New paper: Computing Scattering Resonances

Jonathan Ben-Artzi, Marco Marletta and Frank Rösler submitted the paper Computing Scattering Resonances in which they ask the question:

Does there exist a universal algorithm for computing the resonances of Schrödinger operators with complex potentials?

Resonances are special modes that are nearly eigenvalues, in the sense that corresponding to them there are states that do not belong to the functional space (typically because they are not sufficiently localized). More precisely, let H_q=-\Delta+q be a Schrödinger operator with q:\mathbb{R}^d\to\mathbb{C} compactly supported and let \chi be a smooth cutoff function that is identically 1 on the support of q . Then

Definition (resonance). A resonance of H_q is defined to be a pole of the meromorphic operator-valued function z\mapsto(I+q(-\Delta-z^2)^{-1}\chi)^{-1} .

This paper provides an affirmative answer to the above question. The only a priori information required is the size of the support of q . With this knowledge at hand, this paper provides an algorithm (which is also implemented numerically) which only needs to read finitely many pointwise evaluations x\mapsto q(x) . These values are used to construct a approximation of q(-\Delta-z^2)^{-1}\chi which is shown to converge in an appropriate sense as more and more values of q are sampled.

New Marie Skłodowska-Curie Fellowship

We are delighted to announce that our team has been awarded another Marie Skłodowska-Curie Fellowship. Frank Rösler, with the supervision of Jonathan Ben-Artzi, has been successful with his proposal entitled “Computational Complexity in Quantum Mechanics” (COCONUT).

The short description provided in the proposal states: “The goal of this project is to improve our understanding of how to perform computations in quantum mechanics and classify their complexity. This will be achieved by using modern methods from spectral approximation theory in conjunction with the recently introduced Solvability Complexity Index.”

The total value of the award is €212,934.

Talks & Travels

In his second month here at Cardiff Frank Rösler has been invited to the University of Cambridge and Imperial College London to give seminar talks. These visits provided the opportunity to discuss his research with a wider audience and make progress on open questions from his recent research project on spectral approximation.

On 14th December he will attend the next meeting of the South Wales Analysis & Probability Seminar in Swansea.

New team member: Frank Rösler

On 1 October 2018 we were joined by a new team member, Dr Frank Rösler. Frank completed his PhD at Durham University and worked as a Research Assistant in Freiburg (Germany).

He is interested in the spectral theory of non-selfadjoint operators and other operator-theoretic questions in PDE theory. His past projects involved pseudospectra of non-normal Schrödinger Operators and more general resolvent norm estimates of partial differential operators. More recently, he studied problems in Asymptotic Analysis and Homogenisation from an operator-theoretic perspective.

Welcome Frank!