Mark Malamud
Doctor of Physics and Mathematics
Professor,
s.M. Nikol’skii Mathematical Institute
Develop new theories that will eventually become effective in solving old and new problems.
1972
A graduate of the Department of Mathematical Analysis and Theory of Functions of Donetsk National University. Specialty - “Mathematics” (supervisor -Professor E. R. Tsekanovskii).
1971-1976
Worked at the Institute of Industrial Economics of the Ukrainian SSR Academy of Sciences.
1976-1978
Engineer in Donetsk Research Institute of Coal Industry.
1977
Candidate thesis on “On bringing non-selfadjoint operators to the simplest form” was presented.
1978-1994
Assistant (till 1991), associate Professor of the Department of Mathematical Physics of Donetsk Polytechnic Institute.
1991
The academic title of associate Professor was awarded.
1994 - 2007
Associate Professor of the Department of Mathematical Analysis and Theory of Functions of Donetsk National University.
2006
Worked at Michigan State University (MSU, USA).
2007 - 2017
Leading researcher at the Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine.
2010
Doctoral thesis on “Questions of uniqueness, completeness and self-adjointness in boundary value problems for ODE systems” was presented.
2018 - present
Professor of S.M. Nikol’skii Mathematical Institute of RUDN University.
Member of the editorial board of international mathematical journals:
- Mathematische Nachrichten, Germany;
- Methods of Functional Analysis and Topology (MFAT);
- Ukrainian Mathematical Bulletin, Ukraine.
Member of professional/scientific communities:
- American Mathematical Society;
- International Association of Mathematical Physics.
Teaching
M. M. Malamud was reading a number of special courses for master's students from 1994 till 2014:
- The theory of entire functions;
- Inverse problems for the Sturm-Liouville equation;
- The theory of extensions of symmetric operators and applications to boundary tasks;
- The completeness and basis property of root vectors of non-selfadjoint boundary value problems;
- Spectral theory of operators (seminar);
Under the supervision of M. M. Malamud 10 postgraduate students of Donetsk National University defended their candidate theses, and 2 of them defended their doctoral theses.
Science
- Systems of first-order differential equations on a finite interval with a non-degenerate diagonal matrix B at the derivative and summable potential matrix Q having a zero diagonal were analyzed. It was proved that the potential matrix Q is uniquely determined by the monodromy matrix W(λ). In the case B=B*, the minimum number of elements of the matrix W(λ) is indicated, which is sufficient to uniquely determine the matrix Q. For systems of ordinary differential equations (ODE) of the first order, triangular transformation operators were constructed. The inverse problem of restoring the potential matrix from the spectral matrix of the function was solved. The result was applied to describe the spectral types of a Dirac-type system.
- Spectral characteristics of Schrodinger and Dirac operators with point interactions were studied. The one-dimensional symmetric Schrodinger operator HX,α with d - interactions on a discrete set is studied within the theory of extension. Using the apparatus of boundary triples and corresponding Weyl functions, the connection of operators HX,α with one class of Jacobi matrices was found. The connection allowed us to obtain terms of self-adjointness, semiboundedness below, discreteness of the spectrum and discreteness of the negative part of the spectrum of the operator under study.
- Determinants of perturbations and formulas of traces for pairs of dissipative and selfadjoint operators were studied. A method of double operator integrals was developed to prove formulas of traces for compression functions, dissipative operators, unitary operators and selfadjoint operators.
- The absolutely continuous spectrum and the scattering matrix of operators of different classes were studied. A formula for the scattering matrix (S-matrix) was found, expressing it through the limit values of the Weyl function and boundary operators. A formula was found, for elliptic boundary value problems in external domains, that expresses the S-matrix via the Dirichlet-Neumann operator-function. A connection with the Lax-Phillips scattering theory was found.
- Questions of completeness and basis of Riesz systems of root vectors of boundary value problems for ODE systems were studied. A new concept of weakly regular boundary terms for n × n systems of first-order ordinary differential equations with a constant diagonal matrix B at the first derivative was introduced. It was proved that the system of root functions of boundary value problems of this type on a finite interval is complete and minimal. Performing the Riesz basis characteristic for some classes of boundary terms was defined. The Riesz basis characteristic for strictly regular boundary terms was defined for a 2 × 2 Dirac-type system with a summable potential matrix.
- More than 150 scientific articles were published in central international mathematical journals, including Journal of Functional Analysis, Journal of Differential Equations, Annales Henri Poincaré, Advances in Mathematics, Transactions of American Mathematical Society. Monograph Derkach V. A., Malamud M. M. “Theory of extensions of symmetric operators and boundary value problems”, Kyiv, 2017 was published.
Scientific interests
- Inverse spectral problems for ODE systems.
- Spectral theory of Schrodinger and Dirac operators.
- Determinants of indignation and formulas of traces for pairs of non-selfadjoint and selfadjoint operators.
- Scattering matrix for pairs of selfadjoint and non-selfadjoint operators.
- Questions of completeness and basis of non-selfadjoint operators with discrete spectrum.
Broad classes of non-negative Schrodinger operators in K^2 and K^3 with the following properties are indicated:
1. A suitable set of zero measure in R^2(R^3) defines a narrowing of each of these operators, which is a non-negative symmetric operator (Dirichlet problem) with a compact preresolvent.
2. Under some additional conditions on the potential, the Friedrichs expansion of such a narrowing has a continuous (sometimes absolutely continuous) spectrum filling the positive half-axis.
These results provide a solution to the problem of M. S. Birman.
The paper is a survey and concerns with infinite symmetric block Jacobi matrices J with m×m-matrix entries. We discuss several results on general block Jacobi matrices to be either self-adjoint or have maximal as well as intermediate deficiency indices. We also discuss several conditions for J to have discrete spectrum.
This is a short survey related to transformation operators for fractional order ordinary differential operators. The existence of triangular transformation operators is established in the case of analytic coefficients. Some applications of such a class of transformation operators are discussed.
Malamud M.M. Transformation operators for fractional order ordinary differential equations and their applications // Kravchenko, Vladislav V. (ed.) et al., Transmutation operators and applications. Cham: Birkhäuser. Trends Math., 2020, pp. 539-572.
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 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.
The paper is devoted to block Jacobi matrices, that is, tridiagonal Hermitian infinite matrices in which every matrix element is a p×p matrix, p≥1. It is known that the deficiency numbers n± of the corresponding symmetric operators satisfy the inequality 0≤n±≤p. The authors find new classes of such matrices, for which the deficiency numbers take maximal or intermediate values. Applications to one-dimensional Dirac operators with point interactions are given.
The article is devoted to block Jacobi matrices, that is, tridiagonal Hermitian infinite matrices in which each element of the matrix is a matrix p×p, p≥1. It is known that the defect indices n_± of the corresponding symmetric operators satisfy the inequality 0≤n±≤p. The search is carried out for new classes of such matrices for which the defect indices take maximum or intermediate values. Applications to one-dimensional Dirac operators with point interactions are given.
We investigate quantum graphs with an infinite number of vertices and edges without a general restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. The central result is a close relationship between the spectral properties of a quantum graph and the corresponding properties of some weighted discrete Laplacian on the underlying discrete graph. Using this connection together with the spectral theory of (unbounded) discrete Laplacians on infinite graphs, a number of new results on the spectral properties of quantum graphs are proved. Namely, several self-conjugacy results are proved, including a Gaffney type theorem. The problem of lower half-closure is investigated, several spectral estimates are proved and spectral types are studied.
We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a close connection between spectral properties of a quantum graph and the corresponding properties of a certain weighted discrete Laplacian on the underlying discrete graph. Using this connection together with spectral theory of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new results on spectral properties of quantum graphs. Namely, we prove several self-adjointness results including a Gaffney-type theorem. We investigate the problem of lower semiboundedness, prove several spectral estimates (bounds for the bottom of spectra and essential spectra of quantum graphs, CLR-type estimates) and study spectral types.
Infinite quantum graphs with δ-interactions at vertices are studied without any assumptions on the lengths of edges of the underlying metric graphs. A connection between spectral properties of a quantum graph and a certain discrete Laplacian given on a graph with infinitely many vertices and edges is established. In particular, it is shown that these operators are self-adjoint, lower semibounded, nonnegative, discrete, etc. only simultaneously.
Infinite quantum graphs with δ-interactions at vertices are studied without any assumptions on the lengths of edges of the underlying metric graphs. A connection between spectral properties of a quantum graph and a certain discrete Laplacian given on a graph with infinitely many vertices and edges is established. In particular, it is shown that these operators are self-adjoint, lower semibounded, nonnegative, discrete, etc. only simultaneously.
The matrix Schrödinger operator with point interactions on the semiaxis is studied. Using the theory of boundary triplets and the corresponding Weyl functions, we establish a relationship between the spectral properties (deficiency indices, self-adjointness, semiboundedness, etc.) of the operators under study and block Jacobi matrices of certain class.
A new concept of weakly regular boundary conditions for a first order n× n system of ODE with a constant diagonal matrix B at the first derivative is introduced. It is proved that the system of root functions of such type boundary value problems on a finite interval is complete and minimal. Riesz basis property for for certain classes of boundary conditions was also established. For 2×2 Dirac type system with summable potential matrix the Riesz basis property has been established for strictly regular boundary conditions.
The matrix Schrödinger operator with point interactions on the semiaxis is studied. Using the theory of boundary triplets and the corresponding Weyl functions, we establish a relationship between the spectral properties of the operators under study and block Jacobi matrices of certain class.
Trace formulas are obtained for pairs of self-adjoint, maximal dissipative and accumulative, as well as other types of resolvent comparable operators.
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 minimal number of matrix entries of W(λ) sufficient for unique determination of Q is found.
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.
The paper is concerned with the completeness property of root functions of general boundary value problems for n×n first order systems of ordinary differential equations on a finite interval. It is shown that the system of root vectors of the general n×n Dirac type system subject to certain boundary conditions forms a Riesz basis with parentheses. We also show that arbitrary complete dissipative boundary value problem for Dirac type operator with a summable potential matrix admits the spectral synthesis in L2([0,1];Cn). Finally, we apply our results to investigate completeness and the Riesz basis property of the dynamic generator of spatially non-homogenous damped Timoshenko beam model.
Lunyov A., Malamud M.M. On the completeness and Riesz basis property of root subspaces of boundary value problems for first order systems and applications. J. Spectral Theory, v.5 (2015), no 1, 17—70.
https://www.ems-ph.org/journals/show_abstract.php?issn=1664-039X&vol=5&iss=1&rank=2
Trace formulas for pairs of self-adjoint, maximal dissipative and accumulative as well as other types of resolvent comparable operators are obtained. In particular, the existence of a complex-valued spectral shift function for a pair $\{H',H\}$ of maximal accumulative operators has been proved under an additional week assumption on $H$. We investigate also the existence of a real-valued spectral shift function. Moreover, we treat in detail the case of additive trace class perturbations. Assuming that $H$ and $H'= H+V$ are maximal accumulative and $V$ is trace class, it is proved the existence of a summable complex-valued spectral shift function. We also obtain trace formulas for pairs $\{H, H^*\}$ assuming only that $H$ and $ H^*$ are resolvent comparable.
In this case the determinant of the characteristic function of $H$ is involved into trace formulas.
Spectral properties of Schrodinger Hamiltonian with infinitely many point interactions have been investigated. In particular, there indicated special configurations of points such that the non-negative parts of all realizations are absolutely continuous. The important feature of the approach proposed is based on a connection with a class of radial positive-definite functions.
We consider systems of first-order differential equations on a finite interval with a non-degenerate diagonal matrix B with a derivative and a summable potential matrix Q having a zero diagonal. It is proved that the potential matrix Q is uniquely determined by the monodromy matrix W (λ). In the case of B=B^*, the minimum number of elements of the matrix W(λ) is specified, which is sufficient for an unambiguous definition of the matrix Q.
Trace formulas for pairs of self-adjoint, maximal dissipative and accumulative as well as other types of resolvent comparable operators are obtained.
The completeness problem on non-regular boundary value problem for first order ODE on a finite interval is investigated in the paper. The asymptotic behaviour of solutions and of the characteristic determinant, for which the corresponding block matrix structure is ignored, are discussed. Then, results on completeness of the root vectors of the boundary value problem are formulated. Some important geometric properties of the system of root functions for the dynamic generator of the Timoshenko beam model are obtained.
We investigate spectral properties of Gesztesy–Šeba realizations DX,α and DX,β of the 1-D Dirac differential expression D with point interactions on a discrete set X = {x_n }_(n=1)^∞ ⊂ R. Here α := {α_n }_(n=1)^∞ and β≔{β_n }_(n=1)^∞ ⊂ R. The Gesztesy–Šeba realizations DX,α and DX,β are the relativistic counterparts of the corresponding Schrödinger operators HX,α and HX,β with δ- and δ’ -interactions, respectively. We define the minimal operator DX as the direct sum of the minimal Dirac operators on the intervals (xn−1, xn). Then using the regularization procedure for direct sum of boundary triplets we construct an appropriate boundary triplet for the maximal operator D_X^* in the case d∗(X) := inf{|xi − x j|, i ≠j} = 0. It turns out that the boundary operators BX,α and BX,β parameterizing the realizations DX,α and DX,β are Jacobi matrices. These matrices substantially differ from the ones appearing in spectral theory of Schrödinger operators with point interactions. We show that certain spectral properties of the operators DX,α and DX,β correlate with the corresponding spectral properties of the Jacobi matrices BX,α and BX,β, respectively. Using this connection we investigate spectral properties (self-adjointness, discreteness, absolutely continuous and singular spectra) of Gesztesy–Šeba realizations. Moreover, we investigate the non-relativistic limit as the velocity of light c→∞. Most of our results are new even in the case d∗(X) > 0.
Spectral properties of 1-D Schrödinger operators HX,α with local point interactions on a discrete set X = {x_n }_(n=1)^∞ are well studied when d∗ := infn,k∈N |xn − xk| > 0. Our paper is devoted to the case d∗ = 0. We consider HX,α in the framework of extension theory of symmetric operators by applying the technique of boundary triplets and the corresponding Weyl functions. We show that the spectral properties of HX,α like self-adjointness, discreteness, and lower semiboundedness correlate with the corresponding spectral properties of certain classes of Jacobi matrices. Based on this connection, we obtain necessary and sufficient conditions for the operators HX,α to be self-adjoint, lower semibounded, and discrete in the case d∗ = 0. The operators with δ’ -type interactions are investigated too. The obtained results demonstrate that in the case d∗ = 0, as distinguished from the case d∗ > 0, the spectral properties of the operators with δ- and δ’ -type interactions are substantially different.
Closed (in particular selfadjoint ) realizations of second order elliptic differential expression on smooth (bounded or unbounded) domain with compact boundary are considered. Trace ideal properties of power of resolvent differences for these realizations are investigated. Estimates for the number of negative eigenvalues of certain self-adjoint extensions of a non-negative minimal operator are derived. The results extend and improve classical theorems due to Vishik, Povzner, Birman, and Grubb.
It is solved the Krein problem on description of L2-space constructed by an operator-valued (not necessary orthogonal) measure in a Hilbert space.
The spectral theory of operator measures was developed. In particular, the multiplicity function as well as Hellinger types are introduced for an arbitrary operator measure. It is proved that the set of all principal vectors of an arbitrary operator measure in a Hilbert space is massive, i.e. it is a dense G -set. In particular, it is shown that the set of principal vectors of a self-adjoint operator is massive in any cyclic subspace.
It is proved that subspaces realizing Hellinger types exist and form a massive set.
A resolvent criterion of similarity of a non-selfadjoint operator in a Hilbert space with real spectrum was obtained. Certain sufficient conditions for similarity of a non-selfadjoint operator with well defined imaginary part to a selfadjoint one are provided. In particular, it is proved that the operator is similar to a selfadjoint one with absolutely continuous spectrum provided that the $j$-forms of the characteristic function are bounded in both half-planes. Certain sufficient conditions for a triangular non-selfadjoint operator to be similar to a selfadjoint one are presented too. In the dissipative case a criterion of such similarity is given. The latter generalizes the one due to I. Gohberg and M. Krein.
The paper presents conditions for the deficiency indices of the first order Hamiltonian systems on the half-line (and line) to be minimal and to be maximal. Moreover, certain conditions for the deficiency numbers to be intermediate are indicated too. This covers earlier results due to Kac and Krein, de Branges, and Kogan and Rofe-Beketov.
Selfadjointness conditions for the second order four terms Sturm-Liouville matrix equations are also obtained. The latter substantially generalizes the classical Titchmarsh-Sears theorem and covers earlier results of V. Lidskii, M. Krein, de Brange.
A Hermitian operator A with gaps (αj, βj) (1 ⩽ j ⩽ m ⩽ ∞) is studied. The self-adjoint extensions which put exactly kj < ∞ eigenvalues into each gap (αj, βj), in particular (for kj = 0, 1 ⩽ j ⩽ m) the extensions preserving the gaps, are described in terms of boundary conditions. The generalized resolvents of the extensions with the indicated properties are described also. A solvability criterion and description of all the solutions of the Hamburger moment problem with supports in /⋃j=1m(αj,βj) are obtained in terms of the Nevanlinna matrix.