Finite Multiplicity Research Articles

SPECTRAL PROPERTIES OF AN IMPULSIVE STURM–LIOUVILLE OPERATOR WITH COMPLEX PERIODIC COEFFICIENTS

This study focuses on exploring the spectral properties and scattering function of the impulsive operator generated by the Sturm–Liouville equation with complex periodic potentials. We introduce a new approach to investigate the spectral singularities and eigenvalues of the operator. Additionally, we establish the finiteness of eigenvalues and spectral singularities with finite multiplicities under specific conditions.

Some new results on the property (R) under compact perturbations

The property (R) for a bounded linear operator T on a Hilbert space means that the points λ in the approximate point spectrum of T for which T − λI is upper semi-Browder are exactly the isolated points of the spectrum of T which are eigenvalues of finite multiplicity. For an operator T, if T + K satisfies the property (R) for all compact operators K, then T is said to have the stability of the property (R). In this paper, we give new criteria for the fulfillment of the stability of the property (R) for an operator. Moreover, the property (R) for complex symmetric operators is discussed.

A spectral theorem for compact representations and non-unitary cusp forms

We show that a compact representation of either a semisimple Lie group or a totally disconnected group has a filtration with irreducible subquotients of finite multiplicity. In the Lie group case we show the stronger assertion, that it has an orthogonal decomposition into summands of finite lengths. This generalises and simplifies a number of more special spectral theorems in Deitmar and Monheim (Math Z 284(3–4):1199–1210, 2016, https://doi.org/10.1007/s00209-016-1695-9), Müller (Int Math Res Not 9(2):2068–2109, 2011), Venkov (in: Proceedings of the Steklov Institute of Mathematics, no. 4(153), ix+163 pp. (1983), 1982). We apply it to the case of cusp forms, thus settling the spectral theory for the space of non-unitary twisted cusp forms. We finally show that the space of cusp forms is complemented by the space of Eisenstein series.

Threshold approximations for the exponential of a factorized operator family with correctors taken into account

In a Hilbert space H \mathfrak H , consider a family of selfadjoint operators (a quadratic operator pencil) A ( t ) A(t) , t ∈ R t\in \mathbb {R} , of the form A ( t ) = X ( t ) ∗ X ( t ) A(t) = X(t)^* X(t) , where X ( t ) = X 0 + t X 1 X(t) = X_0 + t X_1 . It is assumed that the point λ 0 = 0 \lambda _0=0 is an isolated eigenvalue of finite multiplicity for the operator A ( 0 ) A(0) . Let F ( t ) F(t) be the spectral projection of the operator A ( t ) A(t) for the interval [ 0 , δ ] [0,\delta ] . Approximations for F ( t ) F(t) and A ( t ) F ( t ) A(t)F(t) for | t | ≤ t 0 |t| \leq t_0 (the so-called threshold approximations) are used to obtain approximations in the operator norm on H \mathfrak H for the operator exponential exp ⁡ ( − i τ A ( t ) ) \exp (-i \tau A(t)) , τ ∈ R \tau \in \mathbb {R} . The numbers δ \delta and t 0 t_0 are controlled explicitly. Next, the behavior for small ε > 0 \varepsilon >0 of the operator exp ⁡ ( − i ε − 2 τ A ( t ) ) \exp (-i \varepsilon ^{-2} \tau A(t)) multiplied by the “smoothing factor” ε s ( t 2 + ε 2 ) − s / 2 \varepsilon ^s (t^2 + \varepsilon ^2)^{-s/2} with a suitable s > 0 s>0 is studied. The obtained approximations are given in terms of the spectral characteristics of the operator A ( t ) A(t) near the lower edge of the spectrum. The results are aimed at application to homogenization of the Schrödinger-type equations with periodic rapidly oscillating coefficients.

Spectrum of the [formula omitted]-Laplace operator on zero forms for the quantum quadric [formula omitted]

We study the Laplacian operator Δ∂‾ associated to a Kähler structure (Ω(•,•),κ) for the Heckenberger–Kolb differential calculus of the quantum quadrics Oq(QN), which is to say, the irreducible quantum flag manifolds of types Bn and Dn. We show that the eigenvalues of Δ∂‾ on zero forms tend to infinity and have finite multiplicity.

Multiplicity of topological systems

Abstract We define the topological multiplicity of an invertible topological system $(X,T)$ as the minimal number k of real continuous functions $f_1,\ldots , f_k$ such that the functions $f_i\circ T^n$ , $n\in {\mathbb {Z}}$ , $1\leq i\leq k,$ span a dense linear vector space in the space of real continuous functions on X endowed with the supremum norm. We study some properties of topological systems with finite multiplicity. After giving some examples, we investigate the multiplicity of subshifts with linear growth complexity.

Mathematical Modeling of Eigenvibrations of the Shallow Shell with an Attached Oscillator

For the problem of eigenvibrations of the shallow shell with an attached oscillator, a new symmetric variational statement in the Hilbert space was proposed. It was established that there exist a sequence of positive eigenvalues of finite multiplicity with a limit point at infinity and the corresponding complete orthonormal system of eigenvectors. The problem was approximated by the mesh scheme of the finite element method with Hermite finite elements. Theoretical error estimates for the approximate solutions were proved. The theoretical findings were verified by the results of numerical experiments.

The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula

For a Hermitian Lie group G of tube type we find the contribution of the holomorphic discrete series to the Plancherel decomposition of the Whittaker space L2(G/N,ψ), where N is the unipotent radical of the Siegel parabolic subgroup and ψ is a certain non-degenerate unitary character on N. The holomorphic discrete series embeddings are constructed in terms of generalized Whittaker vectors for which we find explicit formulas in the bounded domain realization, the tube domain realization and the L2-model of the holomorphic discrete series. Although L2(G/N,ψ) does not have finite multiplicities in general, the holomorphic discrete series contribution does.Moreover, we obtain an explicit formula for the formal dimensions of the holomorphic discrete series embeddings, and we interpret the holomorphic discrete series contribution to L2(G/N,ψ) as boundary values of holomorphic functions on a domain Ξ in a complexification GC of G forming a Hardy type space H2(Ξ,ψ).

Finite element modeling of the eigenvibrations of the square plate with attached oscillator

For the problem on eigenvibrations of the plate with an attached oscillator, the new symmetric linear variational statement is proposed. It is established the existence of the sequence of positive eigenvalues of finite multiplicity with limit point at infinity and the corresponding complete orthonormal system of eigenvectors. The new symmetric scheme of the finite element method with Hermite finite elements is formulated. Error estimates consistent with the solution smoothness for the approximate eigenvalues and approximate eigenvectors are proved. The results of numerical experiments illustrating the influence of the solution smoothness on the computation accuracy are presented.

Finite Multiplicities Beyond Spherical Spaces

Abstract Let $G$ be a real reductive algebraic group, and let $H\subset G$ be an algebraic subgroup. It is known that the action of $G$ on the space of functions on $G/H$ is “tame” if this space is spherical. In particular, the multiplicities of the space ${\mathcal {S}}(G/H)$ of Schwartz functions on $G/H$ are finite in this case. In this paper, we formulate and analyze a generalization of sphericity that implies finite multiplicities in ${\mathcal {S}}(G/H)$ for small enough irreducible representations of $G$.

Structurally Unstable Quadratic Vector Fields of Codimension Two: Families Possessing One Finite Saddle-Node and a Separatrix Connection

This paper is part of a series of works whose ultimate goal is the complete classification of phase portraits of quadratic differential systems in the plane modulo limit cycles. It is estimated that the total number may be around 2000, so the work to find them all must be split in different papers in a systematic way so to assure the completeness of the study and also the non intersection among them. In this paper we classify the family of phase portraits possessing one finite saddle-node and a separatrix connection and determine that there are a minimum of 77 topologically different phase portraits plus at most 16 other phase portraits which we conjecture to be impossible. Along this paper we also deploy a mistake in the book (Artés et al. in Structurally unstable quadratic vector fields of codimension one, Birkhäuser/Springer, Cham, 2018) linked to a mistake in Reyn and Huang (Separatrix configuration of quadratic systems with finite multiplicity three and a M1,10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^0_{1,1}$$\\end{document} type of critical point at infinity. Report Technische Universiteit Delft, pp 95–115, 1995).

The quasi-similarity and dominant conditions of Weyl spectrum in a complex Hilbert space

The study of spectrum in Hilbert spaces is a very rich in giving more structures of the spectrum and we wish to go in to depth to understand deeper on the structure of the spectrum. Apart from the well-known components of spectrum i.e. spectrum, the approximate point spectrum, the point spectrum and the set of eigenvalues of finite multiplicity: there is need for further study on the Weyl spectrum of an Operator in a complex Hilbert space. To succeed in this study, two conditions to help expose properties of the Weyl spectrum i.e. Quasi-Similarity and the dominant condition will be used.

Spectral Properties of the Operator in the Problem of Oscillations in a Mixture of Viscous Compressible Fluids

We study the problem of normal oscillations of a homogeneous mixture of several viscous compressible fluids that fills a bounded domain in three-dimensional space with infinitely smooth boundary. We prove that the essential spectrum of the problem is a finite set of segments located on the real axis. The remaining spectrum consists of isolated eigenvalues of finite algebraic multiplicity and is located on the real axis, with the possible exception of finitely many complex conjugate eigenvalues. The spectrum of the problem contains a subsequence of eigenvalues with a limit point at infinity and a power-law asymptotic distribution.

Some Spectral Properties of Schrödinger Operators on Semi Axis

The main aim of this work is to investigate some spectral properties of Schrödinger operators on semi axis. We first present the Schrödinger equation with a piecewise continuous potential function q so that the problem differs from the classical Schrödinger problems. Then, we get the Wronskian of two specific solution of the given equation which helps us to create the sets of eigenvalues and spectral singularities. The rest of the paper deals with eigenvalues and spectral singularities. By the help of the analytical properties of Jost solutions and resolvent operator of the Schrödinger operators, we provide sufficient conditions guarenteeing finiteness of eigenvalues and spectral singularities with finite multiplicities.

On Statistical Fluctuations in Collective Flows

In relativistic heavy-ion collisions, event-by-event fluctuations are known to have non-trivial implications. Even though the probability distribution is geometrically isotropic for the initial conditions, the anisotropic εn still differs from zero owing to the statistical fluctuations in the energy profile. On the other hand, the flow harmonics extracted from the hadron spectrum using the multi-particle correlators are inevitably subjected to non-vanishing variance due to the finite number of hadrons emitted in individual events. As one aims to extract information on the fluctuations in the initial conditions via flow harmonics and their fluctuations, finite multiplicity may play a role in interfering with such an effort. In this study, we explore the properties and impacts of such fluctuations in the initial and final states, which both notably appear to be statistical ones originating from the finite number of quanta of the underlying system. We elaborate on the properties of the initial-state eccentricities for the smooth and event-by-event fluctuating initial conditions and their distinct impacts on the resulting flow harmonics. Numerical simulations are performed. The possible implications of the present study are also addressed.

On the spectrum of a mixed-type operator with applications to rotating wave solutions

We study rotating wave solutions of the nonlinear wave equation ∂t2v-Δv+mv=|v|p-2vinR×Bv=0onR×∂B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} \\partial _{t}^2 v - \\Delta v + m v = |v|^{p-2} v &{} \ ext {in }{\\mathbb {R} \ imes {\ extbf{B}}} \\\\ v = 0 &{} \\hbox {on }\\mathbb {R} \ imes \\partial {\ extbf{B}} \\end{array} \\right. \\end{aligned}$$\\end{document}where 2<p<infty , m in mathbb {R} and {textbf{B}} subset mathbb {R}^2 denotes the unit disk. If the angular velocity alpha of the rotation is larger than 1, this leads to a semilinear boundary value problem on textbf{B} involving a mixed-type operator, whose spectrum is related to the zeros of Bessel functions and could generally be badly behaved. Based on new estimates for these zeros, we find values of alpha such that the spectrum only consists of eigenvalues with finite multiplicity and has no accumulation point. Combined with suitable spectral estimates, this allows us to formulate an appropriate indefinite variational setting and find ground state solutions of the reduced equation for p in (2,4). Using a minimax characterization of the ground state energy, we ultimately show that these ground states are nonradial and thus yield nontrivial rotating waves, provided m is sufficiently large.

Dimension‐free square function estimates for Dunkl operators

AbstractDunkl operators may be regarded as differential‐difference operators parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood–Paley square function for Dunkl heat flows in is introduced by employing the full “gradient” induced by the corresponding carré du champ operator and then the boundedness is studied for all . For , we successfully adapt Stein's heat flows approach to overcome the difficulty caused by the difference part of the Dunkl operator and establish the boundedness, while for , we restrict to a particular case when the corresponding Weyl group is isomorphic to and apply a probabilistic method to prove the boundedness. In the latter case, the curvature‐dimension inequality for Dunkl operators in the sense of Bakry–Emery, which may be of independent interest, plays a crucial role. The results are dimension‐free.

Perturbations of spectra and property (R) for upper triangular operator matrices

Let H and K be two complex infinite dimensional separable Hilbert spaces. For T∈B(H), T is said to satisfy property (R) if the complement in the approximate point spectrum of the Browder essential approximate point spectrum coincides with the isolated eigenvalues of finite multiplicity. In this paper, we mainly give the sufficient and necessary conditions for the 2×2 upper triangular operator matrices such that they satisfy property (R) using the features of the elements on the diagonal.

Simple weight modules with finite weight multiplicities over the Lie algebra of polynomial vector fields

AbstractLet𝒲n{{\mathcal{W}}_{n}}be the Lie algebra of polynomial vector fields. We classify simple weight𝒲n{{\mathcal{W}}_{n}}-modulesMwith finite weight multiplicities. We prove that every such nontrivial moduleMis either a tensor module or the unique simple submodule in a tensor module associated with the de Rham complex onℂn{\mathbb{C}^{n}}.

On the spectral properties of the Hilbert transform operator on multi-intervals

Let J,E\subset\mathbb{R} be two multi-intervals with non-intersecting interiors. Consider the operator A\colon L^2( J )\to L^2(E),\quad (Af)(x) = \frac 1\pi\int_J \frac {f(y) d y}{{y-x}}, and let A^\dagger be its adjoint. We introduce a self-adjoint operator \mathscr K acting on L^2(E)\oplus L^2(J) , whose off-diagonal blocks consist of A and A^\dagger . In this paper we study the spectral properties of \mathscr K and the operators A^\dagger A and A A^\dagger . Our main tool is to obtain the resolvent of \mathscr K , which is denoted by \mathscr R , using an appropriate Riemann–Hilbert problem, and then compute the jump and poles of \mathscr R in the spectral parameter \lambda . We show that the spectrum of \mathscr K has an absolutely continuous component [0,1] if and only if J and E have common endpoints, and its multiplicity equals to their number. If there are no common endpoints, the spectrum of \mathscr K consists only of eigenvalues and 0 . If there are common endpoints, then \mathscr K may have eigenvalues imbedded in the continuous spectrum, each of them has a finite multiplicity, and the eigenvalues may accumulate only at 0 . In all cases, \mathscr K does not have a singular continuous spectrum. The spectral properties of A^\dagger A and A A^\dagger , which are very similar to those of \mathscr K , are obtained as well.