Krein's Resolvent Formula Research Articles

Asymptotics of the Trace of the Resolvent of the Laplace Operator on a Metric Graph with Arbitrary Boundary Conditions

We obtain an asymptotic formula for the trace of the resolvent of the Laplacian with arbitrary boundary conditions using the Krein resolvent formula.

Dirac–Krein Systems on Star Graphs

We study the spectrum of a self-adjoint Dirac-Krein operator with potential on a compact star graph $\mathcal G$ with a finite number $n$ of edges. This operator is defined by a Dirac-Krein differential expression with summable matrix potentials on each edge, by self-adjoint boundary conditions at the outer vertices, and by a self-adjoint matching condition at the common central vertex of $\mathcal G$. Special attention is paid to Robin matching conditions with parameter $\tau \in\mathbb R\cup\{\infty\}$. Choosing the decoupled operator with Dirichlet condition at the central vertex as a reference operator, we derive Krein's resolvent formula, introduce corresponding Weyl-Titchmarsh functions, study the multiplicities, dependence on $\tau$, and interlacing properties of the eigenvalues, and prove a trace formula. Moreover, we show that, asymptotically for $R\to \infty$, the difference of the number of eigenvalues in the intervals $[0,R)$ and $[-R,0)$ deviates from some integer $\kappa_0$, which we call dislocation index, at most by $n+2$.

On the Stokes flow computation algorithm based on woodbury formula

ow in nanostructures. An algorithm for Stokes ow computation in cases when there is great variation in the viscosity over a small spatial region is described. This method allows us to overcome computational diculties of the nite-dierence method. The background of the approach is using the Woodbury formula - a discrete analog of the Krein resolvent formula. The particular example of a rectangular domain is considered in detail. The inversion of the discrete Stokes operator is made in analytic form for the case of constant viscosity.

Spectral Asymptotics for Nonsmooth Singular Green Operators

Singular Green operators G appear typically as boundary correction terms in resolvents for elliptic boundary value problems on a domain Ω ⊂ ℝ n , and more generally they appear in the calculus of pseudodifferential boundary problems. In particular, the boundary term in a Krein resolvent formula is a singular Green operator. It is well-known in smooth cases that when G is of negative order −t on a bounded domain, its eigenvalues or s-numbers have the behavior (*)s j (G) ∼ cj −t/(n−1) for j → ∞, governed by the boundary dimension n − 1. In some nonsmooth cases, upper estimates (**)s j (G) ≤ Cj −t/(n−1) are known. We show that (*) holds when G is a general selfadjoint nonnegative singular Green operator with symbol merely Hölder continuous in x. We also show (*) with t = 2 for the boundary term in the Krein resolvent formula comparing the Dirichlet and a Neumann-type problem for a strongly elliptic second-order differential operator (not necessarily selfadjoint) with coefficients in for some q > n.

Boundary data maps and Krein's resolvent formula for Sturm-Liouville operators on a finite interval

We continue the study of boundary data maps, that is, generalizations of spectral parameter dependent Dirichlet-to-Neumann maps for (three-coefficient) Sturm–Liouville operators on the finite interval (a,b) , to more general boundary conditions, began in [8] and [17]. While these earlier studies of boundary data maps focused on the case of general separated boundary conditions at a and b , the present work develops a unified treatment for all possible self-adjoint boundary conditions (i.e., separated as well as non-separated ones). In the course of this paper we describe the connections with Krein’s resolvent formula for self-adjoint extensions of the underlying minimal Sturm–Liouville operator (parametrized in terms of boundary conditions), with some emphasis on the Krein extension, develop the basic trace formulas for resolvent differences of self-adjoint extensions, especially, in terms of the associated spectral shift functions, and describe the connections between various parametrizations of all self-adjoint extensions, including the precise relation to von Neumann’s basic parametrization in terms of unitary maps between deficiency subspaces. Mathematics subject classification (2010): Primary 34B05, 34B27, 34L40; Secondary 34B20, 34L05, 47A10, 47E05.

Krein's resolvent formula for self-adjoint extensions of symmetric second-order elliptic differential operators

Given a symmetric, semi-bounded, second-order elliptic differential operator A on a bounded domain with C1,1 boundary, we provide a Kreĭn-type formula for the resolvent difference between its Friedrichs extension and an arbitrary self-adjoint one.

Resolvents of self-adjoint extensions with mixed boundary conditions

We prove a variant of Krein's resolvent formula expressing the resolvents of self-adjoint extensions through the associated boundary conditions. Applications to solvable quantum-mechanical problems are discussed.

A remark on Krein's resolvent formula and boundary conditions

We prove an analog of Krein's resolvent formula expressing the resolvents of self-adjoint extensions in terms of boundary conditions. Applications to quantum graphs and systems with point interactions are discussed.

On Krein's formula in indefinite metric spaces

In this paper we extend some of the recent results in connection with the Krein resolvent formula which provides a complete description of all canonical resolvents and utilizes Weyl–Titchmarsh functions in the spaces with indefinite metrics. We show that coefficients in Krein's formula can be expressed in terms of analogues of the von Neumann parametrization formulas in the indefinite case. We consider properties of Weyl–Titchmarsh functions and show that two Weyl–Titchmarsh functions corresponding to π-self-adjoint extensions of a densely defined π-symmetric operator are connected via linear-fractional transformation with the coefficients presented in terms of von Neumann's parameters.

On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank

We discuss purely singular finite-rank perturbations of a self-adjoint operator A in a Hilbert space ℋ. The perturbed operators \(\tilde A\) are defined by the Krein resolvent formula \((\tilde A - z)^{ - 1} = (A - z)^{ - 1} + B_z \), Im z ≠ 0, where Bz are finite-rank operators such that dom Bz ∩ dom A = |0}. For an arbitrary system of orthonormal vectors \(\{ \psi _i \} _{i = 1}^{n < \infty } \) satisfying the condition span |ψ i } ∩ dom A = |0} and an arbitrary collection of real numbers \({\lambda}_i \in {\mathbb{R}}^1\), we construct an operator \(\tilde A\) that solves the eigenvalue problem \(\tilde A\psi _i = {\lambda}_i {\psi}_i , i = 1, \ldots ,n\). We prove the uniqueness of \(\tilde A\) under the condition that rank Bz = n.

$ mathcal{H} $ -n -perturbations of Self-adjoint Operators and Krein's Resolvent Formula

Supersingular H-n rank one perturbations of an arbitrary positive self-adjoint operator A acting in the Hilbert space H are investigated. The operator corresponding to the formal expression A(alpha) = A + alpha(phi,.)phi, alpha is an element of R, phi is an element of H-n (A), is determined as a regular operator with pure real spectrum acting in a certain extended Hilbert space H superset of X The resolvent of the operator so defined is given by a certain generalization of Krein's resolvent formula. It is proven that the spectral properties of the operator are described by generalized Nevanlinna functions. The results of [24] are extended to the case of arbitrary integer n greater than or equal to 4.

On Krein's Formula in the Case of Non-densely Defined Symmetric Operators

In this paper we provide some additional results related to Krein's resolvent formula for a non-densely defined symmetric operator. We show that coefficients in Krein's formula can be expressed in terms of analogues of the classical von Neumann formulas. The relationship between two Weyl–Tichmarsh m-functions corresponding to self-adjoint extensions of a non-densely defined symmetric operator is established.