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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/10/1999

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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

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## On Eigenvalues of the sum of two random projections

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/10/2015

Relevância na Pesquisa

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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

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## Sums of Laplace eigenvalues - rotationally symmetric maximizers in the plane

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/09/2010

Relevância na Pesquisa

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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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/10/2011

Relevância na Pesquisa

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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...

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## Nontrivial eigenvalues of the Liouvillian of an open quantum system

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/10/2010

Relevância na Pesquisa

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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

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

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

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## Dependence of Discrete Sturm-Liouville Eigenvalues on Problems

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/03/2007

Relevância na Pesquisa

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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

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## Extremal Eigenvalues and Eigenvectors of Deformed Wigner Matrices

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

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## Finding Eigenvalues the Rupert Way

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

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## New Probabilistic Bounds on Eigenvalues and Eigenvectors of Random Kernel Matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/02/2012

Relevância na Pesquisa

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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.

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## On the multiplicity of eigenvalues of conformally covariant operators

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

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

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

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## Convergence of Dirichlet Eigenvalues for Elliptic Systems on Perturbed Domains

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.

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## Complex Eigenvalues for Binary Subdivision Schemes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/01/2008

Relevância na Pesquisa

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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

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## Eigenvalues of collapsing domains and drift Laplacians

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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}.

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## Eigenvalues of finite rank Bratteli-Vershik dynamical systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/08/2012

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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

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## How Many Numerical Eigenvalues can We Trust?

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/12/2013

Relevância na Pesquisa

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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
$N$.

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## Lowest Eigenvalues of Random Hamiltonians

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/01/2008

Relevância na Pesquisa

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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.

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