Non-self-adjoint operators and their spectra

A 10-hours course consisting of five 2-hours lessons. It will take place at the Basque Center for Applied Mathematics (BCAM).

General objective:
The course aims to present a general overview of spectral theory for non-self-adjoint operators, as well as modern techniques to exclude the presence of their eigenvalues or confine them (as the Birman-Schwinger principle and the multipliers method).

Prerequisite Knowledge:

This course assumes a primer knowledge in linear algebra, real analysis and functional analysis.

Literature:

• Krejčiřík, D. and Siegl, P., 2015. Elements of spectral theory without the spectral theorem. Non- Selfadjoint Operators in Quantum Physics: Mathematical Aspects, pp.241-292.
• Davies, E.B., 2007. Linear operators and their spectra (Vol. 106). Cambridge University Press.
• Kenig, C.E., Ruiz, A. and Sogge, C.D., 1987. Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Mathematical Journal 55(2), pp.329-347.
• Hansmann, M. and Krejčiřík, D., 2022. The abstract Birman-Schwinger principle and spectral stability. Journal d\'Analyse Mathématique, 148(1), pp.361-398.
• Frank, R.L., 2011. Eigenvalue bounds for Schrödinger operators with complex potentials. Bulletin of the London Mathematical Society, 43(4), pp.745-750.
• Fanelli, L., Krejčiřík, D. and Vega, L., 2018. Spectral stability of Schrödinger operators with subordinated complex potentials. Journal of Spectral Theory, 8(2), pp.575-604.
• Kato, T.,1966. Perturbation theory for linear operators. Springer-Verlag New York.