# A Gelfand-Levitan trace formula for generic quantum graphs

## TL;DR

This paper develops a Gelfand-Levitan trace formula applicable to a wide class of quantum graphs with diverse edge lengths and boundary conditions, extending classical results to more general settings.

## Contribution

It introduces a generalized Gelfand-Levitan trace formula for quantum graphs with arbitrary edge lengths and coupling conditions, covering nearly all self-adjoint operators.

## Key findings

- Formulated a trace formula for quantum graphs.
- Proved the formula for generic edge lengths and conditions.
- Extends classical trace formulas to complex quantum graph structures.

## Abstract

We formulate and prove a Gelfand-Levitan trace formula for general quantum graphs with arbitrary edge lengths and coupling conditions which cover all self-adjoint operators on quantum graphs, except for a set of measure zero. The formula is reminiscent of the original Gelfand-Levitan result on the segment with Neumann boundary conditions.