Contents/conteúdo

Mathematics Department Técnico Técnico

Operator Theory, Complex Analysis and Applications Seminar  RSS

Sessions

17/01/2013, 16:30 — 17:30 — Room P3.10, Mathematics Building
, Universidade de Lisboa

Spectral analysis of some non-self-adjoint operators

We give an introduction to the study of one particular class of non-self-adjoint operators, namely 𝒫𝒯-symmetric ones. We explain briefly the physical motivation and describe the classes of operators that are considered. We explain relations between the operator classes, namely their non-equivalence, and mention open problems.

In the second part, we focus on the similarity to self-adjoint operators. On the positive side, we present results on one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. Using functional calculus, closed formulas for the similarity transformation and the similar self-adjoint operator are derived in particular cases. On the other hand, we analyse the imaginary cubic oscillator, which, although being 𝒫𝒯-symmetric and possessing real spectrum, is not similar to any self-adjoint operator. The argument is based on known semiclassical results.

  1. P. Siegl: The non-equivalence of pseudo-Hermiticity and presence of antilinear symmetry, PRAMANA-Journal of Physics, Vol. 73, No. 2, 279-287,
  2. D. Krejcirík, P. Siegl and J. Zelezný: On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators, Complex Analysis and Operator Theory, to appear,
  3. P. Siegl and D. Krejcirík: On the metric operator for imaginary cubic oscillator, Physical Review D, to appear.

Seminar organized in the context of the project PTDC/MAT/121837/2010.

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