Yershova Y. Inverse spectral problems in classical and quantum graphs.

The major part of the thesis deals with the solution to the inverse spectral problem of determining matching conditions at graph vertices based on the spectrum of a second-order differential operator (Laplace or Schrodinger one) defined on a finite compact metric graph. The attempt made here to develop a new mathematical apparatus, well-suited for spectral analysis of quantum graphs of arbitrary structure with matching conditions of delta and delta-prime types at vertices, seems to be quite timely and topical. In particular, the approach developed has allowed to make considerable progress in solving the inverse spectral problem described above. Since there exist tight links between inverse spectral problems for quantum graphs and inverse problems for classical graphs, the conditions for existence of systems of subspaces in a Hilbert space subject to additional conditions on the angles between them are also considered in the thesis.