On the Space of Symmetric Operators with Multiple Ground States

TL;DR

This paper investigates the homological structure of spaces of symmetric (self-adjoint) operators based on the multiplicity of their ground states, focusing on finite-dimensional cases with implications for infinite-dimensional generalizations.

Contribution

It introduces a homological framework for analyzing the stratification of self-adjoint operators by ground state multiplicity, offering new insights into their topological structure.

Findings

Homological structures are characterized for finite-dimensional symmetric operators.

The stratification by ground state multiplicity reveals intricate topological features.

Potential extensions to infinite-dimensional operators are suggested.

Abstract

We study homological structure of the filtrations of the spaces of self-adjoint operators by the multiplicity of the ground state. We consider only operators acting in a finite dimensional complex or real Hilbert space but infinite dimensional generalizations are easily guessed.