Spectrum of a non-self-adjoint operator associated with the periodic heat equation

We study the spectrum of the linear operator L=−θ_∂−?θ_∂(sinθθ_∂) subject to the periodic boundary conditions on θ∈[−π,π]. We prove that the operator is closed in with the domain in for |?|<2, its spectrum consists of an infinite sequence of isolated eigenvalues and the set of corresponding eigenfunctions is complete. By using numerical approximations of eigenvalues and eigenfunctions, we show that all eigenvalues are simple, located on the imaginary axis and the angle between two subsequent eigenfunctions tends to zero for larger eigenvalues. As a result, the complete set of linearly independent eigenfunctions does not form a basis in .     

Singular spectrum of a non-self-adjoint operator

Journal of Mathematical Sciences - The study of the spectral structure of nondissipative operators in Hilbert space, started in the previous papers of the author, is continued. In a model...     

Singular decomposition of a differential operator on a semiaxis

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 6, pp. 34–48, November–December, 1993.     

Isometries of non-self-adjoint operator algebras

We study isometries of certain non-self-adjoint operator algebras by means of the structure of the complete holomorphic vector fields on their unit balls and the associated partial Jordan triple products. We show that isometries of nest sub-algebras of B(H) are of the form T UTW or T UJT^*JW, where U, W are suitable unitary operators and J a fixed involution of H.     

Absolutely continuous and singular subspaces of a non-self-adjoint operator

For a non-self-adjoint operator with a characteristic function that has boundary values almost everywhere on the real axis, we consider some problems concerned with local absolutely continuous and singular subspaces. Bibliography: 39 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 222, 1995, pp. 163–202.     

Quantized Riemann surfaces and semiclassical spectral series for a non-self-adjoint Schrödinger operator with periodic coefficients

We consider a non-self-adjoint Schrödinger operator describing the motion of a particle in a one-dimensional space with an analytic potential iV (x) that is periodic with a real period T and is purely imaginary on the real axis. We study the spectrum of this operator in the semiclassical limit and show that the points of its spectrum asymptotically belong to the so-called spectral graph. We construct the spectral graph and evaluate the asymptotic form of the spectrum. A Riemann surface of the particle energy-conservation equation can be constructed in the phase space. We show that both the spectral graph and the asymptotic form of the spectrum can be evaluated in terms of integrals of the pdx form (where x ∈31 ?/T? and p ∈, ? are the particle coordinate and momentum) taken along basis cycles on this Riemann surface. We use the technique of Stokes lines to construct the asymptotic form of the spectrum.     

Faddeev differential equations as a spectral problem for a nonsymmetric operator

We consider a nonsymmetric matrix operator whose eigenvalue problem is the system of Faddeev differential equations for a three-particle system. For this operator and its adjoint, the resolvents are represented in terms of Faddeev T-matrix components of the three-particle Schrödinger operator. On the basis of these representations, the invariant spaces of the operators under consideration are investigated and their eigenfunctions are determined. The biorthogonality and completeness of the eigenfunction system are proved.We dedicate this paper to the memory of Stanislav Petrovitch Merkuriev, who left us three years ago.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 3, pp. 513–528, June, 1996.     

The dynamical inverse problem for a non-self-adjoint sturm-liouville operator

An approach to the inverse problem (the so-called BC-method) based on boundary-control theory is developed. A procedure of reconstructing a nonsymmetric matrix-function (a potential) given on a semiaxis by a dynamical response operator is described. The results of numerical tests are presented. Bibliography: 6 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 7–21. Translated by T. N. Surkova.     

An inverse spectral problem for a nonnormal first order differential operator

We study the eigenvalues of two restrictions ofB _x +P whereB is the two-by-two matrix that is zero on the diagonal and one off the diagonal andP is a two-by-two matrix of Lipschitz functions on the unit interval. We establish asymptotic forms for their eigenvalues and associated root vectors and demonstrate that these root vectors constitute a Riesz basis inL ²(0, 1)². We show that our forward analysis makes rigorous the attack on the associated inverse problem by M. Yamamoto,Inverse spectral problem for systems of ordinary differential equations of first order, I, J. Fac. Sci. Univ. Tokyo, Sect. 1A, Math. 35, 1988, pp. 519–546. We apply these results to the recovery of the line resistance and leakage conductance of a nonuniform transmission line.Supported by NSF grant DMS-9258312.     

Spectrum of a linear fourth‐order differential operator and its applications

In this paper, we apply the disconjugacy theory and Elias's spectrum theory to study the positivity and the spectrum structure of the linear operator coupled with the clamped beam boundary conditions (1.2). We also study the positivity and the spectrum structure of the more general operator coupled with (1.2). As the applications of our results on positivity and spectrum of fourth‐order linear differential operators, we show the existence of nodal solutions for the corresponding nonlinear problems via Rabinowitz's global bifurcation theorem.     

On the spectral properties of a second-order differential operator with a matrix potential

By applying the method of similar operators to a second-order differential operator with a matrix potential and semiperiodic boundary conditions, we obtain asymptotic estimates for the weighted mean eigenvalue and spectral projections and prove the equiconvergence of spectral expansions.