Dr Petr Siegl - Advances in Spectral Theory for Non-Selfadjoint Operators

IAS Visiting Fellow Dr Petr Siegl delivers a seminar on their research -

Many physical systems can be described by partial differential equations which in turn generate operators between Banach spaces. A well-known illustration of such interplay is quantum mechanics together with the spectral theory of self-adjoint operators in a Hilbert space. However, in several branches of physics like hydrodynamics, damped systems, quantum resonances, superconductivity or balanced loss/gain materials, the occurring PDEs contain non-symmetric terms and thus lead to non-self-adjoint operators. Due to the lack of powerful self-adjoint tools like the spectral theorem or variational principles, the encountered phenomena are very different from the self-adjoint case and include spectral instabilities, divergence of expansions in eigenvectors or non-reliability of approximations. In the first part of the talk, we give an introduction in the non-self-adjoint aspects of spectral theory. In the second part, we present new results on the convergence of the expansions in eigenvectors based on the local form subordination.

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