Two-Spectra Theorem with Uncertainty

TL;DR

This paper extends Borg's two-spectra theorem for Schrödinger operators by incorporating uncertainty in eigenvalue placement, leveraging recent advances in the Uncertainty Principle in Harmonic Analysis.

Contribution

It introduces a version of the two-spectra theorem that accounts for eigenvalue uncertainty, providing a formula for the uncertainty size based on spectral interval lengths.

Findings

Derived a formula for the size of eigenvalue uncertainty.

Proved a generalized two-spectra theorem under uncertainty conditions.

Described pairs of indeterminate operators in the three-interval case.

Abstract

The goal of this paper is to combine ideas from the theory of mixed spectral problems for differential operators with new results in the area of the Uncertainty Principle in Harmonic Analysis (UP). Using recent solutions of Gap and Type Problems of UP we prove a version of Borg's two-spectra theorem for Schr\"odinger operators, allowing uncertainty in the placement of the eigenvalues. We give a formula for the exact 'size of uncertainty', calculated from the lengths of the intervals where the eigenvalues may occur. Among other applications, we describe pairs of indeterminate operators in the three-interval case of the mixed spectral problem. At the end of the paper we discuss further questions and open problems.