Mark Malamud
Doctor of Physics and Mathematics

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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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.
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.