Ambarzumyan Type Theorem for a matrix quadratic Sturm-Liouville equation with energy dependent

TL;DR

This paper extends Ambarzumyan's theorem to matrix quadratic Sturm-Liouville systems, showing that identical spectra to the zero potential case imply zero potential functions.

Contribution

It introduces a generalization of Ambarzumyan's theorem for higher-dimensional matrix quadratic Sturm-Liouville problems.

Findings

Spectrum equality implies zero potential functions p and q.

Extension of classical theorem to matrix differential systems.

Provides conditions under which the potential functions are uniquely determined.

Abstract

Ambarzumyan's theorem for quadratic Sturm-Liouville problem is extended to second order differential systems of dimension d. It is shown that if the spectrum is the same as the spectrum belonging to the zero potential, then the matrix valued functions p and q are both zero