# A. V. Kiselev

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### Trace formulae for Schrödinger operators on metric graphs with applications to recovering matching conditions

Yulia Ershova, Alexander V. Kiselev

Methods Funct. Anal. Topology **20** (2014), no. 2, 134-148

The paper is a continuation of the study started in [8]. Schrödinger operators on finite compact metric graphs are considered under the assumption that the matching conditions at the graph vertices are of $\delta$ type. Either an infinite series of trace formulae (provided that edge potentials are infinitely smooth) or a finite number of such formulae (in the cases of $L_1$ and $C^M$ edge potentials) are obtained which link together two different quantum graphs under the assumption that their spectra coincide. Applications are given to the problem of recovering matching conditions for a quantum graph based on its spectrum.

### Trace formulae for graph Laplacians with applications to recovering matching conditions

Yulia Ershova, Alexander V. Kiselev

Methods Funct. Anal. Topology **18** (2012), no. 4, 343-359

Graph Laplacians on finite compact metric graphs are considered under the assumption that the matching conditions at the graph vertices are of either $\delta$ or $\delta'$ type. In either case, an infinite series of trace formulae which link together two different graph Laplacians provided that their spectra coincide is derived. Applications are given to the problem of reconstructing matching conditions for a graph Laplacian based on its spectrum.