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Complex Eigenvalues of the Parabolic Potential Barrier and Gel'fand Triplet

Shimbori, Toshiki; Kobayashi, Tsunehiro
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 04/10/1999
Relevância na Pesquisa
26.68%
The paper deals with the one-dimensional parabolic potential barrier $V(x)={V_0-m\gamma^2 x^2/2}$, as a model of an unstable system in quantum mechanics. The time-independent Schr\"{o}dinger equation for this model is set up as the eigenvalue problem in Gel'fand triplet and its exact solutions are expressed by generalized eigenfunctions belonging to complex energy eigenvalues ${V_0\mp i\Gammav_n}$ whose imaginary parts are quantized as ${\Gammav_n=(n+1/2)\hslash\gamma}$. Under the assumption that time factors of an unstable system are square integrable, we provide a probabilistic interpretation of them. This assumption leads to the separation of the domain of the time evolution, namely all the time factors belonging to the complex energy eigenvalues ${V_0-i\Gammav_n}$ exist on the future part and all those belonging to the complex energy eigenvalues ${V_0+i\Gammav_n}$ exist on the past part. In this model the physical energy distributions worked out from these time factors are found to be the Breit-Wigner resonance formulas. The half-widths of these physical energy distributions are determined by the imaginary parts of complex energy eigenvalues, and hence they are also quantized.; Comment: 18 pages

On Eigenvalues of the sum of two random projections

Kargin, Vladislav
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 04/05/2012
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We study the behavior of eigenvalues of matrix P_N + Q_N where P_N and Q_N are two N -by-N random orthogonal projections. We relate the joint eigenvalue distribution of this matrix to the Jacobi matrix ensemble and establish the universal behavior of eigenvalues for large N. The limiting local behavior of eigenvalues is governed by the sine kernel in the bulk and by either the Bessel or the Airy kernel at the edge depending on parameters. We also study an exceptional case when the local behavior of eigenvalues of P_N + Q_N is not universal in the usual sense.; Comment: 14 pages

Hadamard Type Variation Formulas for the Eigenvalues of the $\eta$-Laplacian and Applications

Gomes, J. N.; Marrocos, M. A. M.; Mesquita, R. R.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 23/10/2015
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In this paper we consider an analytic family of Riemannian structures on a compact smooth manifold $M$ with boundary. We impose the Dirichlet condition to the $\eta$-Laplacian and show the existence of analytic curves of its eigenfunctions and eigenvalues. We derive Hadamard type variation formulas. As an application we show that for a subset of $C^r$-metrics, $1\leq r<\infty$, all eigenvalues of $\eta$-Laplacian operator are generically simple. Moreover, we consider families of perturbations of domains in $M$ and obtain Hadamard type formulas for the eigenvalues of the $\eta$-Laplacian also in this case. We also establish the generic simplicity of the eigenvalues of the $\eta$-Laplacian operator in this context.; Comment: 10 pages

Sums of Laplace eigenvalues - rotationally symmetric maximizers in the plane

Laugesen, R. S.; Siudeja, B. A.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 27/09/2010
Relevância na Pesquisa
26.68%
The sum of the first $n \geq 1$ eigenvalues of the Laplacian is shown to be maximal among triangles for the equilateral triangle, maximal among parallelograms for the square, and maximal among ellipses for the disk, provided the ratio $\text{(area)}^3/\text{(moment of inertia)}$ for the domain is fixed. This result holds for both Dirichlet and Neumann eigenvalues, and similar conclusions are derived for Robin boundary conditions and Schr\"odinger eigenvalues of potentials that grow at infinity. A key ingredient in the method is the tight frame property of the roots of unity. For general convex plane domains, the disk is conjectured to maximize sums of Neumann eigenvalues.

Half eigenvalues and the Fucik spectrum of multi-point, boundary value problems

Genoud, Francois; Rynne, Bryan P.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 04/10/2011
Relevância na Pesquisa
26.68%
We consider the nonlinear boundary value problem consisting of the equation \tag{1} -u" = f(u) + h, \quad \text{a.e. on $(-1,1)$,} where $h \in L^1(-1,1)$, together with the multi-point, Dirichlet-type boundary conditions \tag{2} u(\pm 1) = \sum^{m^\pm}_{i=1}\alpha^\pm_i u(\eta^\pm_i) where $m^\pm \ge 1$ are integers, $\alpha^\pm = (\alpha_1^\pm, ...,\alpha_m^\pm) \in [0,1)^{m^\pm}$, $\eta^\pm \in (-1,1)^{m^\pm}$, and we suppose that $$ \sum_{i=1}^{m^\pm} \alpha_i^\pm < 1 . $$ We also suppose that $f : \mathbb{R} \to \mathbb{R}$ is continuous, and $$ 0 < f_{\pm\infty}:=\lim_{s \to \pm\infty} \frac{f(s)}{s} < \infty. $$ We allow $f_{\infty} \ne f_{-\infty}$ --- such a nonlinearity $f$ is {\em jumping}. Related to (1) is the equation \tag{3} -u" = \lambda(a u^+ - b u^-), \quad \text{on $(-1,1)$,} where $\lambda,\,a,\,b > 0$, and $u^{\pm}(x) =\max\{\pm u(x),0\}$ for $x \in [-1,1]$. The problem (2)-(3) is `positively-homogeneous' and jumping. Regarding $a,\,b$ as fixed, values of $\lambda = \lambda(a,b)$ for which (2)-(3) has a non-trivial solution $u$ will be called {\em half-eigenvalues}, while the corresponding solutions $u$ will be called {\em half-eigenfunctions}. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having specified nodal properties...

Nontrivial eigenvalues of the Liouvillian of an open quantum system

Nakano, Ruri; Hatano, Naomichi; Petrosky, Tomio
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 25/10/2010
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We present methods of finding complex eigenvalues of the Liouvillian of an open quantum system. The goal is to find eigenvalues that cannot be predicted from the eigenvalues of the corresponding Hamiltonian. Our model is a T-type quantum dot with an infinitely long lead. We suggest the existence of the non-trivial eigenvalues of the Liouvillian in two ways: one way is to show that the original problem reduces to the problem of a two-particle Hamiltonian with a two-body interaction and the other way is to show that diagram expansion of the Green's function has correlation between the bra state and the ket state. We also introduce the integral equations equivalent to the original eigenvalue problem.; Comment: 5 pages, 2 figures, proceedings

Correspondence of the eigenvalues of a non-self-adjoint operator to those of a self-adjoint operator

Weir, John
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
We prove that the eigenvalues of a certain highly non-self-adjoint operator that arises in fluid mechanics correspond, up to scaling by a positive constant, to those of a self-adjoint operator with compact resolvent; hence there are infinitely many real eigenvalues which accumulate only at $\pm \infty$. We use this result to determine the asymptotic distribution of the eigenvalues and to compute some of the eigenvalues numerically. We compare these to earlier calculations by other authors.; Comment: 29 pages, corrections to section 3, added section 5

Dependence of Discrete Sturm-Liouville Eigenvalues on Problems

Zhu, Hao; Sun, Shurong; Shi, Yuming; Wu, Hongyou
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 27/05/2015
Relevância na Pesquisa
26.68%
This paper is concerned with dependence of discrete Sturm-Liouville eigenvalues on problems. Topologies and geometric structures on various spaces of such problems are firstly introduced. Then, relationships between the analytic and geometric multiplicities of an eigenvalue are discussed. It is shown that all problems sufficiently close to a given problem have eigenvalues near each eigenvalue of the given problem. So, all the simple eigenvalues live in so-called continuous simple eigenvalue branches over the space of problems, and all the eigenvalues live in continuous eigenvalue branches over the space of self-adjoint problems. The analyticity, differentiability and monotonicity of continuous eigenvalue branches are further studied.; Comment: 32 Pages, 6 figures

Asymptotics for the number of eigenvalues of three-particle Schr\"{o}dinger operators on lattices

Albeverio, Sergio; Antonio, G. F. Dell; Lakaev, Saidakhmat N.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 07/03/2007
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26.68%
We consider the Hamiltonian of a system of three quantum mechanical particles (two identical fermions and boson)on the three-dimensional lattice $\Z^3$ and interacting by means of zero-range attractive potentials. We describe the location and structure of the essential spectrum of the three-particle discrete Schr\"{o}dinger operator $H_{\gamma}(K),$ $K$ being the total quasi-momentum and $\gamma>0$ the ratio of the mass of fermion and boson. We choose for $\gamma>0$ the interaction $v(\gamma)$ in such a way the system consisting of one fermion and one boson has a zero energy resonance. We prove for any $\gamma> 0$ the existence infinitely many eigenvalues of the operator $H_{\gamma}(0).$ We establish for the number $N(0,\gamma; z;)$ of eigenvalues lying below $z<0$ the following asymptotics $$ \lim_{z\to 0-}\frac{N(0,\gamma;z)}{\mid \log \mid z\mid \mid}={U} (\gamma) .$$ Moreover, for all nonzero values of the quasi-momentum $K \in T^3 $ we establish the finiteness of the number $ N(K,\gamma;\tau_{ess}(K))$ of eigenvalues of $H(K)$ below the bottom of the essential spectrum and we give an asymptotics for the number $N(K,\gamma;0)$ of eigenvalues below zero.; Comment: 25 pages

Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices

Lee, Ji Oon; Schnelli, Kevin
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
We consider random matrices of the form $H = W + \lambda V$, $\lambda\in\mathbb{R}^+$, where $W$ is a real symmetric or complex Hermitian Wigner matrix of size $N$ and $V$ is a real bounded diagonal random matrix of size $N$ with i.i.d.\ entries that are independent of $W$. We assume subexponential decay for the matrix entries of $W$ and we choose $\lambda \sim 1$, so that the eigenvalues of $W$ and $\lambda V$ are typically of the same order. Further, we assume that the density of the entries of $V$ is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is $\lambda_+\in\mathbb{R}^+$ such that the largest eigenvalues of $H$ are in the limit of large $N$ determined by the order statistics of $V$ for $\lambda>\lambda_+$. In particular, the largest eigenvalue of $H$ has a Weibull distribution in the limit $N\to\infty$ if $\lambda>\lambda_+$. Moreover, for $N$ sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for $\lambda>\lambda_+$, while they are completely delocalized for $\lambda<\lambda_+$. Similar results hold for the lowest eigenvalues.; Comment: 47 pages

Finding Eigenvalues the Rupert Way

Grudzien, Colin; Jones, Christopher K. R. T.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
We develop a stability index for the travelling waves of non-linear reaction diffusion equations using the geometric phase induced on the Hopf bundle $S^{2n-1} \subset \mathbb{C}^n$. This can be viewed as an alternative formulation of the winding number calculation of the Evans function, whose zeroes correspond to the eigenvalues of the linearization of reaction diffusion operators about the wave. The stability of a travelling wave can be determined by the existence of eigenvalues of positive real part for the linear operator. Our method for locating and counting eigenvalues is developed from the numerical results presented in Way's "Dynamics in the Hopf bundle, the geometric phase and implications for dynamical systems". We prove the relationship proposed by Way between the phase and eigenvalues for dynamical systems defined on $\mathbb{C}^2$, and with modifications, we extend the method to general dimensions, non-symmetric systems and to boundary value problems. Implementing the numerical method developed by Way, we conclude with open questions inspired from the results

New Probabilistic Bounds on Eigenvalues and Eigenvectors of Random Kernel Matrices

Reyhani, Nima; Hino, Hideitsu; Vigario, Ricardo
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 14/02/2012
Relevância na Pesquisa
26.68%
Kernel methods are successful approaches for different machine learning problems. This success is mainly rooted in using feature maps and kernel matrices. Some methods rely on the eigenvalues/eigenvectors of the kernel matrix, while for other methods the spectral information can be used to estimate the excess risk. An important question remains on how close the sample eigenvalues/eigenvectors are to the population values. In this paper, we improve earlier results on concentration bounds for eigenvalues of general kernel matrices. For distance and inner product kernel functions, e.g. radial basis functions, we provide new concentration bounds, which are characterized by the eigenvalues of the sample covariance matrix. Meanwhile, the obstacles for sharper bounds are accounted for and partially addressed. As a case study, we derive a concentration inequality for sample kernel target-alignment.

On the multiplicity of eigenvalues of conformally covariant operators

Canzani, Yaiza
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
Let $(M,g)$ be a compact Riemannian manifold and $P_g$ an elliptic, formally self-adjoint, conformally covariant operator of order $m$ acting on smooth sections of a bundle over $M$. We prove that if $P_g$ has no rigid eigenspaces (see Definition 2.2), the set of functions $f \in C^\infty(M, R)$ for which $P_{e^fg}$ has only simple non-zero eigenvalues is a residual set in $C^\infty(M,R)$. As a consequence we prove that if $P_g$ has no rigid eigenspaces for a dense set of metrics, then all non-zero eigenvalues are simple for a residual set of metrics in the $C^m$-topology. We also prove that the eigenvalues of $P_g$ depend continuously on $g$ in the $C^m$-topology, provided $P_g$ is strongly elliptic. As an application of our work, we show that if $P_g$ acts on $C^\infty(M)$ (e.g. GJMS operators), its non-zero eigenvalues are generically simple.; Comment: To appear in Annales de l'Institut Fourier

On the process of the eigenvalues of a Hermitian L\'evy process

Pérez-Abreu, Victor; Rocha-Arteaga, Alfonso
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
The dynamics of the eigenvalues (semimartingales) of a L\'{e}vy process $X$ with values in Hermitian matrices is described in terms of It\^{o} stochastic differential equations with jumps. This generalizes the well known Dyson-Brownian motion. The simultaneity of the jumps of the eigenvalues of $X$ is also studied. If $X$ has a jump at time $t$ two different situations are considered, depending on the commutativity of $X(t)$ and $X(t-)$. In the commutative case all the eigenvalues jump at time $t$ only when the jump of $X$ is of full rank. In the noncommutative case, $X$ jumps at time $t$ if and only if all the eigenvalues jump at that time when the jump of $X$ is of rank one.; Comment: Issues raised by referees were considered. To appear in The Fascination of Probability, Statistics and their Applications: Festschrift in Honour of Ole E. Barndorff-Nielsen

Convergence of Dirichlet Eigenvalues for Elliptic Systems on Perturbed Domains

Taylor, Justin L.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
We consider the eigenvalues of an elliptic operator for systems with bounded, measurable, and symmetric coefficients. We assume we have two non-empty, open, disjoint, and bounded sets and add a set of small measure to form the perturbed domain. Then we show that the Dirichlet eigenvalues corresponding to the family of perturbed domains converge to the Dirichlet eigenvalues corresponding to the unperturbed domain. Moreover, our rate of convergence is independent of the eigenvalues. In this paper, we consider the Lam\'{e} system, systems which satisfy a strong ellipticity condition, and systems which satisfy a Legendre-Hadamard ellipticity condition.

Complex Eigenvalues for Binary Subdivision Schemes

Kuehn, Christian
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 21/01/2008
Relevância na Pesquisa
26.68%
Convergence properties of binary stationary subdivision schemes for curves have been analyzed using the techniques of z-transforms and eigenanalysis. Eigenanalysis provides a way to determine derivative continuity at specific points based on the eigenvalues of a finite matrix. None of the well-known subdivision schemes for curves have complex eigenvalues. We prove when a convergent scheme with palindromic mask can have complex eigenvalues and that a lower limit for the size of the mask exists in this case. We find a scheme with complex eigenvalues achieving this lower bound. Furthermore we investigate this scheme numerically and explain from a geometric viewpoint why such a scheme has not yet been used in computer-aided geometric design.; Comment: 7 pages, 2 figures

Eigenvalues of collapsing domains and drift Laplacians

Lu, Zhiqin; Rowlett, Julie
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.68%
By introducing a weight function to the Laplace operator, Bakry and \'Emery defined the "drift Laplacian" to study diffusion processes. Our first main result is that, given a Bakry-\'Emery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new relationship between Dirichlet eigenvalues of domains in $\R^n$ and Neumann eigenvalues of domains in $\R^{n+1}$ and a new maximum principle. Using our main result and maximum principle, we are able to generalize \emph{all the results in Riemannian geometry based on gradient estimates to Bakry-\'Emery manifolds}.

Eigenvalues of finite rank Bratteli-Vershik dynamical systems

Bressaud, Xavier; Durand, Fabien; Maass, Alejandro
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 16/08/2012
Relevância na Pesquisa
26.68%
In this article we study conditions to be a continuous or a measurable eigenvalue of finite rank minimal Cantor systems, that is, systems given by an ordered Bratteli diagram with a bounded number of vertices per level. We prove that continuous eigenvalues always come from the stable subspace associated to the incidence matrices of the Bratteli diagram and we study rationally independent generators of the additive group of continuous eigenvalues. Given an ergodic probability measure, we provide a general necessary condition to be a measurable eigenvalue. Then we consider two families of examples. A first one to illustrate that measurable eigenvalues do not need to come from the stable space. Finally we study Toeplitz type Cantor minimal systems of finite rank. We recover classical results in the continuous case and we prove measurable eigenvalues are always rational but not necessarily continuous.; Comment: Ergodic Theory and Dynamical Systems (2010) 26

How Many Numerical Eigenvalues can We Trust?

Zhang, Zhimin
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 24/12/2013
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When using finite element and finite difference methods to approximate eigenvalues of $2m^{th}$-order elliptic problems, the number of reliable numerical eigenvalues can be estimated in terms of the total degrees of freedom $N$ in resulting discrete systems. The truth is worse than what we used to believe in that the percentage of reliable eigenvalues decreases with an increased $N$, even though the number of reliable eigenvalues increases with $N$.

Lowest Eigenvalues of Random Hamiltonians

Shen, J. J.; Zhao, Y. M.; Arima, A.; Yoshinaga, N.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 14/01/2008
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In this paper we present results of the lowest eigenvalues of random Hamiltonians for both fermion and boson systems. We show that an empirical formula of evaluating the lowest eigenvalues of random Hamiltonians in terms of energy centroids and widths of eigenvalues are applicable to many different systems (except for $d$ boson systems). We improve the accuracy of the formula by adding moments higher than two. We suggest another new formula to evaluate the lowest eigenvalues for random matrices with large dimensions (20-5000). These empirical formulas are shown to be applicable not only to the evaluation of the lowest energy but also to the evaluation of excited energies of systems under random two-body interactions.