Finiteness of the point spectrum of linear operators acting in a Banach space is investigated from point of view of perturbation theory. In the first part of the paper we present an abstract result based on analytical continuation of the resolvent function through continuous spectrum. In the second part, the abstract result is applied to differential operators which can be represented as a diff...

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...

The resolvent Krylov subspace method builds approximations to operator functions f(A) times a vector v. For the semigroup and related operator functions, this method is proved to possess the favorable property that the convergence is automatically faster when the vector v is smoother. The user of the method does not need to know the presented theory and alterations of the method are not necessa...

Saitō’s method has been applied successfully for measuring potentials with compact support in three dimensions. Also potentials have been reconstructed in the sense of distributions using a weak version of the method. Saitō’s method does not depend on the decay of the boundary value of the resolvent operator, but instead on certain Reimann-Lebesgue type properties of convolutions of the kernel ...

Abstract. In this paper we consider the time dependent one-dimensional Schrödinger equation with multiple Dirac delta potentials of different positive strengths. We prove that the classical dispersion property holds. The result is obtained in a more general setting of a Laplace operator on a tree with δ-coupling conditions at the vertices. The proof relies on a careful analysis of the propertie...

In this paper we show how to obtain decay estimates for the damped wave equation on a compact manifold without geometric control via knowledge of the dynamics near the un-damped set. We show that if replacing the damping term with a higher-order complex absorbing potential gives an operator enjoying polynomial resolvent bounds on the real axis, then the “resolvent” associated to our damped prob...

Following an old and simple idea of Takagi we propose a formula for computing the norm of a compact complex symmetric operator. This observation is applied to two concrete problems related to quantum mechanical systems. First, we give sharp estimates on the exponential decay of the resolvent and the single-particle density matrix for Schrödinger operators with spectral gaps. Second, we provide ...

Suppose that λ − T is left-invertible in L(H) for all λ ∈ Ω, where Ω is an open subset of the complex plane. Then an operator-valued function L(λ) is a left resolvent of T in Ω if and only if T has an extension T̃ , the resolvent of which is a dilation of L(λ) of a particular form. Generalized resolvents exist on every open set U , with U included in the regular domain of T . This implies a form...

We investigate the analytic perturbation of generalized inverses. Firstly we analyze the analytic perturbation of the Drazin generalized inverse (also known as reduced resolvent in operator theory). Our approach is based on spectral theory of linear operators as well as on a new notion of group reduced resolvent. It allows us to treat regular and singular perturbations in a unified framework. W...

We prove in this paper resolvent estimates for the boundary values of resolvents of selfadjoint operators on a Krein space: if H is a selfadjoint operator on a Krein space H, equipped with the Krein scalar product 〈·|·〉, A is the generator of a C0−group on H and I ⊂ R is an interval such that: 1) H admits a Borel functional calculus on I, 2) the spectral projection 1lI (H) is positive in the Kr...