Annotation of the results obtained in 2022
I. Homogenization theory • We studied homogenization of nonstationary Schrödinger-type equations in R^d. Let A_ε be a matrix strongly elliptic second-order DO with periodic coefficients depending on x/ε. It is known that the exponential exp(-i t A_ε) converges, as ε → 0, to the exponential of the effective operator A^0 in the operator norm from H^3 to L_2, and the error does not exceed C(1+|t|)ε. We were interested in constructing a more accurate approximation for exp(-i t A_ε) in the (H^s→L_2)-norm (with suitable s), as well as an approximation in the energy norm. We managed to construct such approximations for a "corrected" exponential exp(-i t A_ε) (I + ε K(ε)). Here K(ε) involves some rapidly oscillating factor, a first-order DO, and a smoothing operator. For the "corrected" exponential, an approximation in the (H^6→L_2)-norm with error of order of (1+|t|)^2 ε^2, as well as an approximation in the (H^4→H^1)-norm with error of order of (1+|t|)ε, was obtained. In the general case the results are sharp regarding the norm type and the dependence of estimates on t. However, under some additional assumptions, the results were improved. The results were applied to homogenization of the Cauchy problem for the Schrödinger-type equation (with the operator A_ε) with initial data from a special class. • We studied homogenization of the nonlocal Schrödinger operator Н_ε u (x) = ε^{-d-2} ∫a((x-y)/ε) m(x/ε,y/ε) (u(x) - u(y)) dy in L_2(R^d). It was assumed that a(x) = a(-x) ≥ 0 and the function (1+|x|)^3 a(x) is integrable. The coefficient m(x,y) = m(y,x) is positive definite, bounded, and periodic in each variable. We proved that the resolvent of Н_ε converges in the operator norm in L_2 to the resolvent of the effective operator H^0 = - div g^0 grad and obtained an estimate for the error of sharp order ε. • We studied homogenization near the edge of an inner gap for an elliptic DO A_ε = D g(x/ε) D + ε^{-2} p(x/ε) in R^1 with periodic coefficients. Let ε^{-2} ν be the right edge of some spectral gap of A_ε. We constructed an approximation for the resolvent of A_ε at the point (ε^{-2} ν - δ) in the (L_2→H^1)-norm with an estimate for the error of sharp order ε. • We studied homogenization of a parabolic equation near the edge of an inner gap. In R^d, we consider an elliptic DO A_ε = - div g(x/ε) grad + ε^{-2} p(x/ε) with periodic coefficients. Let ε^{-2} ν be the right edge of some spectral gap of A_ε. Let E_ε be the spectral projection corresponding to the semiaxis [ε^{-2} ν, ∞). We obtained an approximation for the operator exp(-A_ε t) E_ε in the (L_2→L_2)-norm for a fixed t > 0 and small ε with error of sharp order ε (aside from the factor exp(- ε^{-2} ν t)), and also a more accurate approximation with corrector. • We investigated homogenization of a matrix strongly elliptic locally periodic DO A_ε = D* a(x,x/ε) D in a Lipschitz domain with fairly general boundary conditions. It was assumed that the coefficients are Hölder continuous of order s > 0 in the "slow" variable away from a Lipschitz curve Γ in O, where they may have a jump discontinuity; in the "fast" variable, they are periodic and bounded. We also assume that the effective operator enjoys the regularity property that its resolvent is continuous from L_2 to the Lipschitz–Besov space Λ_2^(1+s). We obtained approximations for the resolvent of A_ε in the (L_2→L_2)-norm with error of order ε^s and in the (L_2→H^1)-norm with error of order ε^(s/2). II. Spectral theory of periodic operators and related topics • We consider the Laplace operator in a d-dimensional parallelepiped with periodic boundary conditions. Its spectrum coincides with the set of the quadratic form values of a positive definite matrix A on integer vectors. We consider the case of matrix A with integer entries. It was known that the lengths of gaps in this set are unbounded in dimension d = 2 and are bounded in dimensions d ≥ 4. We showed that for d = 3 they are also bounded. • The Polya conjecture states that the counting function of eigenvalues of the Laplace operator with Dirichlet or Neumann boundary conditions on a bounded domain Ω in R^d, is uniformly bounded by the principal term of its Weyl asymptotic law. We proved the Polya conjecture for the Dirichlet problem and the Neumann problem in a circular sector of arbitrary aperture and also for the Dirichlet problem in a ball of arbitrary dimension. • We consider the equation –Δu(x,y) + V(x,y)u(x,y) = 0 in the half-cylinder [0,∞)x(0,a)^d. The functions u and V are real-valued, u is smooth, and V is bounded. We impose the periodic boundary conditions on the lateral side of the cylinder. In the case d ≥ 3, we constructed an example of a non-trivial solution that decays at the rate exp(-c x^{4/3}). It is known that this decay rate cannot be improved. III. Spectral theory of quasiperiodic difference operators We studied difference quasiperiodic Schrödinger operators in ℓ_2(Z) given by Af(k) = f(k+1) + f(k-1) + v(ωk+θ) f(k), where 0 < ω < 1 and 0 ≤ θ < 1. It was assumed that v is a continuous 1-periodic function, and the frequency ω is irrational. We also studied the operator B in L_2(R) given by Bψ(x) = ψ(x+ω) + ψ(x-ω) + v(x) ψ(x). For almost all θ the spectra of A and B (with the same v) coincide. • Sarnak's model. We considered the operator A with the potential v(x) = λ exp (-2π i x), where λ > 1. Consider the series Σ_(n>1) exp(-ξ / (ω_0 ω_1 ω_2 … ω_(n-1))) / ω_n, where ω_(j+1) = {1/ ω_j}, ω_0 = ω, 2π ξ = ln (λ), and { . } is the fractional part function. Provided that this series converges, it is shown that E(n) = 2 cos (2π (ωn+θ+iξ)), where n is integer, are eigenvalues of A. Moreover, corresponding eigenfunctions are constructed using the monodromization method. A solution of the Schrödinger equation Af = Ef can be expressed through a solution of the renormalized equation with new parameters instead of ω, θ, λ and E. After m renormalizations the solution of the resulting equation for all k on the interval of length of order L_m = 1/(ω_0 ω_1 ω_2 ... ω_(m-1)) determines the solution of the original equation. For E = E(n), for a subsequence m_j → ∞, with a suitable choice of the solutions of the m_j-th equations, the solutions of the original equation on intervals of length of order L_m converge in ℓ_2 to an eigenfunction of A. The described construction of eigenfunctions is new, and we expect that it will allow us to describe their multiscale self-similar structure. • A solution of the Schrödinger equation Bψ = E ψ is called its Bloch solution if it is invariant with respect to the shift by 1 up to multiplication by an ω-periodic factor. We proved that E is in the resolvent set of B if and only if there exist two linearly independent Bloch solutions of this equation exponentially decreasing in opposite directions. By studying Bloch solutions, we showed that operator B lacks a point spectrum. • The Ganeshan–Pixley–Das Sarma model. We investigated the operator B with the potential v(x) = λ / (1+ ε cos(2π x)), where 0 < ε < 1 and 0 < λ << 1. In the space of meromorphic solutions of the equation Bψ = E ψ, a basis is constructed for which the monodromy matrix turns out to be an unimodular meromorphic 1-periodic matrix-valued function having two simple poles on its period and tending to constants as Im x →±∞. The location of the poles is known. These properties of the monodromy matrix and the properties resulting from the fact that v is real on R determine the coefficients of the monodromy matrix up to two constant coefficients, the asymptotics, as λ → 0, of which we have found. These results allow us to asymptotically describe several gaps in the spectrum. • A model problem describing sound propagation in a narrow water wedge near a seashore is considered. Sound propagates in the lower halfplane P. This process is described by the Helmholtz equation ∆ U + k^2 U = 0, where the refraction index k is equal to k_0 > 1 in the "water wedge", a narrow angle inside P adjacent to its boundary, and equal to 1 in the "bottom", the remaining part of P. On the boundary of P, the Dirichlet condition is imposed, and on the water-bottom boundary, the solutions and their normal derivatives are continuous. We study a solution U that in the wedge sufficiently far from its vertex admits, as ε → 0, an asymptotic expansion of the type of an adiabatic normal wave (ε is the tangent of the wedge angle). The leading term is proportional to an eigenfunction of a Sturm–Liouville operator, which depends on the horizontal variable x as a parameter. As x approaches the vertex of the wedge, the corresponding eigenvalue moves to the edge of the continuous spectrum and, having reached it, disappears. We obtained a uniform asymptotics for U in the bottom near the "moment" of the disappearance of the eigenvalue. This asymptotics simplifies in a boundary layer, where the leading term is expressed in terms of the Airy functions and the exponential function of polynomials in the boundary layer coordinates.

 

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Annotation of the results obtained in 2023
I. Homogenization theory • We studied homogenization of hyperbolic equations in R^d. Let A_ε be a matrix elliptic second-order differential operator with periodic coefficients depending on x/ε. It is known that, as ε → 0, the operators cos (t A_ε^{1/2}) and A_ε^{-1/2} sin (t A_ε^{1/2}) converge in the (H^2→L_2)- and (H^1→L_2)-norms, respectively, to similar functions of the effective operator A^0, and the errors do not exceed C(1+|t|)ε. For A_ε^{-1/2} sin (t A_ε^{1/2}), an approximation in the (H^2→H^1)-norm is also known. We have found an approximation for the "corrected" operator cosine cos (t A_ε^{1/2}) (I + ε K(ε)), where K(ε) contains a rapidly oscillating factor, a first-order differential operator and a smoothing operator. For the "corrected" cosine, we found an approximation in the (H^4→L_2)-norm with error of order (1+|t|)^2 ε^2, and also an approximation in the (H^3→H^1)-norm with error of order (1+|t|)ε. For the operator A_ε^{-1/2} sin (t A_ε^{1/2}), an approximation in the (H^3→L_2)-norm with error of order (1+|t|)^2 ε^2 was obtained. In the general case, the results are sharp with respect to the norm type and with respect to the dependence of the estimates on t. However, we were able to improve them under some additional assumptions. The results were applied to homogenization of the Cauchy problem with initial data from a special class. • We studied homogenization of a nonlocal convolution type operator Н_ε u (x) = ε^{-d-2} ∫a((x-y)/ε) m(x/ε,y/ε) (u(x) - u(y)) dy in L_2(R^d). Here a(x) = a(-x) ≥ 0, the function (1+|x|)^4 a(x) is integrable, the function m(x,y) = m(y,x) is positive definite, bounded, and periodic in each variable. The resolvent of Н_ε converges in the operator norm on L_2 to the resolvent of the effective operator H^0 = - div g^0 grad; the error is of order O(ε). We have obtained a more accurate approximation for the resolvent with corrector taken into account and with error of order O(ε^2). • We studied homogenization of matrix elliptic operators A_ε of order 2p with periodic coefficients in R^d. Our goal is to construct an approximation for the resolvent of A_ε in the (L_2→H^p)-norm with error O(ε^p). We have developed an abstract operator-theoretic scheme to solve this problem. In the abstract setting, the required approximation for the resolvent of a nonnegative polynomial operator pencil was obtained. • We studied homogenization of a one-dimensional operator B_ε = D^4 + ε^{-4}V(x/ε). It is assumed that V(x) is a real-valued 1-periodic function of class L_1(0,1), the point λ = 0 is the spectral edge of the operator B = D^4 + V(x), the first band function E_1(k) of the operator B on the period has exactly two minimum points k_0 and -k_0, and E_1(k) behaves as b(k-k_0)^2 and b(k+k_0)^2 near these points, b >0. We have found an approximation for the resolvent (B_ε + I)^{-1} in the (L_2→L_2)-norm with error O(ε^2) via the sum of two resolvents (ε^{-2} b D^2 + I)^{-1} sandwiched between suitable rapidly oscillating factors. • We studied homogenization of matrix operators of the form A_ε = D* a_ε(x) D in a Lipschitz domain O with fairly general boundary conditions. The domain O is divided into Lipschitz domains O_i in which a_ε(x) has the form a(x,x/ε_i(ε)), where ε_i(ε) → 0 as ε → 0. The function a(x,y) is uniformly continuous on each O_i with respect to x and periodic and uniformly bounded with respect to y. It is assumed that on O_i the modulus of continuity of a(x,y) is majorized by a function φ_i(t) → 0 as t → 0. Given 1<p<∞, we denote the domain of the sesquilinear form of A_ε by ω_p^1(O) x ω_p'^1(O), where each of the two sets is a subspace of a suitable Sobolev space that corresponds to the boundary conditions. The resolvents of the operator A_ε and the effective operator A^0 are supposed to be continuous from ω_p^1(O) to ω_p'^1(O)*. It is also assumed that the (L_p(O)→L_p(O_i))-operator modulus of continuity of the gradient of the resolvent of A^0 is estimated by φ_i and that a similar condition holds for (A^0)^* with p replaced by p' and φ_i replaced by a function ψ_i. For the resolvent of A_ε, we obtained approximations in the (L_p→L_p)- and (L_p→W_p^1)-norms with errors of order max_i φ_i(ε_i(ε))^1/p ψ_i(ε_i(ε))^1/p' and max_i φ_i(ε_i(ε))^1/p, respectively. II. Spectral theory of periodic operators and related topics • Pólya's conjecture states that the counting function of the eigenvalues of the Laplace operator in a bounded domain with Dirichlet or Neumann boundary condition satisfies estimate via the leading term of the Weyl asymptotics. a) We have proved an analogue of Pólya's conjecture for the Schrödinger operator with the Aharonov-Bohm magnetic potential in a circle on the plane. b) For the Neumann problem for the Laplace operator in a two-dimensional convex domain we have proved an inequality similar to the Pólya conjecture with a constant that is less the one in the conjecture itself, but greater than that in the known Kröger estimate. c) For the Neumann problem in a ball we have proved the Pólya conjecture for dimensions d = 3 and d ≥ 55. • We consider the equation –Δ u + V u = 0 in R^4. The functions u and V are real-valued, u is smooth, V is bounded. We have constructed an example of a non-trivial solution that decays at the rate exp(-c |x|^{4/3 - ε}) for any ε > 0. It is known that the decay with the rate exp(-N |x|^{4/3}) for any N is impossible. III. Spectral theory of quasiperiodic difference operators • The Sarnak model. We studied a difference Schrödinger operator A with the potential j → λ v(jω+θ), where v(x) = exp (-2π i x), and an irrational frequency 0<ω<1, a coupling constant 0<λ<1, and θ are parameters. We imposed the condition of convergence of the series Σ_{n>1} exp( ξ / Ω(n) ) / ω(n), Ω(n) = ω(n-1) … ω(1) ω(0), ξ = ln( λ), where ω(0) = ω, and ω(n) is the Gaussian transformation of the number ω(n-1), n>0. Using the monodromization method, we have constructed generalized eigenfunctions of A of the form f(j) = exp( i p(ω j+θ)) Φ(ω j+θ), where p does not depend on the integer variable j, and Φ is a 1-periodic function. They are constructed in terms of a (regularized) infinite product of matrices, the convergence rate of which is determined by the convergence rate of the series in the imposed condition. The n-th matrix of the product is constructed in terms of solutions of the n-th difference equation arising in the renormalization process. These equations differ only in the values of constant parameters, and the construction clearly indicates self-similar properties of the generalized eigenfunctions. We studied the point spectrum of A with λ>1, obtained a new condition of its existence, and described the construction of eigenfunctions that clearly reflects their self-similar behavior at infinity. We studied the geometry of the spectrum of the difference Schrödinger operator on the real line with the potential v for different values of λ and ω. • The Ganeshan-Pixly-Das Sarma model. We studied a difference Schrödinger operator on R with a periodic real-analytic potential having two simple complex poles per period and tending to zero away from the real axis. It was proved that, for a sufficiently small coupling constant, there is a basis in the space of solutions of the Schrödinger equation in which the monodromy matrix is a periodic meromorphic function having two simple poles per period and bounded at infinity away from the real axis. All its entries are determined by the ones of its first line. Each of them is determined up to three constants. These constants satisfy relations that make it possible to restore them all from two real parameters. For small values of the coupling constant, asymptotic formulas for these constants were obtained. • Adiabatic evolution. On the semi-axis x>0, a one-dimensional non-stationary Schrödinger equation with a small parameter ε in front of the derivative with respect to time t was considered. The potential is a rectangular potential well of width 1-t. The stationary operator H has continuous spectrum σ_c = [0,∞) and a finite number of negative eigenvalues, which are approaching the edge of σ_c and then disappear in turn. We studied a solution that is close at some moment to the n-th eigenfunction of H. It was previously shown that as long as the n-th eigenvalue exists, the solution is localized inside the potential well. Also the delocalization, the behavior of the solution when the eigenvalue disappears, was described. It has been shown that after that, in a neighborhood of the potential well of the order 1/ε, the solution is asymptotically localized in a layer where ε^{2/3} x~1, has the order ε^{1/3} there, and decreases exponentially, as ε^{2/3} x increases. In the potential well, near the moments of disappearance of the eigenvalues with numbers less than n, the solution becomes of the order ε^{2/3} instead of ε.

 

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