Spectra of definite type in waveguide models

Abstract

We develop an abstract method to identify spectral points of definite type in the spectrum of the operator T 1 ⊗ I 2 + I 1 ⊗ T 2 T_1\otimes I_2 + I_1\otimes T_2 , 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 P T \mathcal {P}\mathcal {T} -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.