Quantum graphs attracted a lot of interest recently. There are several reasons for that. On one hand these models are useful as descriptions of various structures prepared from semiconductor wires, carbon nanotubes, and other substances. On the other hand they provide a tool to study properties of quantum dynamics in situations when the system has a nontrivial geometrical or topological structure. Quantum graph models contain typically free parameters related to coupling of the wave functions at the graph vertices, and to get full grasp of the theory one has to understand their physical meaning. A natural approach to this question is to investigate fat graphs, that is, systems of thin tubes built over the skeleton of a given graph, and to analyze its limit as the tube thickness tends to zero.

We adopt a recent work in Chung, Qian, Uhlmann and Zhao (Inverse Problems, 23 (2007) 309-329) to develop a phase space method for reconstructing pressure wave speed and shear wave speed of an elastic medium from travel time measurements. The method is based on the so-called Stefanov-Uhlmann identity which links two Riemannian metrics with their travel time information. We design a numerical algorithm to solve the resulting inverse problem. The algorithm is a hybrid approach that combines both Lagrangian and Eulerian formulations. In particular the Lagrangian formulation in phase space can take into account multiple arrival times naturally, while the Eulerian formulation for wave speeds allows us to compute the solution in physical space. Numerical examples are shown to validate the method.

We study asymptotical behaviour of resonances for a quantum graph consisting of a finite internal part and external leads placed into a magnetic field, in particular, the question whether their number follows the Weyl law. We prove that the presence of a magnetic field cannot change a non-Weyl asymptotics into a Weyl one and vice versa. On the other hand, we present examples demonstrating that for some non-Weyl graphs the effective size of the graph, and therefore the resonance asymptotics, can be affected by the magnetic field.

In this paper, we study gravitational instantons (i.e., complete hyperkähler 4‑manifolds with faster than quadratic curvature decay). We prove three main theorems: (1) Any gravitational instanton must have one of the following known ends: ALE, ALF, ALG, and ALH. (2) In the ALG and ALH non-splitting cases, it must be biholomorphic to a compact complex elliptic surface minus a divisor. Thus, we confirm a long-standing question of Yau in the ALG and ALH cases. (3) In the ALF‑D_k case, it must have an O(4)‑multiplet.

We study spectral and scattering properties of a spinless quantum particle confined to an infinite planar layer with hard walls containing a finite number of point perturbations. A solvable character of the model follows from the explicit form of the Hamiltonian resolvent obtained by means of Kreins formula. We prove the existence of bound states, demonstrate their properties, and find the on-shell scattering operator. Furthermore, we analyze the situation when the system is put into a homogeneous magnetic field perpendicular to the layer; in that case the point interactions generate eigenvalues of a finite multiplicity in the gaps of the free Hamiltonian essential spectrum.