Quantum mechanics with non-self-adjoint operators: transition from spectra to pseudospectra

In recent years there has been an enormous growth of interest in non-self-adjoint operators in quantum mechanics. The motivation lies not only in the possibility of unconventional representation of physical observables by non-self-adjoint operators, which has been overlooked for almost 70 years since the foundation of quantum mechanics, and in the recent availability of related experiments in optics, but also in the need of new ideas and methods to deal with the mathematical problems.

In the talk I will survey recent developments in the field and propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-self-adjoint operators. As an application, employing microlocal techniques for the location of the pseudospectrum of semiclassical operators to models familiar from recent physical literature, unexpected wild properties of the operators will be revealed.