Develop new theories that will eventually become effective in solving old and new problems.
A graduate of the Department of Mathematical Analysis and Theory of Functions of Donetsk National University. Specialty - “Mathematics” (supervisor -Professor E. R. Tsekanovskii).
Worked at the Institute of Industrial Economics of the Ukrainian SSR Academy of Sciences.
Engineer in Donetsk Research Institute of Coal Industry.
Candidate thesis on “On bringing non-selfadjoint operators to the simplest form” was presented.
Assistant (till 1991), associate Professor of the Department of Mathematical Physics of Donetsk Polytechnic Institute.
The academic title of associate Professor was awarded.
Associate Professor of the Department of Mathematical Analysis and Theory of Functions of Donetsk National University.
Worked at Michigan State University (MSU, USA).
Leading researcher at the Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine.
Doctoral thesis on “Questions of uniqueness, completeness and self-adjointness in boundary value problems for ODE systems” was presented.
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.
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.
- 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.
- 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.