [14,Theorem 3.6]) that each function f ∈ Φ n , n ≥ 2, is strictly X-positive definite for any set X of distinct points in R n . This fact has been heavily exploited in [14] for investigation of certain spectral properties of 2D and 3D Schrödinger operator with a finite number of point interactions. On the other hand, if a radial positive definite function is X-strongly positive definite, then X is necessarily separated (see Proposition 3.21).…”

Asymptotics of Resonances Induced by Point Interactions

“…Several alternative ways for parameterizing of all the self-adjoint extensions of S X can be found in a more recent literature; see e.g. [19][20][21]. Below we follow the strategy of [19] and use some of notations therein.…”

“…[19][20][21]. Below we follow the strategy of [19] and use some of notations therein. According to ([19], Prop.…”

Spectral theory of Schrödinger operators with infinitely many point interactions and radial positive definite functions

“…(ii) In the case of finitely many point interactions (m < ∞) different descriptions of nonnegative realizations has been obtained in [8,27,21].…”

“…(1 − e −β|x j −x k | ) |x j − x k | ≤ εd * (X) −1 for β ∈ (0, β 0 ). (ii) For sets X = {x j } m 1 of finitely many points a description of the ac-spectrum and the point spectrum of self-adjoint realizations of L 3 was obtained by different methods in [4, Theorem 1.1.4] and [21]. For this purpose a connection with radial positive definite functions was exploited for the first time and strong X-positive definiteness of some functions f ∈ Φ 3 was used in [21].…”