Victor Antonovich Sadovnichii

Solvability Theorems for an Inverse Nonself-Adjoint Sturm-Liouville Problem with Nonseparated Boundary Conditions

We prove theorems on the solvability of the inverse Sturm-Liouville problem with nonseparated conditions by two spectra and one eigenvalue and theorems on the unique solvability by two spectra and three eigenvalues. We find exact and approximate solutions of the inverse problems. Related examples and counterexample are given.

Sufficient condition for the hyperbolicity of mappings of the torus

We consider mappings of the m-dimensional torus Tm (m ≥ 2) that are C1-perturbations of linear hyperbolic automorphisms. We obtain sufficient conditions for such mappings to be one-to-one hyperbolic mappings (i.e., Anosov diffeomorphisms). These results are used to study the blue-sky catastrophe related to the vanishing of a saddle-node invariant torus with a quasiperiodic winding in a system of ordinary differential equations.

Periodic solutions of travelling-wave type in circular gene networks

We consider circular chains of unidirectionally coupled ordinary differential equations which are mathematical models of artificial gene networks. We study the problems of the existence and stability of special periodic solutions, the so-called travelling waves, in these chains. We establish that the number of such periodic solutions grows unboundedly as the number of links in the chain grows. However, at most one of these travelling waves can be stable. We give an explicit algorithm for choosing the stable cycle.

Uniqueness of reconstruction of the Sturm–Liouville problem with spectral polynomials in nonseparated boundary conditions

Uniqueness theorems for solutions of inverse Sturm–Liouville problems with spectral polynomials in nonseparated boundary conditions are proved. As spectral data two spectra and finitely many eigenvalues of the direct problem or, in the case of a symmetric potential, one spectrum and finitely many eigenvalues are used. The obtained results generalize the Levinson uniqueness theorem to the case of nonseparated boundary conditions containing polynomials in the spectral parameter.

Inverse Problem for a Differential Operator with Nonseparated Boundary Conditions

Uniqueness theorems for the solution of an inverse problem for a fourth-order differential operator with nonseparated boundary conditions are proved. The spectral data for the problem is specified as the spectrum of the problem itself (or its three eigenvalues) and the spectral data of a system of three problems.

Inverse Sturm–Liouville problem with spectral polynomials in nonsplitting boundary conditions

Theorems on the unique reconstruction of a Sturm–Liouville problem with spectral polynomials in nonsplitting boundary conditions are proved. Two spectra and finitely many eigenvalues (one spectrum and finitely many eigenvalues for a symmetric potential) of the problem itself are used as the spectral data. The results generalize the Levinson uniqueness theorem to the case of nonsplitting boundary conditions containing polynomials in the spectral parameter. Algorithms and examples of solving relevant inverse problems are also presented.

Representations of regularized determinants of exponentials of differential operators by functional integrals

Representations of regularized determinants of elements of one-parameter operator semigroups whose generators are second-order elliptic differential operators by Lagrangian functional integrals are obtained. Such semigroups describe solutions of inverse Kolmogorov equations for diffusion processes. For self-adjoint elliptic operators, these semigroups are often called Schrödinger semigroups, because they are obtained by means of analytic continuation from Schrödinger groups. It is also shown that the regularized determinant of the exponential of the generator (this exponential is an element of a one-parameter semigroup) coincides with the exponential of the regularized trace of the generator.

К пятидесятилетию научной и педагогической деятельности Виктора Павловича Маслова@@@Fiftieth anniversary of research and teaching by Viktor Pavlovich Maslov

Viktor Pavlovich Maslov began teaching in 1957 after defending his candidate dissertation under the supervision of Professor S. V. Fomin. This moment can also be considered the start of his independent research because from then on, his strongest side began appearing, the ability to uncover the mathematics underlying the most important modern problems in the natural sciences and to pose relevant problems whose solutions advance not only mathematics itself but also science as a whole. More than 50 years ago, physicists discovered and constructed microwave masers. To build nanowave (light) lasers, in addition to an active substance that when illuminated would not absorb but amplify the beam passing through it, required developing a reflecting system that would retain only one mode while allowing the rest of the beam to leave the device. The young mathematician Maslov suggested a possible type of such a construction in 1957. It turned out that if a sufficiently narrow planar waveguide is bent slightly, then it becomes possible to ensure that only one mode is retained while the others “jump out” of the waveguide. Maslov showed that bending the axis of a very narrow waveguide results in a certain effective potential (refractive index) in the longitudinal direction. This potential contains one stable term −k/4, where k is the curvature of the waveguide axis. If the axis is straight in some waveguide segment, then the curvature and potential of this segment are zero. Bending the waveguide then results in increasing the curvature (and decreasing the potential −k/4). Next, the waveguide axis is straight again, and the curvature and potential are again equal to zero. A negative potential well is thus formed. The Helmholtz equation for a monochromatic electromagnetic wave in a bent narrow waveguide leads to the one-dimensional Schrödinger equation with the potential describing this well. A discrete spectrum exists in this case. If the bend is small, then the depth of the well is sufficiently small, and the well contains a single eigenvalue and therefore exactly one eigenmode. For a somewhat greater bend, there can be two, three, and so on eigenmodes. A wave trap is thus formed. (We note that the standard quasistationary modes, which can be present in such a system because of reflection from the open ends, are exponentially small compared to the waveguide width-to-length ratio.) Maslov was the speaker at three consecutive seminars headed by A. M. Prokhorov, who was determined to comprehend this theory. Toward the end of the discussion, one of the participants bluntly concluded that it was technologically impossible, for example, to construct a single-mode planar laser based on this theory because techniques for creating thin films and ultranarrow channels in them did not exist. At the same time, Maslov’s patent application was rejected (by the same opponent) on the same grounds. But already in 1959, the world saw the birth of the first planar transistor, and R. Feynman published a paper on the prospects of advancing into the nanoscale domain and uttered his now legendary phrase while speaking at Caltech: “There’s plenty of room at the bottom.” It can be said that Maslov’s work published in 1958 as a short, purely mathematical paper in Doklady Akademiki Nauk SSSR was the first example of a nanostructure suggested by a mathematician. The development of this idea has been very fruitful within mathematics itself. Maslov gave a formal mathematical description of this type of mixed system with deterministic (classical) and simultaneously wave (quantum) behavior in his book Perturbation Theory and Asymptotic Methods (MSU Publ., Moscow, 1965; English translation: Mir, Moscow, 1965). The principal mathematical objects introduced in that book are Lagrangian submanifolds in phase space (so named by Maslov), the “canonical operator” assigning wave functions to such submanifolds, and an integer index and