We develop an abstract method to identify spectral points of definite type in the spectrum of the operator T1 ⊗ I2 + I1 ⊗ T2, applicable in particular for non-self-adjoint waveguide type operators with symmetries. Using the remarkable properties of the spectral points of definite type, we obtain new results on realness of weakly coupled bound states and of low lying essential spectrum in the PT -symmetric waveguide. Moreover, we show that the pseudospectrum has a tame behavior near the low lying essential spectrum and exclude the accumulation of non-real eigenvalues from this part of the essential spectrum. The advantage of our approach is particularly visible when the resolvent of the unperturbed operator cannot be explicitly expressed and most of the mentioned conclusions are extremely hard to prove by direct methods.
- Perturbations of essential spectrum
- PT-symmetric waveguide
- Spectral points of definite and of type π
- Weakly coupled bound states
ASJC Scopus subject areas
- Applied Mathematics