17/01/2013, 16:30 — 17:30 — Sala P3.10, Pavilhão de Matemática
Petr Siegl, 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
-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
-symmetric and possessing real spectrum, is
not similar to any self-adjoint operator. The argument is based on
known semiclassical results.
- P. Siegl: The non-equivalence of pseudo-Hermiticity and
presence of antilinear symmetry, PRAMANA-Journal of Physics, Vol.
73, No. 2, 279-287,
- 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,
- P. Siegl and D. Krejcirík: On the metric operator for
imaginary cubic oscillator, Physical Review D, to appear.