New paper: Computing Scattering Resonances

cardiffquest Frank Rosler, Jonathan Ben-Artzi, New Papers

*** Update: this paper will appear in the Journal of the European Mathematical Society ***

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.

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