Differential Operators and Dynamics of Localized Quantum States on Singular Spaces of Variable Dimension. Relation to the Behaviour of Geodesics and to Certain Problems of Analytic Number Theory

We study properties of evolution equations (for example, Schroedinger or wave equations) on singular spaces. These spaces can be obtained from graphs via replacing vertices by low-dimensional Riemannian manifolds. In particular, the behaviour of localized solutions appears to be connected with global properties of geodesic flows on these manifolds as well as with popular problems of analytic number theory.