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We study resonances generated by rank one perturbations of selfadjoint operators with eigenvalues embedded in the continuous spectrum. Instability of these eigenvalues is analyzed and almost exponential decay for the associated resonant states is exhibited. We show how these results can be applied to Sturm-Liouville operators. Main tools are the Aronszajn-Donoghue theory for rank one perturbations, a reduction process of the resolvent based on Feshbach-Livsic formula, the Fermi golden rule and a careful analysis of the Fourier transform of quasi-Lorentzian functions. We relate these results to sojourn time estimates and spectral concentration phenomena
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For a nonnegative self-adjoint operator $A_0$ acting on a Hilbert space $mathfrak{H}$ singular perturbations of the form $A_0+V, V=sum_{1}^{n}{b}_{ij}<psi_j,cdot>psi_i$ are studied under some additional requirements of symmetry imposed on the initial operator $A_0$ and the singular elements $psi_j$. A concept of symmetry is defined by means of a one-parameter family of unitary operators $sU$ that is motivated by results due to R. S. Phillips. The abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials $V$ and the corresponding self-adjoint realizations of $A_0+V$. The results are applied for the investigation of singular perturbations of the Schr{o}dinger operator in $L_2(dR^3)$ and for the study of a (fractional) textsf{p}-adic Schr{o}dinger type operator with point interactions.
Let $A$ be a self-adjoint operator on a Hilbert space $fH$. Assume that the spectrum of $A$ consists of two disjoint components $sigma_0$ and $sigma_1$. Let $V$ be a bounded operator on $fH$, off-diagonal and $J$-self-adjoint with respect to the orthogonal decomposition $fH=fH_0oplusfH_1$ where $fH_0$ and $fH_1$ are the spectral subspaces of $A$ associated with the spectral sets $sigma_0$ and $sigma_1$, respectively. We find (optimal) conditions on $V$ guaranteeing that the perturbed operator $L=A+V$ is similar to a self-adjoint operator. Moreover, we prove a number of (sharp) norm bounds on variation of the spectral subspaces of $A$ under the perturbation $V$. Some of the results obtained are reformulated in terms of the Krein space theory. As an example, the quantum harmonic oscillator under a PT-symmetric perturbation is discussed.
We show that for any bounded operator $T$ acting on an infinite dimensional Banach space there exists an operator $F$ of rank at most one such that $T+F$ has an invariant subspace of infinite dimension and codimension. We also show that whenever the boundary of the spectrum of $T$ or $T^*$ does not consist entirely of eigenvalues, we can find such rank one perturbations that have arbitrarily small norm. When this spectral condition is not satisfied, we can still find suitable finite rank perturbations of arbitrarily small norm, but not necessarily of rank one.
We show that for any bounded operator $T$ acting on infinite dimensional, complex Banach space, and for any $varepsilon>0$, there exists an operator $F$ of rank at most one and norm smaller than $varepsilon$ such that $T+F$ has an invariant subspace of infinite dimension and codimension. A version of this result was proved in cite{T19} under additional spectral conditions for $T$ or $T^*$. This solves in full generality the quantitative version of the invariant subspace problem for rank-one perturbations.
In this article we consider asymptotics for the spectral function of Schrodinger operators on the real line. Let $P:L^2(mathbb{R})to L^2(mathbb{R})$ have the form $$ P:=-tfrac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order differential operator with certain modified almost periodic structure. We show that the kernel of the spectral projector, $mathbb{1}_{(-infty,lambda^2]}(P)$ has a full asymptotic expansion in powers of $lambda$. In particular, our class of potentials $W$ is stable under perturbation by formally self-adjoint first order differential operators with smooth, compactly supported coefficients. Moreover, it includes certain potentials with dense pure point spectrum. The proof combines the gauge transform methods of Parnovski-Shterenberg and Sobolev with Melroses scattering calculus.
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