Inverse spectral problems for Dirac operators with summable matrix-valued potentials

We consider the direct and inverse spectral problems for Dirac operators on $(0,1)$ with matrix-valued potentials whose entries belong to $L_p(0,1)$, $p\in[1,\infty)$. We give a complete description of the spectral data (eigenvalues and suitably introduced norming matrices) for the operators under consideration and suggest a method for reconstructing the potential from the corresponding spectral data.Comment: 32 page

Schrödinger operators with δ and δ′-potentials supported on hypersurfaces

Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity

Unique determination of a system by a part of the monodromy matrix

First-order ODE systems on a finite interval with nonsingular diagonal matrix B multiplying the derivative and integrable off-diagonal potential matrix Q are considered. It is proved that the matrix Q is uniquely determined by the monodromy matrix W(λ). In the case B = B*, the minimum number of matrix entries of W(λ) sufficient to uniquely determine Q is found

Transformation Operators for Fractional Order Ordinary Differential Equations and Their Applications

The survey is concerned with triangular transformation operators for fractional order α = n − ε ordinary differential equations. We discuss the existence of transformation operators in the case of holomorphic coefficients. Similarity between such operators and the simplest fractional differentiation D0α is discussed too. Applications to the unique determination of the operator from n spectra of boundary value problems are given. Applications to the completeness property of certain boundary value problems for such equations are considered. © 2020, Springer Nature Switzerland AG

О сингулярном спектре конечномерных возмущений (к теории Ароншайна-Донохью-Каца)

The main results of the Aronszajn-Donoghue-Kac theory are extended to the case of n-dimensional (in the resolvent sense) perturbations à of an operator A0 = A0* defined on a Hilbert space H. Applying technique of boundary triplets we describe singular continuous and point spectra of extensions AB of a symmetric operator A acting in H in terms of the Weyl function M(·) of the pair {A, A0} and boundary n-dimensional operator B = B*. Assuming that the multiplicity of singular spectrum of A0 is maximal it is established orthogonality of singular parts EsAв and EsAo of the spectral measures EAв and EAo of the operators AB and A0, respectively. It is shown that the multiplicity of singular spectrum of special extensions of direct sums A = A(1) ⊕ A(2) cannot be maximal as distinguished from multiplicity of the absolutely continuous spectrum. In particular, it is obtained a generalization of the Kac theorem on multiplicity of singular spectrum of Schrodinger operator on the line as well as its clarification. The multiplicity of singular spectrum of special extensions of direct sums A = A(1) ⊕ A(2) are investigated. In particular, it is shown that it cannot be maximal as distinguished from multiplicity of the absolutely continuous spectrum. This result generalizes the Kac theorem on multiplicity of singular spectrum of Schrodinger operator on the line and clarifies it.Основные результаты теории Ароншайна-Донохью-Каца распространяются на n-мерные (в резольвентном смысле) возмущения à = Ã* оператора A0 = A0* в гильбертовом пространстве H. Применяя технику граничных троек, мы описываем сингулярный непрерывный и точечный спектры расширений AB симметрического оператора A в терминах функции Вейля M(·) пары {A, A0} и граничного n-мерного оператора B = B*. Устанавливается ортогональность сингулярных частей ЕsАв и ЕsАо спектральных мер ЕАв и ЕАо операторов AB и A0 при условии, что кратность сингулярного спектра оператора A0 максимальна. Исследована кратность сингулярного спектра специальных расширений прямых сумм А = А(1) ⊕ А(2). В частности, показано, что она не может быть максимальной в отличие от кратности абсолютно непрерывного спектра. Этот результат обобщает теорему Каца о кратности сингулярного спектра оператора Шрёдингера на оси и уточняет её

On Singular Spectrum of Finite-Dimensional Perturbations (toward the Aronszajn–Donoghue–Kac Theory)

Abstract: The main results of the Aronszajn–Donoghue–Kac theory are extended to the case of n-dimensional (in the resolvent sense) perturbations A of an operator A0 = A0 * defined on a Hilbert space H. By applying the technique of boundary triplets, the singular continuous and point spectra of extensions AB of a symmetric operator A are described in terms of the Weyl function M(.) of the pair {A, A0} and an n-dimensional boundary operator B = B*. Assuming that the multiplicity of the singular spectrum of A0 is maximal, we establish the orthogonality of the singular parts EAB S and EA0 Sof the spectral measure EAB and EA0 of the operators AB and A0, respectively. The multiplicity of the singular spectrum of special extensions of direct sums A = A(1) + A(2) is investigated. In particular, it is shown that this multiplicity cannot be maximal, as distinguished from the multiplicity of the absolutely continuous spectrum. This result generalizes and refines the Kac theorem on the multiplicity of the singular spectrum of the Schrödinger operator on the line. © 2019, Pleiades Publishing, Ltd

Об однозначном определении системы по части матрицы монодромии

Рассматриваются системы дифференциальных уравнений первого порядка на конечном интервале с невырожденной диагональной матрицей B при производной и суммируемой потенциальной матрицей Q, имеющей нулевую диагональ. Доказывается, что потенциальная матрица Q однозначно определяется по матрице монодромии W(λ). В случае B=B∗ указано минимальное число элементов матрицы W(λ), достаточное для однозначного определения матрицы Q

Spectral theory of fractional order integration operators, their direct sums, and similarity problem to these operators of their weak perturbations

This survey is concerned with the spectral theory of Volterra operators An = ⊕nj bjJαj, αj > 0, which are direct sums of multiples of fractional order Riemann- Liouville operators Jαj. We discuss the lattices of invariant and hyperinvariant subspaces of operators An, as well as their commutants, double commutants, and other operator algebras related to An. We describe the sets of extended eigenvalues and the corresponding eigenvectors of the operators Jα. The Gohberg-Krein conjecture on equivalence of unicellularity and cyclicity properties of a dissipative Volterra operator is also discussed. The problem of the similarity of the Volterra integral operators to the operators Jα is discussed too. © 2019 Walter de Gruyter GmbH, Berlin/Boston

Stability of spectral characteristics of boundary value problems for 2 × 2 Dirac type systems. Applications to the damped string

The paper is concerned with the stability property under perturbation Q→Q˜ of different spectral characteristics of a boundary value problem associated in L2([0,1];C2) with the following 2×2 Dirac type equation LU(Q)y=−iB−1y′+Q(x)y=λy,B=(b100b2),b1<0<b2,y=col(y1,y2), with a potential matrix Q∈Lp([0,1];C2×2) and subject to the regular boundary conditions Uy:={U1,U2}y=0. If b2=−b1=1 this equation is equivalent to one dimensional Dirac equation. Our approach to the spectral stability relies on the existence of the triangular transformation operators for system (0.1) with Q∈L1, which was established in our previous works. The starting point of our investigation is the Lipshitz property of the mapping Q→KQ±, where KQ± are the kernels of transformation operators for system (0.1). Namely, we prove the following uniform estimate: ‖KQ±−KQ˜±‖X∞,p2+‖KQ±−KQ˜±‖X1,p2⩽C⋅‖Q−Q˜‖p,Q,Q˜∈Up,r2×2,p∈[1,∞], on balls Up,r2×2 in Lp([0,1];C2×2). It is new even for Q˜=0. Here X∞,p2, X1,p2 are the special Banach spaces naturally arising in such problems. We also obtained similar estimates for Fourier transforms of KQ±. Both of these estimates are of independent interest and play a crucial role in the proofs of all spectral stability results discussed in the paper. For instance, as an immediate consequence of these estimates we get the Lipshitz property of the mapping Q→ΦQ(⋅,λ), where ΦQ(x,λ) is the fundamental matrix of the system (0.1). Assuming the spectrum ΛQ={λQ,n}n∈Z of LU(Q) to be asymptotically simple, denote by FQ={fQ,n}|n|>N a sequence of corresponding normalized eigenvectors, LU(Q)fQ,n=λQ,nfQ,n. Assuming boundary conditions (BC) to be strictly regular, we show that the mapping Q→ΛQ−Λ0 sends Lp([0,1];C2×2) either into ℓp′ or into the weighted ℓp-space ℓp({(1+|n|)p−2}); we also establish its Lipshitz property on compact sets in Lp([0,1];C2×2), p∈[1,2]. The proof of the second estimate involves as an important ingredient inequality that generalizes classical Hardy-Littlewood inequality for Fourier coefficients. It is also shown that the mapping Q→FQ−F0 sends Lp([0,1];C2×2) into the space ℓp′(Z;C([0,1];C2) of sequences of continuous vector-functions, and has the Lipshitz property on compacts sets in Lp([0,1];C2×2), p∈[1,2]. Certain modifications of these spectral stability results are also proved for balls Up,r2×2 in Lp([0,1];C2×2), p∈[1,2]. Note also that the proof of the Lipshitz property of the mapping Q→FQ−F0 involves the deep Carleson-Hunt theorem for maximal Fourier transform, while the proof of this property for the mapping Q→ΛQ−Λ0 relies on the estimates of the classical Fourier transform and is elementary in character. We apply our previous results to the damped string equation to establish the Riesz basis property and the asymptotic behavior of the eigenvalues of the corresponding dynamic generator under the assumptions d∈L1[0,ℓ], ρ∈W1,1[0,ℓ] on the damping coefficient and the density of the string, that are weaker than previously treated in the literature. We also establish Lipshitz dependence on d and ρ in ℓp-spaces of the remainders in the asymptotic formula for the eigenvalues. © 2022 Elsevier Inc