Perturbation of the spectrum of a differential operator under a bounded perturbation of the potential

We consider a perturbation of a differential operator by a bounded measurable function. We prove that if a certain mean of this function has a limit, then the sequence of Cesaro means of the perturbation of the spectrum has the same limit.     

On the reconstruction of a convolution perturbation of the Sturm-Liouville operator from the spectrum

We consider the sum of the Sturm-Liouville operator and a convolution operator. We study the inverse problem of reconstructing the convolution operator from the spectrum. This problem is reduced to a nonlinear integral equation with a singularity. We prove the global solvability of this nonlinear equation, which permits one to show that the asymptotics of the spectrum is a necessary and sufficient condition for the solvability of the inverse problem. The proof is constructive.     

On the spectrum of the Schrödinger operator perturbed by a rapidly oscillating potential

We study the spectrum of the one-dimensional Schrödinger operator perturbed by a rapidly oscillating potential. The oscillation period is a small parameter. We find explicitly the essential spectrum and study the existence of the discrete spectrum. Complete asymptotic expansions of the eigenvalues and corresponding eigenfunctions are constructed.     

Stationary and periodic solutions of the operator Riccati equation under a random perturbation

Sufficient conditions are presented for the existence of stationary and periodic solutions of the operator Riccati equation under a random perturbation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 5, pp. 609–615, May, 1993.     

Asymptotics of spectrum under infinitesimally form-bounded perturbation

LetA1 be a selfadjoint operator with discrete spectrum and known distribution function of its spectrumN(r,A). SupposeB is a (nonselfadjoint) operator that is form-bounded with respect toA with relative bound zero. If in addition thenN(r,A+B)=N(r,A)(1+o(1)), whereA+B is the operator defined as form sum. The applications to the Schrödinger operator with polynomially growing potential and to the third boundary value problem for the second order elliptic operator are given.Research supported by the Israel Ministries of Science and Absorption     

Conditions under which an operator obtained as a perturbation of multiplication by the independent variable is spectral

Conditions are studied under which the perturbed operator of multiplication by the independent variable is spectral in the sense of Dunford and Bade.Translated from Matematicheskie Zametki, Vol. 13, No. 2, pp. 289–296, February, 1973.     

On perturbation theory for points of discrete spectrum of dissipative operator functions

We consider the operator function L(α, θ) = A(α) ? θR of two complex arguments, where A(α) is an analytic operator function defined in some neighborhood of a real point α ₀ ∈ ? and ranging in the space of bounded operators in a Hilbert space ?. We assume that A(α) is a dissipative operator for real α in a small neighborhood of the point α ₀ and R is a bounded positive operator; moreover, the point α ₀ is a normal eigenvalue of the operator function L(α, θ ₀) for some θ ₀ ∈ ?, and the number θ ₀ is a normal eigenvalue of the operator function L(α ₀ θ). We obtain analogs and generalizations of well-known results for self-adjoint operator functions A(α) on the decomposition of α- and θ-eigenvalues in neighborhoods of the points α ₀ and θ ₀, respectively, and on the representation of the corresponding eigenfunctions by series.     

Nonlinear homogeneous perturbation of the discrete spectrum of a self-adjoint operator and a new Constrained Saddle Point Theorem

We prove a Critical Point Theorem for C¹ functionals on the unit sphere of a separable Hilbert space H which improves a previous result of ours. This is applied in nonlinear eigenvalue theory to study the effect of suitably restricted homogeneous perturbations upon the discrete spectrum of a bounded self-adjoint operator in H.     

On the spectrum of difference Schrödinger operator with a fractal potential

Moscow Civil Engineering Institute. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 26, No. 2, pp. 77–79, April–June, 1992.     

Diminishing of the spectrum of an operator under extension

A negative answer is given to the second question posed on page 210 of the book Complex Analysis — 199 Research Problems, Lecture Notes in Math., No. 1043, Springer, Berlin (1984).Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 45, pp. 96–97, 1986.     

Resolvent estimates and spectrum of the Dirac operator with periodic potential

Some estimates are given of the norm of the resolvent of the Dirac operator on ann-dimensional torus (n 2) for complex values of the quasimomentum. It is shown that the spectrum of the periodic Dirac operator with potential 3$$ " align="middle" border="0"> , >3, is absolutely continuous.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 1, pp. 3–22, April, 1995.