arXiv subjects
Publications and source records attributed to O. A. Veliev.
In this paper we find conditions on the PT-symmetric non-self-adjoint Dirac operator L(Q) with matrix-valued coefficients Q whose entries are PT-symmetric periodic functions for which the number of gaps in the real part of its spectrum is finite.
In this paper, we consider the spectrum of the one dimensional non-self-adjoint Dirac operator with matrix-valued coefficients whose entries are PT-symmetric periodic functions. We prove that the spectrum of the considered operator contains a large part of the real axis.
In this paper, we study the spectrality of the non-self-adjoint Dirac operator L(Q) with a complex-valued periodic matrix potential Q. We establish a condition on the off-diagonal elements of the matrix Q under which L(Q) is an asymptotically spectral operator. Moreover, we derive a condition on Q that ensures the spectrality of this operator. Finally, we consider the spectral expansion in these cases.
In this paper, we construct the spectral expansion for the one dimensional non-self-adjoint Dirac operator L(Q) with a complex-valued periodic matrix potential Q. To this end, we study in detail asymptotic formulas for the Bloch eigenvalues and Bloch functions that are uniform with respect to the complex quasimomentum, as well as the essential spectral singularities of L(Q).
In this paper, we briefly explain the spectral expansion problem for differential operators defined on the entire real line, generated by a differential expression with periodic, complex-valued coefficients.
In this paper, we establish a condition on the coefficients of differential operators generated in the space of square-integrable functions on the entire real line by an ordinary differential expression with periodic, complex-valued coefficients, under which the operator is a spectral operator in the sense of Dunford [1].
In this paper we consider the one-dimensional Schrodinger operator L(q) with a periodic real and locally integrable potential q. We study the bands and gaps in the spectrum and explicitly write out the first and second terms of the asymptotic formulas for the length of the gaps in the spectrum.
In this paper we consider the Bloch eigenvalues and spectrum of the non-self-adjoint differential operator L generated by the differential expression of odd order n with the periodic PT-symmetric coefficients, where n>1. We study the localizations of the Bloch eigenvalues and the structure of the spectrum. Moreover, we find conditions on the norm of the coefficients under which the spectrum of L coincides with the real line
We study the spectrum of the differential operator T generated by the differential expression of order n>2 with the m by m PT-symmetric periodic matrix coefficients. The case when m and n are the odd numbers was investigated in [8]. In this paper, we consider the all remained cases: (a) n is an odd number and m is an even number, (b) n is an even number and m is an arbitrary positive integer. We find conditions on the coefficients under which the spectrum of T in cases (a) and (b) contains respectively large real and positive numbers.
In this paper we investigate the spectrum of the differential operators generated by the ordinary differential expression of odd order with PT-symmertic periodic matrix coefficients
In this paper, we consider the band functions, Bloch functions and spectrum of the self-adjoint differential operator L with periodic matrix coefficients. Conditions are found for the coefficients under which the number of gaps in the spectrum of the operator L is finite
In this paper we consider the continuity of the band functions and Bloch functions of the differential operators generated by the differential expressions with periodic matrix coefficients.
In this paper we consider the spectrum of the self-adjoint differential operator L generated by the differential expression of order n with the m by m periodic matrix coefficients, where n and m are respectively odd and even integers and n>1. We prove that the number of gaps in the spectrum of L is finite and find explicit estimation in term of coefficients for the number of the gaps. Moreover, we find a condition on the norms of the coefficients for which the spectrum is real axis. Besides we investigate the bands of the spectrum and prove that most of the real axis is overlapped by m bands.
In this article we consider the one-dimensional Schrodinger operator L(Q) with a Hermitian periodic m by m matrix potential Q. We investigate the bands and gaps of the spectrum and prove that the main part of the positive real axis is overlapped by m bands. Moreover, we find a condition on the potential Q for which the number of gaps in the spectrum of L(Q) is finite.
In this paper we construct the spectral expansion for the non-self-adjoint differential operators generated in the space of vektor functions by the ordinary differential expression of arbitrary order with the periodic matrix coefficients by using the essential spectral singularities, singular quasimomenta and the series with parenthesis.
In this article we obtain asymptotic formulas for the Bloch eigenvalues of the operator generated by a system of Schrodinger equations with periodic PT-symmetric complex-valued coefficients. Then using these formulas we classify the spectrum of this operator and find a condition on the coefficients for which the spectrum contains a half line.
In this paper we give a complete description of the spectral analysis of the Schrodinger operator L(V) with the optical potentil. First we consider the Bolch eigenvalues and spectrum of L(V). Then using it we investigate spectral singularities and essential spectral singularities (ESS). We prove that the operator L(V) has no ESS and has ESS respectively if and only if V is not a critical point and V is a critical point. Using it we classify the spectral expansion in term of the critical points. Finally we discuss the critical points, formulate some conjectures and describe the changes of the spectrum of L(V) when V changes.
In this paper we investigate the non-self-adjoint operator H generated in all real line by the Mathieu-Hill equation with a complex-valued potential. We find a necessary and sufficient conditions on the potential for which H has no spectral singularity at infinity and it is an asymptotically spectral operator. Moreover, we give a detailed classification, stated in term of the potential, for the form of the spectral decomposition of the operator H by investigating the essential spectral singularities.