Boundary-value problems for the Schröinger equation with rapidly oscillating and delta-liked potentials

This paper deals with boundary-value problems on the closed interval [a, b] for the Schrödinger equation with potential of the form q(x, μ ^?1 x) + ε ^?1 Q(ε ^?1 x), where q(x, ζ) is a 1-periodic (in ζ) function, Q(ξ) is a compactly supported function, 0 ∈ (a, b), and μ, ε are small positive parameters. The solutions of these boundary-value problemsup to O(ε +μ) are constructed by combining the homogenization method and the method of matching asymptotic expansions.     

On the optimal robust solution of IVPs with noisy information

We investigate the optimal solution of systems of initial-value problems with smooth right-hand side functions f from a Hölder class \(F^{r,\varrho }_{\text {reg}}\), where r ≥ 0 is the number of continuous derivatives of f, and ? ∈ (0, 1] is the Hölder exponent of rth partial derivatives. We consider algorithms that use n evaluations of f, the ith evaluation being corrupted by a noise δ_i of deterministic or random nature. For δ ≥ 0, in the deterministic case the noise δ_i is a bounded vector, ∥δ_i∥≤δ. In the random case, it is a vector-valued random variable bounded in average, (E(∥δ_i∥^q))^1/q ≤ δ, q ∈ [1, + ∞). We point out an algorithm whose L_p error (p ∈ [0, + ∞]) is O(n ^{? (r + ?)} + δ), independently of the noise distribution. We observe that the level n ^{? (r + ?)} + δ cannot be improved in a class of information evaluations and algorithms. For ε > 0, and a certain model of δ-dependent cost, we establish optimal values of n(ε) and δ(ε) that should be used in order to get the error at most ε with minimal cost.     

Modified Kantorovich theorem and asymptotic approximations of solutions to singularly perturbed systems of ordinary differential equations

The functional equation f(x,ε) = 0 containing a small parameter ε and admitting regular and singular degeneracy as ε → 0 is considered. By the methods of small parameter, a function x _n ⁰(ε) satisfying this equation within a residual error of O(ε ⁿ⁺¹) is found. A modified Newton’s sequence starting from the element x _n ⁰(ε) is constructed. The existence of the limit of Newton’s sequence is based on the NK theorem proven in this work (a new variant of the proof of the Kantorovich theorem substantiating the convergence of Newton’s iterative sequence). The deviation of the limit of Newton’s sequence from the initial approximation x _n ⁰(ε) has the order of O(ε ⁿ⁺¹), which proves the asymptotic character of the approximation x _n ⁰(ε). The method proposed is implemented in constructing an asymptotic approximation of a system of ordinary differential equations on a finite or infinite time interval with a small parameter multiplying the derivatives, but it can be applied to a wider class of functional equations with a small parameters.     

Asymptotic formula for the convolution of a generalized divisor function

An asymptotic formula is obtained for the sum of terms σ _it (n)σ_-it (N - n) (t is real) over 0 < n < N with a remainder estimated by O _ε((1+|t|)^1+ε N ^3/4+ε) for any ε > 0. As a consequence, Porter’s result on a power scale for the average number of steps in the Euclidean algorithm is improved.     

Two-Parameter Asymptotics in a Bisingular Cauchy Problem for a Parabolic Equation

The Cauchy problem for a quasilinear parabolic equation with a small parameter ε at the highest derivative is considered. The initial function, which has the form of a smoothed step, depends on a “stretched” variable x/ρ, where ρ is another small parameter. This problem statement is of interest for applications as a model of propagation of nonlinear waves in physical systems in the presence of small dissipation. In the case corresponding to a compression wave, asymptotic solutions of the problem are constructed in the parameters ε and ρ independently tending to zero. It is assumed that ε/ρ → 0. Far from the line of discontinuity of the limit solution, asymptotic solutions are constructed in the form of series in powers of ε and ρ. In a small domain of linear approximation, an asymptotic solution is constructed in the form of a series in powers of the ratio ρ/ε. The coefficients of the inner expansion are determined from a recursive chain of initial value problems. The asymptotics of these coefficients at infinity is studied. The time of reconstruction of the scale of the internal space variable is determined.     

Uniform asymptotic expansions for Laguerre polynomials and related confluent hypergeometric functions

Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials \(L_{n}^{(\alpha )}(x)\), as well as complementary confluent hypergeometric functions. The expansions are valid for n large and α small or large, uniformly for unbounded real and complex values of x. The new expansions extend the range of computability of \(L_{n}^{(\alpha )}(x)\) compared to previous expansions, in particular with respect to higher terms and large values of α. Numerical evidence of their accuracy for real and complex values of x is provided.     

The existence of semiclassical states for some <Emphasis Type="Italic">p</Emphasis>-Laplacian equation with critical exponent

In this paper, we study the existence of semiclassical states for some p-Laplacian equation. Under given conditions and minimax methods, we show that this problem has at least one positive solution provided that ε ≤ E; for any m ∈ ?, it has m pairs solutions if ε ≤ E _m , where E, E_m are sufficiently small positive numbers. Moreover, these solutions are closed to zero in W^1,p(? ^N ) as ε → 0.     

Operator renewal theory and mixing rates for dynamical systems with infinite measure

We develop a theory of operator renewal sequences in the context of infinite ergodic theory. For large classes of dynamical systems preserving an infinite measure, we determine the asymptotic behaviour of iterates L ⁿ of the transfer operator. This was previously an intractable problem.Examples of systems covered by our results include (i) parabolic rational maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly expanding interval maps with indifferent fixed points.In addition, we give a particularly simple proof of pointwise dual ergodicity (asymptotic behaviour of \(\sum_{j=1}^{n}L^{j}\)) for the class of systems under consideration.In certain situations, including Pomeau-Manneville intermittency maps, we obtain higher order expansions for L ⁿ and rates of mixing. Also, we obtain error estimates in the associated Dynkin-Lamperti arcsine laws.     

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation(NLDE) in the nonrelativistic limit regime, involving a small dimensionless parameter 0 ε≤ 1 which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e., there are propagating waves with wavelength O(ε~2) and O(1) in time and space,respectively. We begin with the conservative Crank-Nicolson finite difference(CNFD) method and establish rigorously its error estimate which depends explicitly on the mesh size h and time step τ as well as the small parameter 0 ε≤ 1. Based on the error bound, in order to obtain ‘correct' numerical solutions in the nonrelativistic limit regime, i.e., 0 ε■ 1, the CNFD method requests the ε-scalability: τ = O(ε~3) and h= O(ε~(1/2)). Then we propose and analyze two numerical methods for the discretization of NLDE by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and timesplitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their ε-scalability is improved to τ = O(ε~2) and h = O(1) when 0 ε■1. Extensive numerical results are reported to confirm our error estimates.     

Asymptotics of the Velocity Potential of an Ideal Fluid Flowing around a Thin Body

We consider the Neumann problem outside a small neighborhood of a planar disk in the three-dimensional space. The surface of this neighborhood is assumed to be smooth, and its thickness is characterized by a small parameter ε. A uniform asymptotic expansion of the solution of this problem with respect to ε is constructed by the matching method. Since the problem turned out to be bisingular, an additional inner asymptotic expansion in the so-called stretched variables is constructed near the edge of the disk. A physical interpretation of the solution of this boundary value problem is the velocity potential of a laminar flow of an ideal fluid around a thin body, which is the neighborhood of the disk. It is assumed that this flow has unit velocity at a large distance from the disk, which is equivalent to the following condition for the potential: u(x₁, x₂, x₃, ε) = x₃+O(r^?2) as r → ∞, where r is the distance to the origin. The boundary condition of this problem is the impermeability of the surface of the body: ?u/?n = 0 at the boundary. After subtracting x₃ from the solution u(x₁, x₂, x₃, ε), we get a boundary value problem for the potential ?(x₁, x₂, x₃, ε) of the perturbed motion. Since the integral of the function ??/?n over the surface of the body is zero, we have ?(x₁, x₂, x₃, ε) = O(r^?2) as r → ∞. Hence, all the coefficients of the outer asymptotic expansion with respect to ε have the same behavior at infinity. However, these coefficients have growing singularities at the approach to the edge of the disk, which implies the bisingularity of the problem.     

Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients

Let O ? R ^d be a bounded domain of class C ^1,1. Let 0 < ε - 1. In L ₂(O;C ⁿ ) we consider a positive definite strongly elliptic second-order operator B _D,ε with Dirichlet boundary condition. Its coefficients are periodic and depend on x/ε. The principal part of the operator is given in factorized form, and the operator has lower order terms. We study the behavior of the generalized resolvent (B _D,ε ? ζQ ₀(·/ε))^?1 as ε → 0. Here the matrix-valued function Q ₀ is periodic, bounded, and positive definite; ζ is a complex-valued parameter. We find approximations of the generalized resolvent in the L ₂(O;C ⁿ )-operator norm and in the norm of operators acting from L ₂(O;C ⁿ ) to the Sobolev space H ¹(O;C ⁿ ) with two-parameter error estimates (depending on ε and ζ). Approximations of the generalized resolvent are applied to the homogenization of the solution of the first initial-boundary value problem for the parabolic equation Q ₀(x/ε)? _t v _ε (x, t) = ?(B _D,ε v _ε )(x, t).     

Limit Distribution of Eigenvalues for Random Hankel and Toeplitz Band Matrices

Consider real symmetric, complex Hermitian Toeplitz, and real symmetric Hankel band matrix models where the bandwidth b _N →∞ but b _N /N→b∈[0,1] as N→∞. We prove that the distributions of eigenvalues converge weakly to universal symmetric distributions γ _T (b) and γ _H (b). In the case b>0 or b=0 but with the addition of \(b_{N}\geq CN^{\frac{1}{2}+\epsilon_{0}}\) for some positive constants ε ₀ and C, we prove the almost sure convergence. The even moments of these distributions are the sums of some integrals related to certain pair partitions. In particular, when the bandwidth grows slowly, i.e., b=0, γ _T (0) is the standard Gaussian distribution, and γ _H (0) is the distribution |x|exp?(?x ²). In addition, from the fourth moments, we know that γ _T (b) are different for different b, γ _H (b) different for different \(b\in[0,\frac{1}{2}]\), and γ _H (b) different for different \(b\in [\frac{1}{2},1]\).     

On Homogenization for Non-Self-Adjoint Periodic Elliptic Operators on an Infinite Cylinder

We consider an operator A^ε on L₂(\({\mathbb{R}^{{d_1}}} \times {T^{{d_2}}}\)) (d₁ is positive, while d₂ can be zero) given by A^ε = ?div A(ε^?1x₁,x₂)?, where A is periodic in the first variable and smooth in a sense in the second. We present approximations for (A^ε ? μ)^?1 and ?(A^ε ? μ)^?1 (with appropriate μ) in the operator norm when ε is small. We also provide estimates for the rates of approximation that are sharp with respect to the order.     

Besicovitch cylindrical transformation with a Hölder function

For any γ ∈ (0, 1) and ε > 0, we construct a cylindrical cascade with a γ-Hölder function over some rotation of the circle. This transformation has the Besicovitch property; i.e., it is topologically transitive and has discrete orbits. The Hausdorff dimension of the set of points of the circle that have discrete orbits is greater than 1 ? γ ? ε.     

Homogenization of hyperbolic equations

We consider a self-adjoint matrix elliptic operator A _{ε, ε} > 0, on L ₂(R ^d ;C ⁿ ) given by the differential expression b(D)*g(x/ε)b(D). The matrix-valued function g(x) is bounded, positive definite, and periodic with respect to some lattice; b(D) is an (m × n)-matrix first order differential operator such that m ≥ n and the symbol b(ξ) has maximal rank. We study the operator cosine cos(τA _ε ^1/2 ), where τ ∈ R. It is shown that, as ε → 0, the operator cos(τA _ε ^1/2 ) converges to cos(τ(A ⁰)^1/2) in the norm of operators acting from the Sobolev space H ^s (R ^d ;C ⁿ ) (with a suitable s) to L ₂(R ^d ;C ⁿ ). Here A ₀ is the effective operator with constant coefficients. Sharp-order error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the hyperbolic equation ? _τ ² u _ε (x, τ) = ?A _ε u _ε (x, τ).     

Homogenization of a nonstationary model equation of electrodynamics

In L ₂(?₃;?₃), we consider a self-adjoint operator ? _ε , ε > 0, generated by the differential expression curl η(x/ε)^?1 curl??ν(x/ε) div. Here the matrix function η(x) with real entries and the real function ν(x) are periodic with respect to some lattice, are positive definite, and are bounded. We study the behavior of the operators cos(τ? _ε ^1/2 ) and ? _ε ^?1/2 sin(τ? _ε ^1/2 ) for τ ∈ ? and small ε. It is shown that these operators converge to cos(τ(?⁰)^1/2) and (?₀)^?1/2 sin(τ(?₀)^1/2), respectively, in the norm of the operators acting from the Sobolev space H ^s (with a suitable s) to ?₂. Here ?⁰ is an effective operator with constant coefficients. Error estimates are obtained and the sharpness of the result with respect to the type of operator norm is studied. The results are used for homogenizing the Cauchy problem for the model hyperbolic equation ? _τ ² v _ε = ?? _ε v _ε , div v _ε = 0, appearing in electrodynamics. We study the application to a nonstationary Maxwell system for the case in which the magnetic permeability is equal to 1 and the dielectric permittivity is given by the matrix η(x/ε).     

Skew inverse power series rings over a ring with projective socle

A ring R is called a right PS-ring if its socle, Soc(R _R ), is projective. Nicholson and Watters have shown that if R is a right PS-ring, then so are the polynomial ring R[x] and power series ring R[[x]]. In this paper, it is proved that, under suitable conditions, if R has a (flat) projective socle, then so does the skew inverse power series ring R[[x ^?1; α, δ]] and the skew polynomial ring R[x; α, δ], where R is an associative ring equipped with an automorphism α and an α-derivation δ. Our results extend and unify many existing results. Examples to illustrate and delimit the theory are provided.     

Basic expansions of solutions to the sixth Painlevé equation in the generic case

We consider the sixth Painlevé equation for generic values of its four complex parameters. By methods of power geometry, we obtain those asymptotic expansions of solutions to the equation near the singular point x = 0 for which the order of the first term is less than unity. We refer to these expansions as basic expansions. They form 10 families and include expansions of four types, namely, power, power-logarithmic, complicated, and exotic. All other asymptotic expansions of solutions to the equation near the three singular points x = 0, x = 1, and x = ∞ can be computed from the basic expansions with the use of symmetries of the equation. Most of these expansions are new.     

On homogenization for non-self-adjoint locally periodic elliptic operators

In this note we consider the homogenization problem for a matrix locally periodic elliptic operator on R ^d of the form A ^ε = ?divA(x, x/ε)?. The function A is assumed to be Hölder continuous with exponent s ∈ [0, 1] in the “slow” variable and bounded in the “fast” variable. We construct approximations for (A ^ε ? μ)^?1, including one with a corrector, and for (?Δ) ^s/2(A ^ε ? μ)^?1 in the operator norm on L ₂(R ^d ) ⁿ . For s ≠ 0, we also give estimates of the rates of approximation.     

Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions

Let D be an open connected subset of the complex plane C with sufficiently smooth boundary ?D. Perturbing the Cauchy problem for the Cauchy–Riemann system ??u = f in D with boundary data on a closed subset S ? ?D, we obtain a family of mixed problems of the Zaremba-type for the Laplace equation depending on a small parameter ε ∈ (0, 1] in the boundary condition. Despite the fact that the mixed problems include noncoercive boundary conditions on ?D\S, each of them has a unique solution in some appropriate Hilbert space H ⁺(D) densely embedded in the Lebesgue space L ₂(?D) and the Sobolev–Slobodetski? space H ^1/2?δ(D) for every δ > 0. The corresponding family of the solutions {u _ε} converges to a solution to the Cauchy problem in H ⁺(D) (if the latter exists). Moreover, the existence of a solution to the Cauchy problem in H ⁺(D) is equivalent to boundedness of the family {u _ε} in this space. Thus, we propose solvability conditions for the Cauchy problem and an effective method of constructing a solution in the form of Carleman-type formulas.