Finite Rank Perturbations Research Articles

Exact hydrodynamic manifolds for the linear Boltzmann BGK equation I: spectral theory

We perform a complete spectral analysis of the linear three-dimensional Boltzmann BGK operator resulting in an explicit transcendental equation for the eigenvalues. Using the theory of finite-rank perturbations, we confirm the existence of a critical wave number kcrit\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k_{\ extrm{crit}}$$\\end{document} which limits the number of hydrodynamic modes in the frequency space. This implies that there are only finitely many isolated eigenvalues above the essential spectrum at each wave number, thus showing the existence of a finite-dimensional, well-separated linear hydrodynamic manifold as a combination of invariant eigenspaces. The obtained results can serve as a benchmark for validating approximate theories of hydrodynamic closures and moment methods and provides the basis for the spectral closure operator.

On a nonlocal integral operator commuting with the Laplacian and the Sturm–Liouville problem: Low rank perturbations of the operator

We reformulate all general real coupled self-adjoint boundary value problems as integral operators and show that they are all finite rank perturbations of the free space Green’s function on the real line. This free space Green’s function corresponds to the nonlocal boundary value problem proposed earlier by Saito [Appl. Comput. Harmon. Anal. 25, 68–97 (2008)]. We prove these perturbations to be polynomials of rank up to 4. They encapsulate in a fundamental way the corresponding boundary conditions.

Finite rank perturbations of normal operators: hyperinvariant subspaces and a problem of Pearcy

Finite rank perturbations of normal operators: hyperinvariant subspaces and a problem of Pearcy

On the Fourier asymptotics of absolutely continuous measures with power-law singularities

We prove sharp estimates on the time-average behavior of the squared absolute value of the Fourier transform of some absolutely continuous measures that may have power-law singularities, in the sense that their Radon–Nikodym derivatives diverge with a power-law order. We also discuss an application to spectral measures of finite-rank perturbations of the discrete Laplacian.

Spectral analysis and hydrodynamic manifolds for the linearized Shakhov model

We perform a complete spectral analysis of the linearized Shakhov model involving two relaxation times τfast and τslow. Our results are based on spectral functions derived from the theory of finite-rank perturbations, which allows us to derive an explicit expression for the critical wave number kcrit limiting the number of discrete eigenvalues above the essential spectrum together with the existence of a finite-dimensional slow manifold defining non-local hydrodynamics. We discuss the merging of hydrodynamic modes as well as the existence of second sound and the appearance of ghost modes beneath the essential spectrum in dependence of the Prandtl number.

Tamed stability of finite difference schemes for the transport equation on the half-line

In this paper, we prove that, under precise spectral assumptions, some finite difference approximations of scalar leftgoing transport equations on the positive half-line with numerical boundary conditions are ℓ 1 \ell ^1 -stable but ℓ q \ell ^q -unstable for any q > 1 q>1 . The proof relies on the accurate description of the Green’s function for a particular family of finite rank perturbations of Toeplitz operators whose essential spectrum belongs to the closed unit disk and with a simple eigenvalue of modulus 1 1 embedded into the essential spectrum.

Finite rank perturbations of heavy-tailed Wigner matrices

One-rank perturbations of Wigner matrices have been closely studied: let [Formula: see text] with [Formula: see text] symmetric, [Formula: see text] i.i.d. with centered standard normal distributions, and [Formula: see text] It is well known [Formula: see text] the largest eigenvalue of [Formula: see text] has a phase transition at [Formula: see text]: when [Formula: see text] [Formula: see text] whereas for [Formula: see text] [Formula: see text] Under more general conditions, the limiting behavior of [Formula: see text] appropriately normalized, has also been established: it is normal if [Formula: see text] or the convolution of the law of [Formula: see text] and a Gaussian distribution if [Formula: see text] is concentrated on one entry. These convergences require a finite fourth moment, and this paper considers situations violating this condition. For symmetric distributions [Formula: see text] heavy-tailed with index [Formula: see text] the fluctuations are shown to be universal and dependent on [Formula: see text] but not on [Formula: see text] whereas a subfamily of the edge case [Formula: see text] displays features of both the light- and heavy-tailed regimes: two limiting laws emerge and depend on whether [Formula: see text] is localized, each presenting a continuous phase transition at [Formula: see text] respectively. These results build on our previous work which analyzes the asymptotic behavior of [Formula: see text] in the aforementioned subfamily.

Finite rank perturbations of normal operators: Spectral idempotents and decomposability

We prove that a large class of finite rank perturbations of diagonal operators and, in general, of diagonalizable normal operators of multiplicity one acting boundedly on a separable, infinite dimensional complex Hilbert space are decomposable operators in the sense of Colojoară and Foiaş [1]. Consequently, every operator T in such a class has a rich spectral structure and plenty of non-trivial closed hyperinvariant subspaces which extends, in particular, previous theorems of Foiaş, Jung, Ko and Pearcy [5–7], Fang and J. Xia [3] and the authors [8,9] on an open question posed by Pearcy in the seventies.

Overlaps between eigenvectors of spiked, correlated random matrices: From matrix principal component analysis to random Gaussian landscapes.

We consider pairs of Gaussian orthogonal ensemble matrices which are correlated with each other and subject to additive and multiplicative rank-one perturbations. We focus on the regime of parameters in which the finite-rank perturbations generate outliers in the spectrum of the matrices. We investigate the statistical correlation (i.e., the typical overlap) between the eigenvectors associated to the outlier eigenvalues of each matrix in the pair, as well as the typical overlap between the outlier eigenvector of one matrix with the eigenvectors in the bulk of the spectrum of the other matrix. We discuss implications of these results for the signal recovery problem for spiked matrices, as well as for problems of high-dimensional random landscapes.

Dispersive estimates for the Schrödinger equation with finite rank perturbations

In this paper, we investigate dispersive estimates for the time evolution of HamiltoniansH=−Δ+∑j=1N〈⋅,φj〉φjinRd,d≥1, where each φj satisfies certain smoothness and decay conditions. We show that, under a spectral assumption, there exists a constant C=C(N,d,φ1,…,φN)>0 such that‖e−itH‖L1−L∞≤Ct−d2,fort>0. As far as we are aware, this seems to provide the first study of L1−L∞ estimates for finite rank perturbations of the Laplacian in any dimension.We first deal with rank one perturbations (N=1). Then we turn to the general case. By using an Aronszajn-Krein type formula for finite rank perturbations, we reduce the problem to the rank one case and solve it in a unified manner. Moreover, we show that in some specific situations, the constant C(N,d,φ1,…,φN) grows polynomially in N. Finally, as an application, we are able to extend the results to N=∞ and deal with some trace class perturbations.

Eigenvalue Distribution of a High-Dimensional Distance Covariance Matrix With Application

We introduce a new random matrix model called distance covariance matrix in this paper, whose normalized trace is equivalent to the distance covariance. We first derive a deterministic limit for the eigenvalue distribution of the distance covariance matrix when the dimensions of the vectors and the sample size tend to infinity simultaneously. This limit is valid when the vectors are independent or weakly dependent through a finite-rank perturbation. It is also universal and independent of the details of the distributions of the vectors. Furthermore, the top eigenvalues of this distance covariance matrix are shown to obey an exact phase transition when the dependence of the vectors is of finite rank. This finding enables the construction of a new detector for such weak dependence where classical methods based on large sample covariance matrices or sample canonical correlations may fail in the considered high-dimensional framework.

Spectral analysis for finite rank perturbation of diagonal operator in non-archimedean Banach space of countable type

In this paper we are concerned with the spectral analysis for the classes of finte rank perturbations of diagonal operators in the form A = D+F where D is a diagonal operator and F = u’1 ⊗v’1+...+u’m ⊗vm is an operator of finite rank in the non-archimedean Banach space of countable type. We compute the spectrum of A using the theory of Fredholm operators in non archimedean setting and the concept of essential spectrum for linear operators.

Spectral measures for derivative powers via matrix-valued Clark theory

The theory of finite-rank perturbations allows for the determination of spectral information for broad classes of operators using the tools of analytic function theory. In this work, finite-rank perturbations are applied to powers of the derivative operator, providing a full account from self-adjoint boundary conditions to computing aspects of the operators' matrix-valued spectral measures. In particular, the support and weights of the Clark (spectral) measures are computed via the connection between matrix-valued contractive analytic functions and matrix-valued nonnegative measures through the Herglotz Representation Theorem. For operators associated with several powers of the derivative, explicit formulae for these measures are included. While eigenfunctions and eigenvalues for these operators with fixed boundary conditions can often be computed using direct methods from ordinary differential equations, this approach provides a more complete picture of the spectral information.

Finite rank perturbations of normal operators: Spectral subspaces and Borel series

We characterize the spectral subspaces associated to closed sets of rank-one perturbations of diagonal operators and, in general, of diagonalizable normal operators of multiplicity one acting boundedly on a separable, infinite dimensional complex Hilbert space by means of functional equations involving Borel series. As a particular instance, if T=DΛ+u⊗v is a rank-one perturbation of a diagonalizable normal operator DΛ with respect to a basis E=(en)n≥1 and the vectors u and v have Fourier coefficients (αn)n≥1 and (βn)n≥1 with respect to E, respectively, it is shown that T has non-trivial closed invariant subspaces provided that either (αn)n≥1∈ℓ1 or (βn)n≥1∈ℓ1. Likewise, analogous results hold for finite rank perturbations of DΛ. Moreover, such operators T have non-trivial closed hyperinvariant subspaces whenever they are not a scalar multiple of the identity extending previous theorems of Foiaş, Jung, Ko and Pearcy [8] and of Fang and J. Xia [6] on an open question of at least forty years.

Sharp stability for finite difference approximations of hyperbolic equations with boundary conditions

Abstract In this article, we consider a class of finite rank perturbations of Toeplitz operators that have simple eigenvalues on the unit circle. Under a suitable assumption on the behavior of the essential spectrum, we show that such operators are power bounded. The problem originates in the approximation of hyperbolic partial differential equations with boundary conditions by means of finite difference schemes. Our result gives a positive answer to a conjecture by Trefethen, Kreiss and Wu that only a weak form of the so-called uniform Kreiss–Lopatinskii condition is sufficient to imply power boundedness.

Kernels of perturbed Toeplitz operators in vector-valued Hardy spaces

Recently, Liang and Partington (Integr Equ Oper Theory 92(4): 35, 2020) show that kernels of finite-rank perturbations of Toeplitz operators are nearly invariant with finite defect under the backward shift operator acting on the scalar-valued Hardy space. In this article, we provide a vectorial generalization of a result of Liang and Partington. As an immediate application, we identify the kernel of perturbed Toeplitz operator in terms of backward shift-invariant subspaces in various important cases by applying the recent theorem (see Chattopadhyay et al. in Integr Equ Oper Theory 92(6): 52, 2020, Theorem 3.5 and O’Loughlin in Complex Anal Oper Theory 14(8): 86, 2020, Theorem 3.4) in connection with nearly invariant subspaces of finite defect for the backward shift operator acting on the vector-valued Hardy space.

Higher order local Dirichlet integrals and de Branges-Rovnyak spaces

We investigate expansive Hilbert space operators T that are finite rank perturbations of isometric operators. If the spectrum of T is contained in the closed unit disc D‾, then such operators are of the form T=U⊕R, where U is isometric and R is unitarily equivalent to the operator of multiplication by the variable z on a de Branges-Rovnyak space H(B). In fact, the space H(B) is defined in terms of a rational operator-valued Schur function B. In the case when dim⁡ker⁡T⁎=1, then H(B) can be taken to be a space of scalar-valued analytic functions in D, and the function B has a mate a defined by |B|2+|a|2=1 a.e. on ∂D. We show the mate a of a rational B is of the form a(z)=a(0)p(z)q(z), where p and q are appropriately derived from the characteristic polynomials of two associated operators. If T is a 2m-isometric expansive operator, then all zeros of p lie in the unit circle, and we completely describe the spaces H(B) by use of what we call the local Dirichlet integral of order m at the point w∈∂D.

Generalized eigenfunctions for quantum walks via path counting approach

We consider the time-independent scattering theory for time evolution operators of one-dimensional two-state quantum walks. The scattering matrix associated with the position-dependent quantum walk naturally appears in the asymptotic behavior at the spatial infinity of generalized eigenfunctions. The asymptotic behavior of generalized eigenfunctions is a consequence of an explicit expression of the Green function associated with the free quantum walk. When the position-dependent quantum walk is a finite rank perturbation of the free quantum walk, we derive a kind of combinatorial construction of the scattering matrix by counting paths of quantum walkers. We also mention some remarks on the tunneling effect.

Spectra of Jacobi Operators via Connection Coefficient Matrices

We address the computational spectral theory of Jacobi operators that are compact perturbations of the free Jacobi operator via the asymptotic properties of a connection coefficient matrix. In particular, for Jacobi operators that are finite-rank perturbations we show that the computation of the spectrum can be reduced to a polynomial root finding problem, from a polynomial that is derived explicitly from the entries of a connection coefficient matrix. A formula for the spectral measure of the operator is also derived explicitly from these entries. The analysis is extended to trace-class perturbations. We address issues of computability in the framework of the Solvability Complexity Index, proving that the spectrum of compact perturbations of the free Jacobi operator is computable in finite time with guaranteed error control in the Hausdorff metric on sets.

Finite Rank Perturbations of Linear Relations and Matrix Pencils

We elaborate on the deviation of the Jordan structures of two linear relations that are finite-dimensional perturbations of each other. We compare their number of Jordan chains of length at least n. In the operator case, it was recently proved that the difference of these numbers is independent of n and is at most the defect between the operators. One of the main results of this paper shows that in the case of linear relations this number has to be multiplied by n+1 and that this bound is sharp. The reason for this behavior is the existence of singular chains. We apply our results to one-dimensional perturbations of singular and regular matrix pencils. This is done by representing matrix pencils via linear relations. This technique allows for both proving known results for regular pencils as well as new results for singular ones.