On the Weyl solution of the 1-dim Schrödinger operator with inverse fourth power potential

Abstract

We consider the one dimensional Schrödinger operator with potential 1/x 4 on the half line. It is known that a generalized Titchmarsh-Weyl function can be associated to it. For other strongly singular potentials in some previous works it was possible to give an operator theoretic interpretation of this fact. However, for the present potential we show that such an interpretation does not exist.

Boundary Value Problems, Weyl Functions, and Differential Operators

“…The determination of the Kreȋn-von Neumann extension and other nonnegative extensions can be found in [212,350]. For singular perturbations associated with Sturm-Liouville operators, see for instance [331] and the later papers [9,28,29,124,171,182,282,315,316,479,480,481,482,507,531,532,549,550]; for δ-point interactions we refer to [8,293]. Special properties of the Titchmarsh-Weyl coefficient have been studied in many papers; we just mention [130,292,366,367].…”

A Novel Exact Solution of the 2+1-Dimensional Radial Dirac Equation for the Generalized Dirac Oscillator with the Inverse Potentials

“…From the phenomenological point of view, singular potential as a very useful form of anharmonicity is used in many aspects of physics [42][43]. The inverse quartic potential has also been investigated in many different questions by lots of authors [44][45][46]. Now, the purpose of our study is to obtain the analytical properties of the scattering amplitude about singular potential.…”

“…It is easy to find that the degree n polynomial solutions of the differential equation ( 44 (45) Substituting equation (45) into equation (44) and applying the Bethe ansatz method, the analytical expressions of the energy spectrum and the wave functions could be given…”