We embed general boundary value problems for the time-harmonic Maxwell equations into the elliptic boundary value theory. This is achieved by introducing two new scalar functions to the electromagnetic field and imposing additional boundary conditions, after which the problem becomes elliptic. The results are applied to general problems for Maxwell's equations in bounded and unbounded domains, as well as to the transmission problem with inhomogeneities on the right-hand side of the equations and at all boundaries. Relations between the inhomogeneities of the elliptic problem are established that provide a one-to-one correspondence between the solutions of Maxwell's problem and the solutions of the elliptic boundary value problem. As an application, the existence of an orthogonal basis from the eigenfunctions of the general Maxwell's problem is justified.

We consider acoustic wave propagation through a periodic array of small inclusions of arbitrary shape. The inclusion size a is much smaller than the array period, while the wavelength is fixed. This problem is a singular perturbation of the problem without inclusions, and its solution does not converge to an unperturbed wave when a → 0. However, the dispersion surface is close to the one for the inclusionless problem, and this implies the absence of global gaps in any fixed interval (ε, ε−1 ) of the time frequency ω if a is small enough. The notion of local gaps, which depends on the choice of a wave vector k, will be introduced and studied.

We investigate the nonstationary parabolic Anderson problem ∂u∂t=ϰLu(t,x)+ξt(x)u(t,x),u(0,x)≡1,(t,x)∈[0,∞)×Zd where ϰL denotes a nonlocal Laplacian and ξt(x) is a correlated white-noise potential. The irregularity of the solution is linked to the upper spectrum of certain multiparticle Schrödinger operators that govern the moment functions mp(t,x1,x2,⋯,xp)=⟨u(t,x1)u(t,x2)⋯u(t,xp)⟩. First, we establish a weak form of intermittency under broad assumptions on L and on a positive-definite noise correlator B=B(x). We then examine strong intermittency, which emerges from the existence of a positive eigenvalue in a related lattice Schrödinger-type operator with potential B. Here, B does not have to be positive definite but must satisfy ∑B(x)≥0. The presence of such an eigenvalue intensifies the growth properties of the second moment m2, revealing a more pronounced intermittent regime.

The paper aims to study the spectral properties of elliptic operators with highly inhomogeneous coefficients and related issues concerning wave propagation in high-contrast media. A unified approach to solving problems in bounded domains with Dirichlet or Neumann boundary conditions, as well as in infinite periodic media, is proposed. For a small parameter ε > 0 characterizing the contrast of the components of the medium, the analyticity of the eigenvalues and eigenfunctions is established in a neighborhood of ε = 0. Effective operators corresponding to ε = 0 are described.

We investigate the behavior of waves in a periodic medium containing small soft inclusions or cavities of arbitrary shape, such that the homogeneous Dirichlet conditions are satisfied at the boundary. The leading terms of Bloch waves, their dispersion relations, and low frequency cutoff are rigorously derived. Our approach reveals the existence of exceptional wave vectors for which Bloch waves are comprised of clusters of perturbed plane waves that propagate in different directions. We demonstrate that for these exceptional wave vectors, no Bloch waves propagate in any one specific direction.

The propagation of acoustic waves in a medium with a periodic array of small inclusions of arbitrary shape is considered. The inclusion size a is much smaller than the array period. We show that global gaps do not exist if a is small enough. We introduce and study the concept of local gaps, which depends on the choice of the wave vector k . We analytically determine the location of local gaps for the Dirichlet and transmission problems.

Part 1 The stationary phase method: on asymptotic expansions the stationary phase method the stationary phase method, the multidimensional case the problem of waves on the surface of a liquid the asymptotic behaviour of the Fourier transform of a function concentrated on a smooth closed surface. Part 2 The WKB method for ordinary differential equations: the asymptotic behaviour of solutions of a homogeneous equation the scattering problem the asymptotic behaviour of solutions of boundary-value problems. Part 3 Partial differential equations of the first order and characteristics for equations of higher order: quasilinear partial differential equations of the first order general partial differential equations of the first order the Hamilton-Jacobi equation example propagation of light waves in an inhomogenous medium characteristic surfaces for differential operators of high order, connection with the well-posedness of the Cauchy problem the search for characteristic surfaces. Part 4 Propagation of discontinuities, problems with rapidly oscillating data: the Leibniz formula problems with rapidly oscillating initial data discontinuous solutions of equations. Part 5 The Maslov canonical operator: the problem of scattering of a plane wave in an inhomogenous medium the Lagrange maniford the precanonical operator the canonical operator, construction of a formal asymptotic solution a field in an isotropic medium with parabolic wave front more general problems. Part 6 Elliptic problems in a bounded domain: Sobolev-Slobodetskii spaces elliptic problems elliptic problems with a parameter inversion of a finitely-meromorphic Fredholm family of operators. Part 7 Equations and systems with constant coefficients in Rn: equations with a non-zero characteristic polynomial equations and systems of the type of the Helmholtz equation radiation conditions the principle of limiting absorption. Part 8 Elliptic equations with variable coefficients and boundary-value problems in the exterior of a bounded domain: solubility and a priori estimated of solutions of exterior boundary-value problems the principle of limiting absorption for exterior problems. Part 9 Analytic properties of the resolvent of operators that depend polynomially on a parameter: equations with constant coefficients equations with variable coefficients and problems in the interior of a bounded domain the asymptotic behaviour of solutions of exterior problems for small frequencies. Part 10 Short-wave asymptotic behaviour of solutions of stationary problems and the asymptotic behaviour of solutions of hyperbolic equations as t introduction short-wave asymptotic behaviour as t of solutions of mixed problems. Part 11 Quasiclasical approximations in stationary scattering problems: the asymptotic behaviour of the solution of the scattering problem and the amplitude of the scattering proof of theorems 1 and 2. Part contents...

We present a theoretical study of the propagation of acoustic waves in a 3D infinite medium containing a periodic array of small identical inclusions of arbitrary shape with transmission conditions on their interfaces. The inclusion size a is much smaller than the array period. We present the dispersion relation and show that there are exceptional frequencies for which the solution is a cluster of waves propagating in several different directions. Different clusters may contain waves with the same direction, and the frequencies of the waves depend on the clusters but not on the direction of waves. We show that global gaps do not exist if a is small enough. The notion of local gaps which depends on the choice of the wavevector k, is introduced and discussed. The location of local gaps for a medium with a simple cubic lattice of identical inclusions is determined.

The research explores a high irregularity, commonly referred to as intermittency, of the solution to the non-stationary parabolic Anderson problem: ∂ u/∂ t = ϰℒu(t,x) + ξ_t(x)u(t,x) with the initial condition u(0,x) ≡ 1, where (t,x) ∈ [0,∞)×ℤ^d. Here, ϰℒ denotes a non-local Laplacian, and ξ_t(x) is a correlated white noise potential. The observed irregularity is intricately linked to the upper part of the spectrum of the multiparticle Schrödinger equations for the moment functions m_p(t,x_1,x_2,⋯,x_p) = ⟨ u(t,x_1)u(t,x_2)⋯ u(t,x_p)⟩. In the first half of the paper, a weak form of intermittency is expressed through moment functions of order p≥ 3 and established for a wide class of operators ϰℒ with a positive-definite correlator B=B(x)) of the white noise. In the second half of the paper, the strong intermittency is studied. It relates to the existence of a positive eigenvalue for the lattice Schrödinger type operator with the potential B. This operator is associated with the second moment m_2. Now B is not necessarily positive-definite, but ∑ B(x)≥ 0.

We consider the discrete Schr\”odinger operator $H=-\Delta+V$ with a sparse potential $V$ and find conditions guaranteeing either existence of wave operators for the pair $H$ and $H_0=-\Delta$, or presence of dense purely point spectrum of the operator $H$ on some interval $[\lambda_0,0]$ with $\lambda_0<0$.

Simon’s results on the negative spectrum of recurrent Schrödinger operators ( d = 1 , 2 d=1,2 ) are extended to a wider class of potentials and to non-local operators. An example of L 1 − L^1- potental is constructed for which the essential spectrum of two dimensional Schrödinger operator covers the whole axis. Some counterexamples are provided for transient operators ( d ≥ 3 d\geq 3 ) showing that the assumptions on the potential for the validity of the Cwikel-Lieb-Rozenblum estimate can’t be improved significantly.

We consider the propagation of acoustic, electromagnetic, and elastic waves in a one-dimensional periodic two-component material. Accurate asymptotic formulas are provided for the group velocity as a function of the material parameters when the concentration of scatterers is small or the characteristic impedances of the two media differ substantially. In the latter case, it is shown that the minimum group velocity occurs when the volume fractions of the components of the material are equal. In both asymptotic cases, we show that the leading terms of the group velocity do not depend on frequency. Thus slowdown is frequency-independent and is not related to the resonance phenomena.

We consider a mean-field model of a polymer with a spherically symmetric finitely supported potential. We describe how the typical size of the polymer depends on the two parameters: the temperature, which approaches the critical value, and the length of the polymer chain, which goes to infinity.

We obtain the asymptotics, as $t + |x| \rightarrow \infty$, of the fundamental solution to the heat equation with a compactly supported potential. It is assumed that the corresponding stationary operator has at least one positive eigenvalue. Two regions with different types of behavior are distinguished: Inside a certain conical surface in the $(t,x)$ space, the asymptotics is determined by the principal eigenvalue and the corresponding eigenfunction; outside of the conical surface, the main term of the asymptotics is a product of a bounded function and the fundamental solution of the unperturbed operator, with the contribution from the potential becoming negligible if $|x|/t \rightarrow \infty$. A formula for the global asymptotics, as $t + |x| \rightarrow \infty$, of the solution in the entire half-space $t > 0$ is provided. In probabilistic terms, the result describes the asymptotics of the density of particles in a branching diffusion with compactly supported branching and killing potentials.

We consider acoustic wave propagation through a periodic array of the inclusions of arbitrary shape. The inclusion size is much smaller than the array period while the wavelength is fixed. We derive and rigorously justify the dispersion relation for general frequencies and show that there are exceptional frequencies for which the solution is a cluster of waves propagating in different directions with different frequencies so that the dispersion relation cannot be defined uniquely. Examples are provided for the spherical inclusions.

We consider exterior elliptic problems with coefficients stabilizing at infinity and study the critical value β_cr of the coupling constant (the coefficient at the potential) that separates operators with a discrete spectrum and those without it. The dependence of β_cr on the boundary condition and on the distance between the boundary and the support of the potential is described. The discrete spectrum of a non-symmetric operator with the FKW boundary condition (that appears in diffusion processes with traps) is also investigated.

We consider the propagation of acoustic time-harmonic waves in a homogeneous media containing periodic lattices of spherical or cylindrical inclusions. It is assumed that the wavelength has the order of the periods of the lattice while the radius $a$ of inclusions is small. A new approach is suggested to derive the complete asymptotic expansions of the dispersion relations in two and three-dimensional cases as $a \to 0$ and evaluate explicitly several first terms. Our method is based on the reduction of the original singularly perturbed (by inclusions) problem to the regular one. The Neumann, Dirichlet and transmission boundary conditions are considered. The effective wave speed is obtained as a function of the wave frequency, the filling fraction of the inclusions, and the physical properties of the constituents of the mixture. Dependence of asymptotic formulas obtained in the paper on geometric and material parameters is illustrated by graphs.

Stanislav Molchanov was born on 21 December 1940 in the village of Snetinovo in the Ivanovo Oblast. His mother, Nina Grigorievna Molchanova, was an elementary school teacher. His father, Alexei Pavlovich Molchanov, was an accountant at a collective farm. As a son of a priest who had been persecuted after the Russian revolution, Alexei Molchanov was able to enroll at University (the Pedagogical Institute in the city of Ivanovo) only at the end of the 1940s, due to his status as a wounded veteran of WWII. After he graduated from the university, Alexei Molchanov started teaching physics at School no. 54 in the settlement of Nerl’ in the Teikovo District of the Ivanovo Oblast. Nina Molchanova moved to work at the same school as a teacher of German. After spending five years at a local school in Snetinovo, Molchanov moved to Nerl’ with his parents, where he graduated from School no. 54 with a gold medal. While in high school, he studied on his own some parts of elementary mathematics and introductory calculus not included in the high school program, using brochures from the school mathematical library and a book by A. Ya. Khinchin. In 1958 Molchanov enrolled in the Faculty of Mechanics and Mathematics at the Lomonosov Moscow State University. He received straight A grades, and, starting from his third year, he was a recipient of the Lenin Scholarship. He participated actively in the seminars of Prof. E. B. Dynkin, his scientific advisor. As a participant of the seminar, he carried out his first research work. After graduating from the university, Molchanov stayed there as a graduate student from 1963 till 1966. He was a member of Dynkin’s school, which exemplified a high level of general mathematical education: not only probability theory and stochastic processes, but also partial differential equations, Riemannian geometry, Lie groups and algebras, and so on.

We study spectral properties of convolution operators L and their perturbations H = L+ v(x) by compactly supported potentials. Results are applied to determine the front propagation of a population density governed by operator H with a compactly supported initial density provided that H has positive eigenvalues. If there is no positive spectrum, then the stabilization of the population density is proved.