Elliptic equations in hilbert space and associated spectral problems

Problems are formulated for abstract higher-order elliptic equations on the semiaxis and on a finite interval and general theorems for the Fredholm solvability and exact solvability of these equations given emission conditions to infinity are proved. A classification of the real spectrum of the pencil associated with the equation is presented, and possible rules for rigorous selection of the segment of its eigenelements and associated elements formulated. Completeness, minimality, and the basis property of the fundamental solutions of the equation in the solution space, along with the properties of the derivative chains of the eigenelements and associated elements of the pencil that correspond to problems on the semiaxis and on a finite interval are studied.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 140–224, 1989.     

Bounds on Variation of Spectral Subspaces under J-Self-adjoint Perturbations

Let A be a self-adjoint operator on a Hilbert space . Assume that the spectrum of A consists of two disjoint components σ₀ and σ₁. Let V be a bounded operator on , off-diagonal and J-self-adjoint with respect to the orthogonal decomposition where and are the spectral subspaces of A associated with the spectral sets σ₀ and σ₁, 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 the 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 -symmetric perturbation is discussed. This work was supported by the Deutsche Forschungsgemeinschaft (DFG), the Heisenberg-Landau Program, and the Russian Foundation for Basic Research.     

A system of functions

A system of functions $$f_k (x) = \sum\nolimits_{i = 1}^r a _i \varphi _\iota (x)^k + b_i \overline {\varphi _\iota } (x)^k , k = 1,2,...$$ is considered on the interval [0,l]. Under certain conditions on the? _i(x), it is proved that the system 1 ∪ {f_k(x)} _k=1 ^∞ is complete in the space L_p(0,l). In the case r=1 it is proved, under certain additional assumptions, that the system {f_k(x)} _k=0 ^∞ is minimal.     

An operator model for the oscillation problem of liquids on an elastic bottom

This paper deals with the problem of small oscillations in a liquid layer of finite depth under the assumption that the bottom is an elastic medium. The system of equations corresponding to the problem is written out and explained. The main aim of the paper is to recast these equations in the form

On the interpolation of analytic mappings

Let (E ₀,E ₁) and (H ₀,H ₁) be two pairs of complex Banach spaces densely and continuously embedded into each other, E ₁ ? E ₀ and H ₁ ? H ₀ and also let $\left\| x \right\|_{E_0 } \leqslant \left\| x \right\|_{E_1 } $ . By E _θ = [E ₀, E ₁]_θ and H _θ = [H ₀, H ₁]_θ we denote the spaces obtained by the complex interpolation method for θ ∈ [0, 1], and by B _θ(0,R) we denote an open ball of radius R in the space E _θ. Let Φ: B ₀(0,R) → H ₀ be an analytic mapping taking B ₁(0,R) into H ₁, and let the estimates $\left\| {\Phi (x)} \right\|_{H_\theta } \leqslant C_\theta \left\| x \right\|_{H_\theta } for allx \in B_\theta (0,R)$ hold for θ = 0, 1. Then, for all θ ∈ (0, 1), the mapping Φ takes the ball B _θ(0,r) of radius r ∈ (0,R) in the space E _θ into H _θ and $\left\| {\Phi (x)} \right\|_{H_\theta } \leqslant C_0^{1 - \theta } C_1^\theta \frac{R} {{R - r}}\left\| x \right\|_{E_\theta } ,x \in B_\theta (0,r). $ .     

Inverse problems for Sturm—Liouville operators with potentials in Sobolev spaces: Uniform stability

Two inverse problems for the Sturm-Liouville operator Ly = s-y″ + q(x)y on the interval [0, fy] are studied. For θ ⩾ 0, there is a mapping F:W ₂^θ → l _B ^θ, F(σ) = {s _k }₁^∞, related to the first of these problems, where W ₂^∞ = W ₂^∞[0, π] is the Sobolev space, σ = ∫ q is a primitive of the potential q, and l _B ^θ is a specially constructed finite-dimensional extension of the weighted space l ₂^θ, where we place the regularized spectral data s = {s _k }₁^∞ in the problem of reconstruction from two spectra. The main result is uniform lower and upper bounds for ∥σ - σ₁∥_θ via the l _B ^θ-norm ∥s − s₁∥_θ of the difference of regularized spectral data. A similar result is obtained for the second inverse problem, that is, the problem of reconstructing the potential from the spectral function of the operator L generated by the Dirichlet boundary conditions. The result is new even for the classical case q ∈ L ₂, which corresponds to θ = 1.     

Spectral Decomposition of Symmetric Operator Matrices

The authors study symmetric operator matrices in the product of Hilbert spaces H = H₁×H₂, where the entries are not necessarily bounded operators. Under suitable assumptions the closure L_o exists and is a selfadjoint operator in H. With L_o, the closure of the transfer function is considered. Under the assumption that there exists a real number β < inf p(A) such that M(β)<< 0, it follows that β ε p(L_o). Applying a factorization result of A.I. Virozub and V.I. Matsaev [VM] to the holomorphic operator function M(λ, the_spectral subspaces of L_o corresponding to the intervals ] — ∞, β] and [β, ∞[ and the restrictions of L_o to these subspaces are characterized. Similar results are proved for operator matrices which are symmetric in a Krein space.