This paper touches upon several traditional topics of 1D linear complex analysis including distribution of zeros of entire functions, completeness problem for complex exponentials and for other families of special functions, some problems of spectral theory of selfadjoint differential operators. Their common feature is the close relation to the theory of complex Fourier transform of compactly s...

We study low lying eigenvalues for non-selfadjoint semiclassical differential operators, where symmetries play an important role. In the case of the Kramers-Fokker-Planck operator, we show how the presence of certain supersymmetric and PT -symmetric structures leads to precise results concerning the reality and the size of the exponentially small eigenvalues in the semiclassical (here the low t...

We describe methods which have been used to analyze the spectrum of non-self-adjoint differential operators, emphasizing the differences from the self-adjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the opera...

The pseudospectra of non-selfadjoint linear ordinary differential operators with variable coefficients are considered. It is shown that when a certain winding number or twist condition is satisfied, closely related to Hörmander’s commutator condition for partial differential equations, 3-pseudoeigenfunctions of such operators for exponentially small values of 3 exist in the form of localized wa...

We continue the spectral analysis of differential operators with complex coefficients, extending some results for Sturm-Liouville to higher order operators. give conditions essential spectrum be empty, and operator have compact resolvent. Conditions are given on coefficients resolvent Hilbert-Schmidt. These new even real i.e., selfadjoint case. Asymptotic is a central tool. See also https://ejd...