Eigensystem of an $L^2$-perturbed harmonic oscillator is an unconditional basis

On the Similarity of Sturm–Liouville Operators with Non-Hermitian Boundary Conditions to Self-Adjoint and Normal Operators

“…separation distance of eigenvalues tends to infinity, that is used for the proof of the existence of similarity transformations [11]. Recent results on basis properties for perturbations of harmonic oscillator type operators [1,30,3] give a possibility to investigate the structure of similarity transformation in these cases as well. Another step is to extend the results e.g.…”

“…1,2 (−a, a) in the space of uniformly continuous functions C 0 [−a, a]. An elementary idea of the proof of the embedding can be also used to show that the boundary terms represent a relatively bounded perturbation of the form associated with the Neumann Laplacian (i.e., c ± = 0).…”

Equiconvergence of spectral decompositions of 1D Dirac operators with regular boundary conditions

“…It is well-known that if f is a function of bounded variation on [0, π] then its Fourier series with respect to the system {e imx , m ∈ 2Z} converges point-wise to 1 2 [f (x−0)+f (x+0)] for x ∈ (0, π), and to 1 2 [f (π −0)+f (0+0)] for x = 0, π. More precisely, the following holds.…”

“…See [1] for the proof of a particular version of this criterion which is good enough for Lemma 30. .…”