Spectral Analysis of One-Dimensional Dirac System with Summable Potential and Sturm- Liouville Operator with Distribution Coefficients

We consider one-dimensional Dirac operatorLP,U with Birkhoff regular boundary conditions and summable potential P(x) on[0, ]. We introduce strongly and weakly regular operators. In both cases, asymptotic formulas for eigenvalues are found. In these formulas, we obtain main asymptotic terms and estimates for the second term. We specify these estimates depending on the functional class of the potential: Lp[0,] with p [1,2] and the Besov space Bp,p'[0,] with p [1,2] and (0,1/p). Additionally, we prove that our estimates are uniform on balls Pp,R Then we get asymptotic formulas for normalized eigenfunctions in the strongly regular case with the same residue estimates in uniform metric on x [0,]. In the weakly regular case, the eigenvalues 2n and 2n+1 are asymptotically close and we obtain similar estimates for two-dimensional Riesz projectors. Next, we prove the Riesz basis property in the space (L2[0,])2 for a system of eigenfunctions and associated functions of an arbitrary strongly regular operatorLP,U. In case of weak regularity, the Riesz basicity of two-dimensional subspaces is proved. In parallel with theLP,U operator, we consider the SturmLiouville operator Lq,U generated by the differential -y'' + q(x)y expressionwith distribution potential q of first-order singularity (i.e., we assume that the primitive u = q(1) belongs to L2[0, ]) and Birkhoff-regular boundary conditions. We reduce to this case -(1y')'+i(y)'+iy'+0y, operators of more general form where '1,,0(-1)L2and 10. For operator Lq,U, we get the same results on the asymptotics of eigenvalues, eigenfunctions, and basicity as for operator LP,U . Then, for the Dirac operator LP,U, we prove that the Riesz basis constant is uniform over the ballsPp,R for p1 or 0. The problem of conditional basicity is naturally generalized to the problem of equiconvergence of spectral decompositions in various metrics. We prove the result on equiconvergence by varying three indices: fL[0,] (decomposable function), PL[0,] (potential), and Sm-Sm00,m in L[0,] (equiconvergence of spectral decompositions in the corresponding norm). In conclusion, we prove theorems on conditional and unconditional basicity of the system of eigenfunctions and associated functions of operator LP,U in the spaces L[0,],2, and in various Besov spaces Bp,q[0,].

Solving the Boundary Value Problems for Differential Equations with Fractional Derivatives by the Method of Separation of Variables

This paper is devoted to solving boundary value problems for differential equations with fractional derivatives by the Fourier method. The necessary information is given (in particular, theorems on the completeness of the eigenfunctions and associated functions, multiplicity of eigenvalues, and questions of the localization of root functions and eigenvalues are discussed) from the spectral theory of non-self-adjoint operators generated by differential equations with fractional derivatives and boundary conditions of the Sturm–Liouville type, obtained by the author during implementation of the method of separation of variables (Fourier). Solutions of boundary value problems for a fractional diffusion equation and wave equation with a fractional derivative are presented with respect to a spatial variable.

Completeness Theorem for Eigenparameter Dependent Dissipative Dirac Operator with General Transfer Conditions

This paper deals with a singular (Weyl’s limit circle case) non-self-adjoint (dissipative) Dirac operator with eigenparameter dependent boundary condition and finite general transfer conditions. Using the equivalence between Lax-Phillips scattering matrix and Sz.-Nagy-Foiaş characteristic function, the completeness of the eigenfunctions and associated functions of this dissipative operator is discussed.

On One Problems of Spectral Theory for Ordinary Differential Equations of Fractional Order

The present paper is devoted to the spectral analysis of operators induced by fractional differential equations and boundary conditions of Sturm-Liouville type. It should be noted that these operators are non-self-adjoint. The spectral structure of such operators has been insufficiently explored. In particular, a study of the completeness of systems of eigenfunctions and associated functions has begun relatively recently. In this paper, the completeness of the system of eigenfunctions and associated functions of one class of non-self-adjoint integral operators corresponding boundary value problems for fractional differential equations is established. The proof is based on the well-known Theorem of M.S. Livshits on the spectral decomposition of linear non-self-adjoint operators, as well as on the sectoriality of the fractional differentiation operator. The results of Dzhrbashian-Nersesian on the asymptotics of the zeros of the Mittag-Leffler function are used.

On Stability of Basis Property of Root Vectors System of the Sturm-Liouville Operator with an Integral Perturbation of Conditions in Nonstrongly Regular Samarskii-Ionkin Type Problems

We study a question on stability and instability of basis property of system of eigenfunctions and associated functions of the double differentiation operator with an integral perturbation of Samarskii-Ionkin type boundary conditions.

Basis Properties of Eigenfunctions of Second-Order Differential Operators with Involution

We study the basis properties of systems of eigenfunctions and associated functions for one kind of generalized spectral problems for a second-order ordinary differential operator.