Embedded eigenvalues and Neumann–Wigner potentials for relativistic Schrödinger operators

Abstract

The existence of potentials for relativistic Schrödinger operators allowing eigenvalues embedded in the essential spectrum is a long-standing open problem. We construct NeumannWigner type potentials for the massive relativistic Schrödinger operator in one and three dimensions for which an embedded eigenvalue exists. We show that in the non-relativistic limit these potentials converge to the classical Neumann-Wigner and Moses-Tuan potentials, respectively. For the massless operator in one dimension we construct two families of potentials, different by the parities of the (generalized) eigenfunctions, for which an eigenvalue equal to zero or a zero-resonance exists, dependent on the rate of decay of the corresponding eigenfunctions. We obtain explicit formulae and observe unusual decay behaviours due to the non-locality of the operator.

Absence of Embedded Eigenvalues for Non-Local Schrödinger Operators

“…For non-local Schrödinger operators the occurrence of zero or strictly positive eigenvalues just begins to be studied. In [24] we have constructed Neumann-Wigner type potentials for the operator (−∆ + m 2 ) 1/2 − m, for sufficiently large m > 0, giving rise to embedded eigenvalues equal to √ 1 + m 2 − m. Also, we have shown that the operators we obtained converge to classical cases of Neumann-Wigner type potentials in the non-relativistic limit. We have also obtained two families of fractional Schrödinger operators in one-dimension for the case m = 0 for which zero-eigenvalues occur.…”

Dirac Operators with Operator Data of Wigner-von Neumann Type

“…For example, Behncke [3] established criteria for subordinate solutions in both the Schrödinger and Dirac settings, and Naboko [23] demonstrated dense point spectrum in the absolutely continuous spectrum of Dirac operators and deduced the same for Schrödinger operators as a special case. While there are some considerations of Dirac operators with Wigner-von Neumann type operator data in the literature (e.g., [5,22]), such considerations seem to be rare. Here we continue the work of Lukić on spectral type charcterization of models with Wigner-von Neumann type data in the context of the half-line Dirac operator.…”

Bulk behaviour of ground states for relativistic Schrödinger operators with compactly supported potentials

“…Classic papers include [62,13,48] on the square-root Klein-Gordon equation, [64,37,26,27] on the properties of the spectrum, stability of the matter [51,32,33,50], and eigenfunction decay [16]. More recent developments further addressed low-energy scattering theory [55], embedded eigenvalues and Neumann-Wigner type potentials [54], decay rates when magnetic potentials and spin are included [38], a relativistic Kato-inequality [39], Carleman estimates and unique continuation [56,31], or nonlinear relativistic Schrödinger equations [25,60,2]. Given its relationship with random processes with jumps, the V = 0 case has received much attention also in potential theory [57,36,15].…”