Sequence Of Eigenvalues Research Articles

Three-term asymptotic formula for large eigenvalues of the two-photon quantum Rabi model

We prove that the spectrum of the two-photon quantum Rabi Hamiltonian consists of two eigenvalue sequences (E_{m}^{+})_{m=0}^{\infty} , (E_{m}^{-})_{m=0}^{\infty} satisfying a three-term asymptotic formula with the remainder estimate O(m^{-1}\ln m) when m tends to infinity. Our asymptotic formula can be written so that the third term is given by an explicit oscillatory integral and an explicit remainder estimate.

Multiplicity results for nonlocal critical elliptic problems

AbstractWe prove new multiplicity results for some nonlocal critical growth elliptic problems in bounded domains. More specifically, we show that the problems considered here have arbitrarily many solutions for all sufficiently large values of a certain parameter . In particular, the number of solutions goes to infinity as . We also give an explicit lower bound on in order to have a given number of solutions. This lower bound is in terms of a sequence of eigenvalues of the associated nonlocal elliptic operator. The proofs are based on an abstract critical point theorem.

Behavior of large eigenvalues for the two-photon asymmetric quantum Rabi model

The asymptotic behavior of large eigenvalues is studied for the two-photon quantum Rabi model with a finite bias. It is proved that the spectrum of this Hamiltonian model consists of two eigenvalue sequences { E n + } n = 0 ∞ \lbrace E_n^+\rbrace _{n=0}^{\infty } , { E n − } n = 0 ∞ \lbrace E_n^-\rbrace _{n=0}^{\infty } , and their large n n asymptotic behavior with error term O ⁡ ( n − 1 / 2 ) \operatorname {O}(n^{-1/2}) is described. The principal tool is the method of near-similarity of operators introduced by G. V. Rozenblum and developed in works of J. Janas, S. Naboko, and E. A. Yanovich (Tur).

Asymptotic distribution of nodal intersections for ARW against a surface

We investigate Gaussian Laplacian eigenfunctions (Arithmetic Random Waves) on the three-dimensional standard flat torus, in particular the asymptotic distribution of the nodal intersection length against a fixed regular reference surface. Expectation and variance have been addressed by Maffucci [Ann. Henri Poincaré 20(11), 3651–3691 (2019)] who found that the expected length is proportional to the square root of the eigenvalue times the area of the surface, while the asymptotic variance only depends on the geometry of the surface, the projected lattice points being equidistributed on the two-dimensional unit sphere in the high-energy limit. He also noticed that there are “special” surfaces, so-called static, for which the variance is of smaller order; however he did not prescribe the precise asymptotic law in this case. In this paper, we study second order fluctuations of the nodal intersection length. Our first main result is a Central Limit Theorem for “generic” surfaces, while for static ones, a sphere or a hemisphere e.g., our main results are a non-Central Limit Theorem and a precise asymptotic law for the variance of the nodal intersection length, conditioned on the existence of so-called well-separated sequences of Laplacian eigenvalues. It turns out that, in this regime, the nodal area investigated by Cammarota [Trans. Am. Math. Soc. 372(5), 3539–3564 (2019)] is asymptotically fully correlated with the length of the nodal intersections against any sphere. The main ingredients for our proofs are the Kac-Rice formula for moments, the chaotic decomposition for square integrable functionals of Gaussian fields, and some arithmetic estimates that may be of independent interest.

Localization of eigenfunctions in the Dirichlet beaker

AbstractWe construct the asymptotics of the eigenpairs of the Dirichlet problem for the Laplace operator in a thin‐walled beaker and prove the localization effect for the functions near the bottom edge, a smooth closed contour, of the beaker. The main asymptotic terms are described by the eigenpairs of an ordinary differential equation on the edge and by the single eigenvalue belonging to the discrete spectrum of the Dirichlet Laplacian in an ‐shaped infinite waveguide. The corresponding eigenfunctions are shown to decay exponentially at some distance from the edge. Also, we find the asymptotics of eigenvalue sequences generated by planar Dirichlet problems on the bottom and walls of the limit beaker of zero thickness. Open questions related to other sequences of eigenvalues are discussed.

Behaviour of large eigenvalues for the asymmetric quantum Rabi model

We prove that the spectrum of the asymmetric quantum Rabi model consists of two eigenvalue sequences ( E m + ) m = 0 ∞ , ( E m − ) m = 0 ∞ , satisfying a two-term asymptotic formula with error estimate of the form O ( m − 1 / 4 ), when m tends to infinity.

The allow sequence of distinct eigenvalues for a sign pattern

A sign pattern $\mathcal{A}$ is a matrix with entries in $\{+,-,0\}$. This article introduces the allow sequence of distinct eigenvalues for an $n\times n$ sign pattern $\mathcal{A}$, defined as $q_{\rm seq}(\mathcal{A})=\langle q_1,\ldots,q_n\rangle$, with $q_k=1$ if there exists a real matrix with exactly $k$ distinct eigenvalues having pattern $\mathcal{A}$, and $q_k=0$ otherwise. For example, $q_{\rm seq}(\mathcal{A})=\langle 0,\ldots,0,1\rangle$ is equivalent to $\mathcal{A}$ requiring all distinct eigenvalues, while $q_{\rm seq}(\mathcal{A})=\langle 1,0,\ldots,0\rangle$ is equivalent to the digraph of $\mathcal{A}$ being acyclic. Relationships between the allow sequence for $\mathcal{A}$ and composite cycles of the digraph of $\mathcal{A}$ are explored to identify zeros in the sequence, while methods based on Jacobian matrices are developed to identify ones in the sequence. When $\mathcal{A}$ is an $n\times n$ irreducible sign pattern, the possible sequences for $q_{\rm seq}(\mathcal{A})$ are completely determined when $n\leq 4$ and when the sequence has at least $n-4$ trailing zeros for $n\geq 5$.

On a family of finitely many point interaction Hamiltonians free of ultraviolet pathologies

Abstract We re-investigate the entire family of many center point interaction Hamiltonians. Under the assumption of exchange symmetry with respect to the point positions, we show that a large sub-family of point interaction Hamiltonian operators does not become either singular or trivial when the positions of two or more scattering centers tend to coincide. In this sense, they appear to be renormalized by default as opposed to the point interaction Hamiltonians usually considered in the literature. Functions in their domains satisfy regularized boundary conditions which turn out to be very similar to the ones proposed recently in many-body quantum mechanics to define three-particle system Hamiltonians with contact interactions bounded from below. In the two-center case, we study the behavior of the negative eigenvalues as a function of the center distance. The result is used to analyze a formal Born–Oppenheimer approximation of a three-particle system with two heavy bosons and one light particle. We demonstrate that this simplified model describes a stable system (no ‘fall to the center’ problem is present). Furthermore, in the unitary limit, the energy spectrum is characterized by an infinite sequence of negative energy eigenvalues accumulating at zero according to the geometrical Efimov law.

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.

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.

Convergence in Total Variation for nonlinear functionals of random hyperspherical harmonics

Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit d-dimensional sphere (d≥2). We study the convergence in Total Variation distance for their nonlinear statistics in the high energy limit, i.e., for diverging sequences of Laplace eigenvalues. Our approach takes advantage of a recent result by Bally, Caramellino and Poly (2020): combining the Central Limit Theorem in Wasserstein distance obtained by Marinucci and Rossi (2015) for Hermite-rank 2 functionals with new results on the asymptotic behavior of their Malliavin-Sobolev norms, we are able to establish second order Gaussian fluctuations in this stronger probability metric as soon as the functional is regular enough. Our argument requires some novel estimates on moments of products of Gegenbauer polynomials that may be of independent interest, which we prove via the link between graph theory and diagram formulas.

Rigorous derivation of the Efimov effect in a simple model

We consider a system of three identical bosons in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document} with two-body zero-range interactions and a three-body hard-core repulsion of a given radius a>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ a > 0$$\\end{document}. Using a quadratic form approach, we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of a. In particular, this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, i.e., we show that the Hamiltonian has an infinite sequence of negative eigenvalues En\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_n$$\\end{document} accumulating at zero and fulfilling the asymptotic geometrical law En+1/En→e-2πs0forn→+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\;E_{n+1} / E_n \\; \\rightarrow \\; e^{-\\frac{2\\pi }{s_0}}\\,\\; \\,\ ext {for} \\,\\; n\\rightarrow +\\infty $$\\end{document} holds, where s0≈1.00624\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_0\\approx 1.00624$$\\end{document}.

Multipliers on Spaces of Holomorphic Functions

We consider multipliers on the space of holomorphic functions of one variable H(Omega ),,Omega subset mathbb {C} open, that is, linear continuous operators for which all monomials are eigenvectors. If zero belongs to Omega and Omega is a domain these operators are just multipliers on the sequences of Taylor coefficients at zero. In particular, Hadamard multiplication operators are multipliers. In the case of Runge open sets we represent all multipliers via a kind of multiplicative convolutions with analytic functionals and characterize the corresponding sequences of eigenvalues as moments of suitable analytic functionals. We also identify which topology should be put on the subspace of analytic functionals in order for that isomorphism to become a topological isomorphism, when the space of multipliers inherits the topology of uniform convergence on bounded sets from the space of all endomorphisms on H(Omega ). We provide one more representation of multipliers via suitable germs of holomorphic functions with Laurent or Taylor coefficients equal to the eigenvalues of the operator. We also discuss a special case, namely, when Omega is convex.

Eigenvalues of the Laplace-Beltrami operator on a prolate spheroid

The Laplace-Beltrami operator on a prolate spheroid admits a sequence of eigenvalues. These eigenvalues are determined by a singular Sturm-Liouville problem. Properties of the eigenvalues are obtained using the minimum-maximum principle and the Prüfer angle. In particular, eigenvalues are approximated by those of generalized matrix eigenvalue problems, and their behavior is studied when the eccentricity of the spheroid approaches 0 or 1.

Electromagnetic Steklov eigenvalues: existence and distribution in the self-adjoint case

In previous works, it was suggested to use Steklov eigenvalues for Maxwell equations as target signature for nondestructive testing, and it was recognized that this eigenvalue problem cannot be reformulated as a standard eigenvalue problem for a compact operator. Consequently, a modified eigenvalue problem with the desired properties was proposed. We report that apart for a countable set of particular frequencies, the spectrum of the original self-adjoint eigenvalue problem consists of three disjoint parts: The essential spectrum consisting of the origin, an infinite sequence of positive eigenvalues which accumulate only at infinity and an infinite sequence of negative eigenvalues which accumulate only at zero. The analysis is based on a suitable topological decomposition, a representation of the operator as block operator and Schur-factorizations. For each Schur-complement, the existence of an infinite sequence of eigenvalues is proven via an intermediate value technique. The modified eigenvalue problem arises as limit of one Schur-complement.

Eigenvalue problems for p-div-curl systems

We consider eigenvalue problems for a general p-div-curl system. In particular, we consider a nonlinear eigenvalue problem similar to the standard Dirichlet eigenvalue problem for the p-Laplacian, but due to the presence of the curl, we see that eigenfunctions of the general p-div-curl operator do not enjoy nice regularity as in the p-Laplacian case. We first show that weak solutions of a particular nonlinear p-div-curl eigenvalue problem split into a good and bad part, where the good part is not even divergence free. We then show that the general p-div-curl system admits a Ljusternik-Schnirelman sequence of eigenvalues. Finally, we show a general chain rule for the curl and divergence.

The Uniform Lipschitz Continuity of Eigenvalues of Sturm–Liouville Problems with Respect to the Weighted Function

The present paper is concerned with the uniform boundedness of the normalized eigenfunctions of Sturm–Liouville problems and shows that the sequence of eigenvalues is uniformly local Lipschitz continuous with respect to the weighted functions.

Machine Learning Methods for Inferring the Number of UAV Emitters via Massive MIMO Receive Array

To provide important prior knowledge for the direction of arrival (DOA) estimation of UAV emitters in future wireless networks, we present a complete DOA preprocessing system for inferring the number of emitters via a massive multiple-input multiple-output (MIMO) receive array. Firstly, in order to eliminate the noise signals, two high-precision signal detectors, the square root of the maximum eigenvalue times the minimum eigenvalue (SR-MME) and the geometric mean (GM), are proposed. Compared to other detectors, SR-MME and GM can achieve a high detection probability while maintaining extremely low false alarm probability. Secondly, if the existence of emitters is determined by detectors, we need to further confirm their number. Therefore, we perform feature extraction on the the eigenvalue sequence of a sample covariance matrix to construct a feature vector and innovatively propose a multi-layer neural network (ML-NN). Additionally, the support vector machine (SVM) and naive Bayesian classifier (NBC) are also designed. The simulation results show that the machine learning-based methods can achieve good results in signal classification, especially neural networks, which can always maintain the classification accuracy above 70% with the massive MIMO receive array. Finally, we analyze the classical signal classification methods, Akaike (AIC) and minimum description length (MDL). It is concluded that the two methods are not suitable for scenarios with massive MIMO arrays, and they also have much worse performance than machine learning-based classifiers.

The discrete spectrum of the Neumann-Poincaré operator in 3D elasticity

For the Neumann-Poincaré (double layer potential) operator in the three-dimensional elasticity we establish asymptotic formulas for eigenvalues converging to the points of the essential spectrum and discuss geometric and mechanical meaning of coefficients in these formulas. In particular, we establish that for any body, there are infinitely many eigenvalues converging from above to each point of the essential spectrum. On the other hand, if there is a point where the boundary is concave (in particular, if the body contains cavities) then for each point of the essential spectrum there exists a sequence of eigenvalues converging to this point from below. The reasoning is based upon the representation of the Neumann-Poincaré operator as a zero order pseudodifferential operator on the boundary and the earlier results by the author on the eigenvalue asymptotics for polynomially compact pseudodifferential operators.

The infimum eigenvalue for degenerate p(x)-biharmonic operator with the Hardy potentiel

The aim of this article is to study the existence of at least one unbounded nondecreasing sequence of nonnegative eigenvalues (λk)k≥1 for a class of elliptic Navier boundary value problems involving the degenerate p(·)-biharmonic operator with q(x)-Hardy inequality by using the variational technique based on the Ljusternik-Schnirelmann theory on C1-manifolds and the theory of the variable exponent Lebesgue spaces. Also, we obtain the positivity of the infimum eigenvalue for the problem.