Poster Blitz

Quantum Resonances in Relativistic Systems

Significant amounts of research have been completed on mathematical quantum resonances in the non-relativistic setting. This is achieved by studying the spectra of the perturbed Schrödinger operator.

However success in generalizing these results to the relativistic setting has been limited. In this poster I have outlined qualitatively some of the results of my research in this field thus far.

Symplectic non-squeezing theorem in Hilbert space

Inverse problems for a fractional conductivity equation

We show global uniqueness in two inverse problems for a fractional conductivity equation: an unknown conductivity in a bounded domain is uniquely determined by measurements of solutions taken in arbitrary open, possibly disjoint subsets of the exterior. The results are based on a reduction from the fractional conductivity equation to the fractional Schrödinger equation.

Inhomogeneous Neumann Problem for the fractional Laplace operator

3D Electrical Impedancetomography

The Schrodinger equation on star-graphs under general coupling conditions

Operator error estimates for homogenization of elliptic and parabolic systems

We consider a matrix strongly elliptic second order differential operator acting in a bounded domain with the Dirichlet boundary condition. The operator is self-adjoint. Coefficients are periodic and oscillate rapidly. We study the behavior of solutions of the corresponding elliptic and parabolic systems in the small period limit. The results can be written as approximations of the resolvent and the semigroup in L2→L2 and L2→H1 operator norms. So, the estimates of this type are called operator error estimates in homogenization theory. The talk is based on a joint work with T. A. Suslina.

Unique continuation for the Helmholtz equation. Stability estimates and numerical analysis

We consider the unique continuation problem for the Helmholtz equation that arises, e.g. in inverse boundary value problems for the wave equation. Following previous work by Isakov we prove conditional Hölder stability estimates with constants independent of the wave number, when the solution is continued along a convex surface. The main tools we employ are Carleman estimates and semiclassical analysis.

We then introduce a stabilized finite element method and prove convergence with the order given by the conditional stability, and with explicit dependence on the wave number.

A nonlocal maximum principle

A resolvent estimate for the magnetic Schrödinger operator in the presence of short and long-range

Tensor tomography on Cartan–Hadamard manifolds

We study the geodesic x-ray transform on Cartan–Hadamard manifolds, generalizing the x-ray transforms on Euclidean and hyperbolic spaces that arise in medical and seismic imaging. We prove solenoidal injectivity of this transform acting on functions and tensor fields of any order. The functions are assumed to be exponentially decaying if the sectional curvature is bounded, and polynomially decaying if the sectional curvature decays at infinity.

Propagators on spacetimes of low regularity

Boundary behaviour of certain pure-jump Markov processes on sets