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In this paper, we prove the -boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent parameter , where , and our boundary conditions are nonhomogeneous Neumann on upper boundary and Dirichlet on lower boundary. We want to emphasize that we can choose 0 0 arbitrarily, although usual parabolic theorem tells us that we must choose a large γ0 > 0 for given 0

We prove resolvent estimates for self-adjoint operators of the form on , , where is a semi-classical parameter and , , is a real-valued potential. The potential is supposed to have very little regularity with respect to the radial variable, only. As a consequence, we obtain a region free of resonances in the case when V is of compact support.

In this paper we consider a boundary problem for a parameter-elliptic, multi-order system of differential equations defined over a bounded region in and under limited smoothness assumptions as well as under boundary conditions which include those of Dirichlet. Information is then derived concerning the asymptotic behaviour of the trace of a power of the resolvent of the Hilbert space operator, in general non-selfadjoint, induced by the boundary problem under null boundary conditions. This information will then be used in a subsequent work to derive various results pertaining to the asymptotic behaviour of the eigenvalues of this operator.

In this paper we consider a resolvent problem of the Stokes operator with some boundary condition in the half space, which is obtained as a model problem arising in evolution free boundary problems for viscous, incompressible fluid flow. We show standard resolvent estimates in the Lq framework (1 < q < ∞), applying some kernel estimates to concrete solution formulas. The Volevich trick in [21] plays a fundamental role in estimating solutions (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Let H and T: H H denote a separable Hilbert space and an operator in a Schatten-von Neumann ideal Sp(H), respectively. Consider the resolvent operator (λI - T)−1, where I is the identity operator and λ belongs to the resolvent set of T. Some sharp bounds for the uniform operator norm of (λI - T)−1 are derived in some situations of particular interest for certain applications, namely when only partial or minimal information about the spectrum is available. The results obtained may also be regarded as generalizations of Carleman's inequality for quasinilpotent operators.

This paper is concerned with the standard Lp estimate of solutions to the resolvent problem for the Stokes operator on an infinite layer with ‘Neumann–Dirichlet-type’ boundary condition. Copyright © 2004 John Wiley & Sons, Ltd.

We investigate analytical properties of a measure geometric Laplacian which is given as the second derivative w.r.t. two atomless finite Borel measures μ and ν with compact supports supp μ ⊂ supp ν on the real line. This class of operators includes a generalization of the well-known Sturm-Liouville operator as well as of the measure geometric Laplacian given by . We obtain for this differential operator under both Dirichlet and Neumann boundary conditions similar properties as known in the classical Lebesgue case for the euclidean Laplacian like Gaus-Green-formula, inversion formula, compactness of the resolvent and its kernel representation w.r.t. the corresponding Green function. Finally we prove nuclearity of the resolvent and give two representations of its trace.

In this paper we are concerned with the complete spectral analysis for the operator = in the space Lp() ( denoting the unit circle), where is the characteristic function of some arc of , is the singular integral operator with Cauchy kernel and is a Carleman shift operator which satisfies the relations 2 = I and = ±, where the sign + or — is taken in dependence on whether is a shift operator on Lp() preserving or changing the orientation of . This includes the identification of the Fredholm and essential parts of the spectrum of the operator , the determination of the defect numbers of — λI, for λ in the Fredholm part of the spectrum, as well as a formula for the resolvent operator.

Mourre method of commutators is used to get low energy resolvent bound for an abstract operatorin a Hilbert space and for a second order variable coefficient elliptic operator in Rd, d⩾3. Copyright © 2002 John Wiley & Sons, Ltd.

In this note we study the connection between the spectra of the products AB and BA of unbounded closed operators A and B acting in Banach spaces. Under the condition that the resolvent sets of these products are not empty we show that the spectra of AB and BA coincide away from zero and prove the commutation relation . Further, we prove statements concerning the relationship between the spectra of the operator AB and the block operator matrix .

Considered is the problem of adjusting a positive definite operator in order to obtain a partially zero inverse. Using generalized determinants the problem is settled for operator matrices with Hilbert Schmidt off diagonal entries. For Fredholm integral operators an approximation result is obtained.

Let (sj)∞j=0 be a sequence of real numbers such that the Hankel matrices (si+j)∞0, (Si+j+i)∞0 have finite numbers of negative eigenvalues. The indefinite moment problem with the moments Sj (j = 0,1,2, …) and the corresponding Stieltjes string are investigated. We use the approach via the Krein — Langer extension theory of symmetric operators in spaces with indefinite metric. In the framework of this approach a description of L– resolvents of a class of symmetric operators in Krein space and a simple formula for the calculation of the L– resolvent matrix in terms of boundary operators are given.

Some elementary properties of a general linear operator, which is defined on a Hilbert space but is not necessarily self-adjoint, are briefly reviewed: the classical canonical form, the bi-orthogonality theorem, the bivariational principle, the spectral resolution of the resolvent, and the properties of exponential forms

Standard Lippmann–Schwinger theory does not apply to decay of metastable or unstable states due to failure of the adiabatic hypothesis. In the Fock–Tani representation, discrete unstable states can be described by state vectors orthogonal to the asymptotic states representing their decay fragments, and a decay/formation interaction is exhibited explicitly as a portion of the total interaction Hamiltonian. This allows a straightforward derivation of a generalized Lippmann–Schwinger “half-collision” differential decay cross section without the need of projection operators. It reduces in first order to the product of a resonance line shape by a Golden Rule matrix element squared. In the general case the line shape is non-Lorentzian and the matrix element factor contains final-state interaction contributions. The exact expression is expected to be applicable to a variety of processes such as predissociation, autoionization, and Auger effect. The derivation employs the van Hove resolvent formalism to exhibit the dependence of the cross section on the complex self energy of the decaying state.

The Slater–Koster resolvent formalism of exciton theory, as proposed originally by Takeuti, has been applied to calculate charge transfer exciton states and to investigate hypochromism in polynucleotides. As a first step, spatially well localized ab initio Wannier functions (WFS) are calculated at the Hartree–Fock level using a two-phase (inter- and intramolecular) localization procedure for the Fourier transformation of the Bloch functions. The single particle energies, entering the Green's function of the polymer, are corrected for electron correlation effects with the help of second order Moller–Plesset (MP) perturbation theory. The interelectronic matrix elements, used in the MP calculation as well as in solving the resolvent problem for the excitons, are calculated in terms of the WFS. Singlet- and triplet-excitonic dispersions, oscillator strengths, the possible affects of ions, hydration, and aperiodicity on the exciton spectrum are discussed.

A perturbation approach based on resolvent technique and Pade approximants is proposed. The eigenvalue of interest is part of a solution of two nonlinear algebraic equations. The nonlinear equations are arrived at by considering two different expression of the expectation value of the resolvent of an outer projection of the Hamiltonian. The first expression is based on the spectral resolution of the resolvent, and the second one is obtained by a power series expansion analogous to that applied in the derivation of the energy expression in the Brillouin–Wigner perturbation theory. The truncated power series is extrapolated by Pade approximants of type II. The method is tested on a CI calculation of the energy of the lowest 1Σ state of the B2 molecule.

If a quantummechanical potential is introduced the calculation of the quantummechanical binary distribution function for a system with Coulomb interaction is reduced to the well known mathematical formalism of classical mechanics in the case nλ3 ≪ 1. The two-particle quantummechanical potential is determined by the two particle Slater sum. In this paper we calculated the two particle Slater sum using a development according to e2 and the resolvent formalism. From the binary distribution function the correlation energy as well as the free energy were determined up to the order e4. Symmetry effects were taken into account.