Operator In L2 Research Articles

On the quadratic closeness of eigenfunctions of «unperturbed» and «perturbed» differentiation operators on an interval

The article considers the eigenvalue problem of the differentiation operator, when the spectral param- eter is also present in the boundary condition with an integral perturbation, where the integrand has the property of limited variation and has a value of unity at the ends of the segment [-1, 1]. The derivatives with respect to the time variable included in the boundary condition naturally arise when solving (by the Fourier method) initial boundary value problems for evolution equations. The conjugate operator is constructed. It is shown that the spectral questions of the conjugate operator have a similar structure. The characteristic determinant of the original direct spectral problem with an integral perturbation of the boundary condition and for the eigenvalue problem of a loaded first-order differential equation on a segment with a periodic boundary condition, which is an entire analytical function of the spectral parameter, is constructed. Based on the formula of the characteristic determinant, conclusions are drawn about the asymptotic behavior of the spectrum of the original «perturbed» differentiation operator and the loaded first-order differential equation on the segment. A special feature of the operator under consideration is the non-self-adjointness of the operator in L2(–1,1). The quadratic proximity of the systems of eigen functions of the «unperturbed» and «perturbed» differentiation operators and the Riesz basis property of these systems are proved. In this case, the system of eigenfunctions of the «perturbed» operator is not orthonormal.

The point scatterer approximation for wave dynamics

Given an open, bounded and connected set Ω⊂R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subset \\mathbb {R}^{3}$$\\end{document} and its rescaling Ωε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega _{\\varepsilon }$$\\end{document} of size ε≪1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon \\ll 1$$\\end{document}, we consider the solutions of the Cauchy problem for the inhomogeneous wave equation (ε-2χΩε+χR3\\Ωε)∂ttu=Δu+f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} (\\varepsilon ^{-2}\\chi _{\\Omega _{\\varepsilon }}+\\chi _{\\mathbb {R}^{3}\\backslash \\Omega _{\\varepsilon }})\\partial _{tt}u=\\Delta u+f \\end{aligned}$$\\end{document}with initial data and source supported outside Ωε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega _{\\varepsilon }$$\\end{document}; here, χS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi _{S}$$\\end{document} denotes the characteristic function of a set S. We provide the first-order ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}-corrections with respect to the solutions of the inhomogeneous free wave equation and give space-time estimates on the remainders in the L∞((0,1/ετ),L2(R3))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{\\infty }((0,1/\\varepsilon ^{\ au }),L^{2}(\\mathbb {R}^{3})) $$\\end{document}-norm. Such corrections are explicitly expressed in terms of the eigenvalues and eigenfunctions of the Newton potential operator in L2(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{2}(\\Omega )$$\\end{document} and provide an effective dynamics describing a legitimate point scatterer approximation in the time domain.

Non-self-adjoint quasi-periodic operators with complex spectrum

We give a precise and complete description on the spectrum for a class of non-self-adjoint quasi-periodic operators acting on ℓ2(Zd) which contains the Sarnak's model as a special case. As a consequence, one can see various interesting spectral phenomena including PT symmetric breaking, the non-simply-connected two-dimensional spectrum in this class of operators. Particularly, we provide new examples of non-self-adjoint operator in ℓ2(Z) whose spectra (actually a two-dimensional subset of C) can not be approximated by the spectra of its finite-interval truncations.

Resolvent expansions of 3D magnetic Schrödinger operators and Pauli operators

We obtain asymptotic resolvent expansions at the threshold of the essential spectrum for magnetic Schrödinger and Pauli operators in dimension three. These operators are treated as perturbations of the Laplace operator in L2(R3) and L2(R3;C2), respectively. The main novelty of our approach is to show that the relative perturbations, which are first order differential operators, can be factorized in suitably chosen auxiliary spaces. This allows us to derive the desired asymptotic expansions of the resolvents around zero. We then calculate their leading and sub-leading terms explicitly. Analogous factorization schemes for more general perturbations, including e.g. finite rank perturbations, are discussed as well.

One-dimensional Schrödinger operator with decaying white noise potential

In this paper, we consider a random one-dimensional Schrödinger operator Hω on the half line which has, as its potential term, Gaussian white noise multiplied by a decaying factor. Although the potential term is not an ordinary function, but a distribution, it is possible to realize Hω as a symmetric operator in L2([0, ∞); dt) as was pointed out by the present author [Minami, Lect. Notes Math. 1299, 298 (1986)], and it will be shown that Hω is actually self-adjoint with probability one. When the white noise in Hω is replaced by random functions of a specific type, [Kotani and Ushiroya, Commun. Math. Phys. 115, 247 (1988)] made a precise analysis of the positive part of the spectrum. According to them, if the decaying factor is not square integrable, the positive part of the spectrum typically consists of singular continuous and dense pure-point parts, which are separated by a threshold number. On the other hand, the positive part of the spectrum is purely absolutely continuous when the decaying factor is square integrable. In this work, we shall focus on the case of square integrable decaying factor, and prove the absolute continuity of the positive part of the spectrum of Hω. We shall further prove that the negative part of the spectrum of Hω is discrete, with no accumulation points other than 0.

Dynamics of unbounded linear operators

We apply the well-known and also the newly introduced notions from bounded linear dynamics to unbounded linear operators. We present a hypermixing criterion similar to that given for bounded linear operators and then we show that the derivative operator in H2, the Laplacian operator in L2(Ω), and all unbounded weighted translations in Lp(0,∞) and C0[0,∞) are hypermixing.

On the spectrum of convolution operator with a potential

This paper focuses on the spectral properties of a bounded self-adjoint operator in L2(Rd) being the sum of a convolution operator with an integrable convolution kernel and an operator of multiplication by a continuous potential converging to zero at infinity. We study both the essential and the discrete spectra of this operator. It is shown that the essential spectrum of the sum is the union of the essential spectrum of the convolution operator and the image of the potential. We then provide a number of sufficient conditions for the existence of discrete spectrum and obtain lower and upper bounds for the number of discrete eigenvalues. Special attention is paid to the case of operators possessing countably many points of the discrete spectrum. We also compare the spectral properties of the operators considered in this work with those of classical Schrödinger operators.

Ergodic theorems for Cesàro bounded operators in L1

We consider a positive invertible Lamperti operator T with positive inverse. Our result concerns the averages Ak,n, 0≤k,n∈Z, and the ergodic maximal operatorMTf=supk,n≥0⁡|Ak,nf|=supk,n≥0⁡|1k+n+1∑j=−knTjf|, the ergodic Hilbert transform defined by HTf(x)=limn→∞⁡∑j=1n1j(Tjf(x)−T−jf(x)) and the ergodic power functions defined by Pr,Tf(x)=(∑n=0∞|An+1,0f(x)−An,0f(x)|r+|A0,n+1f(x)−A0,nf(x)|r)1/r, for 1<r<+∞. Several authors proved that if the averages are uniformly bounded in Lp, 1<p<∞, then these operators are bounded in Lp[23,24,27,28,21]. Regarding the case p=1, Gillespie and Torrea [12] showed that this condition is not sufficient to assure MT is of weak type (1,1). We provide two sufficient conditions that recall the assumptions in the Dunford-Schwartz theorem: if the averages are uniformly bounded in L1 and in L∞ then MT, HT and Pr,T apply L1 into weak-L1 and the corresponding sequences of functions in L1 converge a.e. and in measure. Furthermore, we reach the same conclusions by assuming a weaker condition: we replace the uniform boundedness of the averages An,n in L∞ by the assumption that, for a fixed p∈(1,∞), the averages associated with a modified operator Tp, related with T, are uniformly bounded in Lp. We end the paper showing examples of nontrivial operators satisfying the assumptions of the two main results.

Spectral analysis of Dirac operators with delta interactions supported on the boundaries of rough domains

Given an open set Ω⊂R3, we deal with the spectral study of Dirac operators of the form Ha,τ = H + Aa,τδ∂Ω, where H is the free Dirac operator in R3 and Aa,τ is a bounded, invertible, and self-adjoint operator in L2(∂Ω)4, depending on parameters (a,τ)∈R×Rn, n ⩾ 1. We investigate the self-adjointness and the related spectral properties of Ha,τ, such as the phenomenon of confinement and the Sobolev regularity of the domain in different situations. Our set of techniques, which is based on the use of fundamental solutions and layer potentials, allows us to tackle the above problems under mild geometric measure theoretic assumptions on Ω.

Absence of absolutely continuous spectrum for generic quasi-periodic Schrödinger operators on the real line

We show that a generic quasi-periodic Schrödinger operator in L2(ℝ) has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schrödinger operator with the resulting potential has empty absolutely continuous spectrum.

Homogenization of a non-stationary periodic Maxwell system in the case of constant permeability

In L2(R3;C3), we consider a selfadjoint operator Lε, ε>0, given by the differential expression μ−1/2curlη(x/ε)−1curlμ−1/2−μ1/2∇ν(x/ε)divμ1/2, where μ is a constant positive matrix, a matrix-valued function η(x) and a real-valued function ν(x) are periodic with respect to some lattice, positive definite and bounded. We study the behavior of the operator-valued functions cos⁡(τLε1/2) and Lε−1/2sin⁡(τLε1/2) for τ∈R and small ε. It is shown that these operators converge to the corresponding operator-valued functions of the operator L0 in the norm of operators acting from the Sobolev space Hs (with a suitable s) to L2. Here L0 is the effective operator with constant coefficients. Also, an approximation with corrector in the (Hs→H1)-norm for the operator Lε−1/2sin⁡(τLε1/2) is obtained. We prove error estimates and study the sharpness of the results regarding the type of the operator norm and regarding the dependence of the estimates on τ. The results are applied to homogenization of the Cauchy problem for the non-stationary Maxwell system in the case where the magnetic permeability is equal to μ, and the dielectric permittivity is given by the matrix η(x/ε).

Nonlocal Problem with Multipoint Perturbations of Dirichlet Conditions for Even-Order Partial Differential Equations with Constant Coefficients

For a partial differential equation of order 2n with constant coefficients in the domain G := {x = (x1,…, xm) : 0 < xj < 1 < ∞, j = 1,…, m, m ϵ ℕ} , we study the problem with conditions that are multipoint perturbations of the Dirichlet boundary conditions by using the Fourier method. To investigate the spectral properties of a multipoint problem, we use the operator of transformation R: L2 (G) → L2 (G) that establishes the relationship RL0 = LR between the self-adjoint operator L0 of the Dirichlet problem and the operator L of multipoint problem. The solution of the problem with homogeneous multipoint conditions is constructed in the form of Fourier series in the system of eigenfunctions of the operator of the problem. Moreover, the conditions for its existence and uniqueness are established.

On an Evolutionary Dynamical System of the First Order with Boundary Control

The work is carried out as part of the program to construct a new functional (so-called wave) model of symmetric operators. It is shown that an abstract evolutionary dynamic system of the first order (with respect to time) with boundary control, which is determined by a symmetric operator L0 : → , is controllable if and only if L0 has no maximal symmetric parts in .

A noncommutative generalisation of a problem of Steinhaus

We extend the Révész and Komlós theorems to arbitrary finite von Neumann algebras, and in doing so solve an open problem of Randrianantoanina, removing the need for hyperfiniteness. The main result is the noncommutative Komlós theorem, which states that every norm-bounded sequence of operators in L1(M), for any finite von Neumann algebra M, admits a subsequence, such that for any further subsequence, the Cesàro averages converge bilaterally almost uniformly. This is a natural extension of Komlós' original result to the noncommutative setting.The necessary techniques which facilitate the proof also allow us to extend the Révész theorem to the noncommutative setting, which gives a similar subsequential law for series over bounded sequences in L2(M).

On the Fourier-Sine and Kontorovich–Lebedev Generalized Convolution Transforms and Their Applications

We study generalized convolutions for the Fourier sine and Kontorovich–Lebedev transforms $$ \left(h\underset{F_s,K}{\ast }f\right)(x) $$ in a two-parameter function space $$ {L}_p^{\alpha, \beta}\left({\mathrm{\mathbb{R}}}_{+}\right) $$ . We obtain several estimates for the norms and prove a Young-type inequality for this generalized convolution. We also impose necessary and sufficient conditions on the kernel h to ensure that the generalized convolution transform $$ {D}_h:f\mapsto {D}_h\left[f\right]=\left(1-\frac{d^2}{dx^2}\right)\left(h\underset{F_s,K}{\ast }f\right)(x) $$ is a unitary operator in L2(ℝ+) (Watson-type theorem) and derive its inverse formula. Finally, we apply these results to an integrodifferential equation and obtain an estimate for the solution in the Lp-norm.

Equivalent metrics on normal composition operators

We define some metrics on the set of all bounded normal composition operators in L2(Σ), and show that these metrics are equivalent with the metric induced by the usual operator norm.

Yang–Baxter integrable dimers on a strip

The dimer model on a strip is considered as a Yang–Baxter integrable six vertex model at the free-fermion point with crossing parameter and quantum group invariant boundary conditions. A one-to-many mapping of vertex onto dimer configurations allows for the solution of the free-fermion model to be applied to the anisotropic dimer model on a square lattice where the dimers are rotated by compared to their usual orientation. In a suitable gauge, the dimer model is described by the Temperley–Lieb algebra with loop fugacity . It follows that the model is exactly solvable in geometries of arbitrary finite size. We establish and solve transfer matrix inversion identities on the strip with arbitrary finite width N. In the continuum scaling limit, in sectors with magnetization Sz, we obtain the conformal weights where . We further show that the corresponding finitized characters decompose into sums of q-Narayana numbers or, equivalently, skew q-binomials. In the particle representation, the local face tile operators give a representation of the fermion algebra and the fermion particle trajectories play the role of nonlocal degrees of freedom. We argue that, in the continuum scaling limit, there exist nontrivial Jordan blocks of rank 2 in the Virasoro dilatation operator L0. This confirms that, with quantum group invariant boundary conditions, the dimer model gives rise to a logarithmic conformal field theory with central charge c = −2, minimal conformal weight and effective central charge . Our analysis of the structure of the ensuing rank 2 modules indicates that the familiar staggered c = −2 modules appear as submodules.

On the description of multidimensional normal Hausdorff operators on Lebesgue spaces

Abstract Hausdorff operators originated from some classical summation methods. Now this is an active research field. In the present article, a spectral representation for multidimensional normal Hausdorff operator is given. We show that normal Hausdorff operator in L 2 ⁢ ( ℝ n ) {L^{2}(\mathbb{R}^{n})} is unitary equivalent to the operator of multiplication by some matrix-valued function (its matrix symbol) in the space L 2 ⁢ ( ℝ n ; ℂ 2 n ) {L^{2}(\mathbb{R}^{n};\mathbb{C}^{2^{n}})} . Several corollaries that show that properties of a Hausdorff operator are closely related to the properties of its symbol are considered. In particular, the norm and the spectrum of such operators are described in terms of the symbol.

The continuous Anderson hamiltonian in d ≤ 3

We construct the continuous Anderson hamiltonian on (−L,L)d driven by a white noise and endowed with either Dirichlet or periodic boundary conditions. Our construction holds in any dimension d≤3 and relies on the theory of regularity structures [13]: it yields a self-adjoint operator in L2((−L,L)d) with pure point spectrum. In d≥2, a renormalisation of the operator by means of infinite constants is required to compensate for ill-defined products involving functionals of the white noise. We also obtain left tail estimates on the distributions of the eigenvalues: in particular, for d=3 these estimates show that the eigenvalues do not have exponential moments.

Perturbation Theory for Solutions to Second Order Elliptic Operators with Complex Coefficients and the Lp Dirichlet Problem

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If L0 = divA0(x)∇ + B0(x). ∇ is a p-elliptic operator satisfying the assumptions of Theorem 1.1 then the Lp Dirichlet problem for the operator L0 is solvable in the upper half-space ℝ+n. In this paper we prove that the Lp solvability is stable under small perturbations of L0. That is if L1 is another divergence form elliptic operator with complex coefficients and the coefficients of the operators L0 and L1 are sufficiently close in the sense of Carleson measures, then the Lp Dirichlet problem for the operator L1 is solvable for the same value of p. As a corollary we obtain a new result on Lp solvability of the Dirichlet problem for operators of the form L = divA(x)∇ + B(x) · ∇ where the matrix A satisfies weaker Carleson condition (expressed in term of oscillation of coefficients). In particular the coefficients of A need no longer be differentiate and instead satisfy a Carleson condition that controls the oscillation of the matrix A over Whitney boxes. This result in the real case has been established by Dindos, Petermichl and Pipher.