Smooth, invariant orthonormal basis for singular potential Schroedinger operators

TL;DR

This paper constructs a smooth, invariant orthonormal basis for singular potential Schrödinger operators, extending previous work by applying Gram-Schmidt orthonormalization and analyzing applications on the positive real line.

Contribution

It introduces a method to obtain an orthonormal basis from a smooth, invariant domain for singular Schrödinger operators, including on the positive real line where integrals are estimated.

Findings

Successfully performed Gram-Schmidt orthonormalization to create an orthonormal basis.

Provided analytical estimates for integrals on the positive real line.

Demonstrated the invariance properties of the basis under derivatives and powers of the coordinate.

Abstract

In a recent contribution we showed that there exists a smooth, dense domain for singular potential Schr\"odinger operators on the real line which is invariant under taking derivatives of arbitrary order and under multiplication by positive and negative integer powers of the coordinate. Moreover, inner products between basis elements of that domain were shown to be easily computable analytically. A task left open was to construct an orthonormal basis from elements of that domain by using Gram-Schmidt orthonormalisation. We perform that step in the present manuscript. We also consider the application of these methods to the positive real line for which one can no longer perform the integrals analytically but for which one can give tight analytical estimates.