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3,000 |
Trace formulae and inverse spectral theory for Schrödinger operators
|
math.SP
|
We extend the well-known trace formula for Hill's equation to general
one-dimensional Schr\"odinger operators. The new function $\xi$, which we
introduce, is used to study absolutely continuous spectrum and inverse
problems.
|
math
|
3,001 |
Singular continuous spectrum is generic
|
math.SP
|
In a variety of contexts, we prove that singular continuous spectrum is
generic in the sense that for certain natural complete metric spaces of
operators, those with singular spectrum are a dense $G_\delta$.
|
math
|
3,002 |
Canonical systems and finite rank perturbations of spectra
|
math.SP
|
We use Rokhlin's Theorem on the uniqueness of canonical systems to find a new
way to establish connections between Function Theory in the unit disk and rank
one perturbations of self-adjoint or unitary operators. In the n-dimensional
case, we prove that for any cyclic self-adjoint operator $A$, operator
$A_\lambda= A + \Sigma_{k=1}^n \lambda_k(\cdot,\phi_k)\phi_k$ is pure point for
a. e. $\lambda=(\lambda_1,\lambda_2,...,\lambda_n) \in\Bbb R^n$ iff operator
$A_\eta=A+\eta(\cdot,\phi_k)\phi_k$ is pure point for a.e.\ $\eta\in\Bbb R$ for
$k=1,2,...,n$. We also show that if $A_\lambda$ is pure point for a.e.\
$\lambda\in \Bbb R^n$ then $A_\lambda$ is pure point for a.e.\ $\lambda\in
\gamma$ for any analytic curve $\gamma\in\Bbb R^n$.
|
math
|
3,003 |
On the evaluation of the norm of an integral operator associated with the stability of one-electron atoms
|
math.SP
|
The norm of an integral operator occurring in the partial wave decomposition
of an operator B introduced by Brown and Ravenhall in a model for relativistic
one-electron atoms is determined. The result implies that B is non-negative and
has no eigenvalue at 0 when the nuclear charge does not exceed a specified
critical value.
|
math
|
3,004 |
On the virial theorem for the relativistic operator of Brown and Ravenhall, and the absence of embedded eigenvalues
|
math.SP
|
A virial theorem is established for the operator proposed by Brown and
Ravenhall as a model for relativistic one-electron atoms. As a consequence, it
is proved that the operator has no eigenvalues greater than $\max(m c^2, 2
\alpha Z - \frac{1}{2})$, where $\alpha$ is the fine structure constant, for
all values of the nuclear charge $Z$ below the critical value $Z_c$: in
particular there are no eigenvalues embedded in the essential spectrum when $Z
\leq 3/4 \alpha$. Implications for the operators in the partial wave
decomposition are also described.
|
math
|
3,005 |
Extremal properties of the first eigenvalue of Schrödinger-type operators
|
math.SP
|
Given a separable, locally compact Hausdorff space $X$ and a positive Radon
measure $m(dx)$ on it, we study the problem of finding the potential $V(x) \ge
0$ that maximizes the first eigenvalue of the Schr\"odinger-type operator
$L+V(x)$; $L$ is the generator of a local Dirichlet form $(a, D[a])$ on $L^2(X,
m(dx))$.
|
math
|
3,006 |
A new approach to inverse spectral theory, II. General real potentials and the connection to the spectral measure
|
math.SP
|
We continue the study of the A-amplitude associated to a half-line
Schrodinger operator, -d^2/dx^2+ q in L^2 ((0,b)), b <= infinity. A is related
to the Weyl-Titchmarsh m-function via m(-\kappa^2) =-\kappa - \int_0^a
A(\alpha) e^{-2\alpha\kappa} d\alpha +O(e^{-(2a -\epsilon)\kappa}) for all
\epsilon > 0. We discuss five issues here. First, we extend the theory to
general q in L^1 ((0,a)) for all a, including q's which are limit circle at
infinity. Second, we prove the following relation between the A-amplitude and
the spectral measure \rho: A(\alpha) = -2\int_{-\infty}^\infty
\lambda^{-\frac12} \sin (2\alpha \sqrt{\lambda})\, d\rho(\lambda) (since the
integral is divergent, this formula has to be properly interpreted). Third, we
provide a Laplace transform representation for m without error term in the case
b<\infty. Fourth, we discuss m-functions associated to other boundary
conditions than the Dirichlet boundary conditions associated to the principal
Weyl-Titchmarsh m-function. Finally, we discuss some examples where one can
compute A exactly.
|
math
|
3,007 |
Removal of the resolvent-like dependence on the spectral parameter from perturbations
|
math.SP
|
The spectral problem (A + V(z))\psi=z\psi is considered with A, a
self-adjoint operator. The perturbation V(z) is assumed to depend on the
spectral parameter z as resolvent of another self-adjoint operator A':
V(z)=-B(A'-z)^{-1}B^{*}. It is supposed that the operator B has a finite
Hilbert-Schmidt norm and spectra of the operators A and A' are separated.
Conditions are formulated when the perturbation V(z) may be replaced with a
``potential'' W independent of z and such that the operator H=A+W has the same
spectrum and the same eigenfunctions (more precisely, a part of spectrum and a
respective part of eigenfunctions system) as the initial spectral problem. The
operator H is constructed as a solution of the non-linear operator equation
H=A+V(H) with a specially chosen operator-valued function V(H). In the case if
the initial spectral problem corresponds to a two-channel variant of the
Friedrichs model, a basis property of the eigenfunction system of the operator
H is proved. A scattering theory is developed for H in the case where the
operator A has continuous spectrum.
|
math
|
3,008 |
On an Analog of Selberg's Eigenvalue Conjecture for SL_3(Z)
|
math.SP
|
Let H be the homogeneous space associated to the group PGL_3(R). Let
X=\Gamma/H where \Gamma=SL_3(Z) and consider the first non-trivial eigenvalue
\lambda_1 of the Laplacian on L^2(X). Using geometric considerations, we prove
the inequality \lambda_1<pi^2/10. Since the continuous spectrum is represented
by the band [1,\infty), our bound on \lambda_1 can be viewed as an analogue of
Selberg's eigenvalue conjecture for quotients of the hyperbolic half space.
|
math
|
3,009 |
The uniqueness of the solution of the Schrodinger equation with discontinuous coefficients
|
math.SP
|
Consider the Schroeodinger equation: - Du(x) - l(x)u + s(x)u = 0, where D is
the Laplacian, l(x) > 0 and s(x) is dominated by l(x). We shall extend the
celebrated Kato's result on the asymptotic behavior of the solution to the case
where l(x) has unbounded discontinuity. The result will be used to establish
the limiting absorption principle for a class of reduced wave operators with
discontinuous coefficients.
|
math
|
3,010 |
A note on one inverse spectral problem
|
math.SP
|
The note contains the proof of the uniqueness theorem for the inverse problem
in the case of $n$-th order differential equation.
|
math
|
3,011 |
An analogue of Borg's uniqueness theorem in the case of indecomposable boundary conditions
|
math.SP
|
An uniqueness theorem for the inverse problem in the case of a second-order
equation defined on the interval [0,1] when the boundary forms contain
combinations of the values of functions at the points 0 and 1 is proved. The
auxiliary eigenvalue problems in our theorem are chose in the same manner as in
Borg's uniqueness theorem are not as in that of Sadovni\v ci$\check \imath $'s.
So number of conditions in our theorem is less than that in Sadovni\v
ci$\check\imath$'s.
|
math
|
3,012 |
The Spectral Shift Operator
|
math.SP
|
We introduce the concept of a spectral shift operator and use it to derive
Krein's spectral shift function for pairs of self-adjoint operators. Our
principal tools are operator-valued Herglotz functions and their logarithms.
Applications to Krein's trace formula and to the Birman-Solomyak spectral
averaging formula are discussed.
|
math
|
3,013 |
On The Eigenvalues of Some Vectorial Sturm-Liouville Eigenvalue Problems
|
math.SP
|
The author tries to derive the asymptotic expression of the large eigevalues
of some vectorial Sturm-Liouville differential equations. A precise description
for the formula of the square root of the large eiegnvalues up to the
$O(1/n)$-term is obtained.
|
math
|
3,014 |
On the Construction of Isospectral Vectorial Sturm-Liouville Differential Equations
|
math.SP
|
The author extends the idea of Jodeit and Levitan for constructing
isospectral problems of the classical scalar Sturm-Liouville differential
equations to the vectorial Sturm-Liouville differential equations. Some
interesting relations are found.
|
math
|
3,015 |
Resolvent estimates of the Dirac operator
|
math.SP
|
We shall investigate the asymptotic behavior of the extended resolvent R(s)
of the Dirac operator as |s| increases to infinity, where s is a real
parameter. It will be shown that the norm of R(s), as a bounded operator
between two weighted Hilbert spaces of square integrable functions on the
3-dimensional Euclidean space, stays bounded. Also we shall show that R(s)
converges 0 strongly as |s| increases to infinity. This result and a result of
Yamada [15] are combined to indicate that the extended resolvent of the Dirac
operator decays much more slowly than those of Schroedinger operators.
|
math
|
3,016 |
On the spectrum of the reduced wave operator with cylindrical discontinuity
|
math.SP
|
Consider the differential operator H = -(1/m(x))L, where L is the
N-dimensional Laplacian, in the weighted Hilbert space of square integrable
functions on N-dimensional Euclidean space with weight m(x)dx. Here m(x) is a
positive step function with a surface S of discontinuity (the separation
surface). So far the stratified media in which the separating surface S
consists of paralell planes have been vigorously studied. Also the case where S
has a cone shape has been discussed. In this work we shall deal with a new type
of discontinuity which we call cylindrical discontinuity. Under this condition
we shall use the limiting absorption method to prove that H is absolute
continuous. Our method is based on a apriori estimates of radiation condition
term.
|
math
|
3,017 |
The reduced wave equation in layered materials
|
math.SP
|
Let H = -(1/m(x))L be the reduced wave operator defined on the N-dimensional
Euclidean space, where \f L is the Laplacian. Here m(x) is a positive step
function with possible countably infinte surfaces of discontinuity (separating
surfaces) under the compatibilty condition (1.12) on each separating surface.
These compatibily condition allows us to treat the cases, among others, the
separating surfaces are cylinders. The case where the separating surface has
only one connected component was discussed in [9]. Also the case where the
separating surface is cone-shaped was considered by Eidus [6] and others ([10],
[11]). We shall prove the limiting absorption principle for H. Also we shall
discuss the case where m(x) is perturbed by a short-range or long-range
function.
|
math
|
3,018 |
Some Applications of the Spectral Shift Operator
|
math.SP
|
The recently introduced concept of a spectral shift operator is applied in
several instances. Explicit applications include Krein's trace formula for
pairs of self-adjoint operators, the Birman-Solomyak spectral averaging formula
and its operator-valued extension, and an abstract approach to trace formulas
based on perturbation theory and the theory of self-adjoint extensions of
symmetric operators.
|
math
|
3,019 |
The Xi Operator and its Relation to Krein's Spectral Shift Function
|
math.SP
|
We explore connections between Krein's spectral shift function
$\xi(\lambda,H_0,H)$ associated with the pair of self-adjoint operators
$(H_0,H)$, $H=H_0+V$ in a Hilbert space $\calH$ and the recently introduced
concept of a spectral shift operator $\Xi(J+K^*(H_0-\lambda-i0)^{-1}K)$
associated with the operator-valued Herglotz function $J+K^*(H_0-z)^{-1}K$,
$\Im(z)>0$ in $\calH$, where $V=KJK^*$ and $J=\sgn(V)$. Our principal results
include a new representation for $\xi(\lambda,H_0,H)$ in terms of an averaged
index for the Fredholm pair of self-adjoint spectral projections
$(E_{J+A(\lambda)+tB(\lambda)}((-\infty,0)),E_J((-\infty,0)))$, $t\in\bbR$,
where $A(\lambda)=\Re(K^*(H_0-\lambda-i0)^{-1}K)$,
$B(\lambda)=\Im(K^*(H_0-\lambda-i0)^{-1}K)$ a.e. Moreover, introducing the new
concept of a trindex for a pair of operators $(A,P)$ in $\calH$, where $A$ is
bounded and $P$ is an orthogonal projection, we prove that $\xi(\lambda,H_0,H)$
coincides with the trindex associated with the pair
$(\Xi(J+K^*(H_0-\lambda-i0)^{-1}K),\Xi(J))$. In addition, we discuss a variant
of the Birman-Krein formula relating the trindex of a pair of $\Xi$-operators
and the Fredholm determinant of the abstract scattering matrix.
We also provide a generalization of the classical Birman-Schwinger principle,
replacing the traditional eigenvalue counting functions by appropriate spectral
shift functions.
|
math
|
3,020 |
Weyl-Titchmarsh M-Function Asymptotics for Matrix-Valued Schrödinger Operators
|
math.SP
|
We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh
matrices associated with general matrix-valued Schr\"odinger operators on a
half-line.
|
math
|
3,021 |
Borg-Type Theorems for Matrix-Valued Schrödinger Operators
|
math.SP
|
A Borg-type uniqueness theorem for matrix-valued Schr\"odinger operators is
proved. More precisely, assuming a reflectionless potential matrix and spectrum
a half-line $[0,\infty)$, we derive triviality of the potential matrix. Our
approach is based on trace formulas and matrix-valued Herglotz representation
theorems. As a by-product of our techniques, we obtain an extension of Borg's
classical result from the class of periodic scalar potentials to the class of
reflectionless matrix-valued potentials.
|
math
|
3,022 |
A new approach to inverse spectral theory, I. Fundamental formalism
|
math.SP
|
We present a new approach (distinct from Gel'fand-Levitan) to the theorem of
Borg-Marchenko that the m-function (equivalently, spectral measure) for a
finite interval or half-line Schr\"odinger operator determines the potential.
Our approach is an analog of the continued fraction approach for the moment
problem. We prove there is a representation for the m-function
m(-\kappa^2) = -\kappa - \int_0^b A(\alpha) e^{-2\alpha\kappa}\, d\alpha +
O(e^{-(2b-\varepsilon)\kappa}).
A on [0,a] is a function of q on [0,a] and vice-versa. A key role is played
by a differential equation that A obeys after allowing x-dependence:
\frac{\partial A}{\partial x} = \frac{\partial A}{\partial \alpha} +
\int_0^\alpha A(\beta, x) A(\alpha -\beta, x)\, d\beta.
Among our new results are necessary and sufficient conditions on the
m-functions for potentials q_1 and q_2 for q_1 to equal q_2 on [0,a].
|
math
|
3,023 |
Monotonicity and Concavity Properties of The Spectral Shift Function
|
math.SP
|
Let $H_0$ and $V(s)$ be self-adjoint, $V,V'$ continuously differentiable in
trace norm with $V''(s)\geq 0$ for $s\in (s_1,s_2)$, and denote by
$\{E_{H(s)}(\lambda)\}_{\lambda\in\bbR}$ the family of spectral projections of
$H(s)=H_0+V(s)$. Then we prove for given $\mu\in\bbR$, that $s\longmapsto
\tr\big (V'(s)E_{H(s)}((-\infty, \mu))\big) $ is a nonincreasing function with
respect to $s$, extending a result of Birman and Solomyak. Moreover, denoting
by $\zeta (\mu,s)=\int_{-\infty}^\mu d\lambda \xi(\lambda,H_0,H(s))$ the
integrated spectral shift function for the pair $(H_0,H(s))$, we prove
concavity of $\zeta (\mu,s)$ with respect to $s$, extending previous results by
Geisler, Kostrykin, and Schrader. Our proofs employ operator-valued Herglotz
functions and establish the latter as an effective tool in this context.
|
math
|
3,024 |
Regularity of dissipative operators
|
math.SP
|
S.G.Krein's conjecture concerning Birkhoff-regularity of dissipative
differential operators has been proved in the even order case. As a byproduct
an existence of the limit of characteristic matrix as in the lower half-plane
has been established. Up to multiplication by a nonvanishing matrix this limit
coincides with the ratio of the matrices of regularity determinants.
|
math
|
3,025 |
On Local Borg-Marchenko Uniqueness Results
|
math.SP
|
We provide a new short proof of the following fact, first proved by one of us
in 1998: If two Weyl-Titchmarsh m-functions, $m_j(z)$, of two Schr\"odinger
operators $H_j = -\f{d^2}{dx^2} + q_j$, j=1,2 in $L^2 ((0,R))$, $0<R\leq
\infty$, are exponentially close, that is, $|m_1(z)- m_2(z)|
\underset{|z|\to\infty}{=} O(e^{-2\Ima (z^{1/2})a})$, 0<a<R, then $q_1 = q_2$
a.e.~on $[0,a]$. The result applies to any boundary conditions at x=0 and x=R
and should be considered a local version of the celebrated Borg-Marchenko
uniqueness result (which is quickly recovered as a corollary to our proof).
Moreover, we extend the local uniqueness result to matrix-valued Schr\"odinger
operators.
|
math
|
3,026 |
An inverse problem for point inhomogeneities
|
math.SP
|
We study quantum scattering theory off $n$ point inhomogeneities ($n\in\bbN$)
in three dimensions. The inhomogeneities (or generalized point interactions)
positioned at $\{\xi_1,...,\xi_n\}\subset\bbR^3$ are modeled in terms of the
$n^2$ (real) parameter family of self-adjoint extensions of
$-\Delta\big|_{C^\infty_0(\bbR^3\backslash\{\xi_1,...,\xi_n\})}$ in
$L^2(\bbR^3)$. The Green's function, the scattering solutions and the
scattering amplitude for this model are explicitly computed in terms of
elementary functions. Moreover, using the connection between fixed energy
quantum scattering and acoustical scattering, the following inverse spectral
result in acoustics is proved: The knowledge of the scattered field on a plane
outside these point-like inhomogeneities, with all inhomogeneities located on
one side of the plane, uniquely determines the positions and boundary
conditions associated with them.
|
math
|
3,027 |
Scattering Spaces and a Decomposition of Continuous Spectral Subspace
|
math.SP
|
We introduce the notion of scattering space $S_b^r$ for $N$-body quantum
mechanical systems, where $b$ is a cluster decomposition with $2\le |b|\le N$
and $r$ is a real number $0\le r\le 1$. Utilizing these spaces, we give a
decomposition of continuous spectral subspace by $S_b^1$ for $N$-body quantum
systems with long-range pair potentials
$V_\alpha^L(x_\alpha)=O(|x_\al|^{-\ep})$. This is extended to a decomposition
by $S_b^r$ with $0\le r\le 1$ for some long-range case. We also prove a
characterization of ranges of wave operators by $S_b^0$.
|
math
|
3,028 |
Spectral Properties of Non-Self-Adjoint Operators in the Semi-classical Regime
|
math.SP
|
We give a spectral description of the semi-classical Schrodinger operator
with a piecewise linear, complex valued potential. Moreover, using these
results, we show how an arbitrarily small bounded perturbation of a
non-self-adjoint operator can completely change the spectrum of the operator.
|
math
|
3,029 |
Uniqueness Results for Matrix-Valued Schrödinger, Jacobi, and Dirac-Type Operators
|
math.SP
|
Let $g(z,x)$ denote the diagonal Green's matrix of a self-adjoint $m\times m$
matrix-valued Schr\"odinger operator $H= -\f{d^2}{dx^2}I_m +Q(x)$ in $L^2
(\bbR)^{m}$, $m\in\bbN$. One of the principal results proven in this paper
states that for a fixed $x_0\in\bbR$ and all $z\in\bbC_+$, $g(z,x_0)$ and
$g^\prime (z,x_0)$ uniquely determine the matrix-valued $m\times m$ potential
$Q(x)$ for a.e.~$x\in\bbR$. We also prove the following local version of this
result. Let $g_j(z,x)$, $j=1,2$ be the diagonal Green's matrices of the
self-adjoint Schr\"odinger operators $H_j=-\f{d^2}{dx^2}I_m +Q_j(x)$ in $L^2
(\bbR)^{m}$. Suppose that for fixed $a>0$ and $x_0\in\bbR$,
$\|g_1(z,x_0)-g_2(z,x_0)\|_{\bbC^{m\times m}}+ \|g_1^\prime (z,x_0)-g_2^\prime
(z,x_0)\|_{\bbC^{m\times m}}
\underset{|z|\to\infty}{=}O\big(e^{-2\Im(z^{1/2})a}\big)$ for $z$ inside a cone
along the imaginary axis with vertex zero and opening angle less than $\pi/2$,
excluding the real axis. Then $Q_1(x)=Q_2(x)$ for a.e.~$x\in [x_0-a,x_0+a]$.
Analogous results are proved for matrix-valued Jacobi and Dirac-type operators.
|
math
|
3,030 |
Analytic continuation and resonance-free regions for Sturm-Liouville pote ntials with power decay
|
math.SP
|
We are concerned with the Sturm-Liouville problem on the half line. We show
that when the potential $q$ is subject only to power decay at infinity the
$L^2$ solution may be continued into a sector of the so-called un-physical
sheet. This gives rise to resonance free regions and the numerical estimation
of resonances outside these regions.
|
math
|
3,031 |
The Pseudospectral Properties of non-self-adjoint Schrödinger Operators in the semi-classical limit
|
math.SP
|
We describe the general qualitative behaviour of the resolvent norm for a
very wide class of non-self-adjoint Schroedinger operators in the
semi-classical regime, as the spectral parameter varies over the complex plane.
|
math
|
3,032 |
Analysis of geometric operators on open manifolds: A groupoid approach
|
math.SP
|
We use algebras of pseudodifferential operators on groupoids to study
geometric operators on non-compact manifolds and singular spaces. The first
step is to establish that the geometric operators are in our algebras. This
then leads to criteria for Fredholmness for geometric operators on suitable
non-compact manifolds, as well as to an inductive procedure to study their
essential spectrum. As an application, we answer a question of Melrose on the
essential spectrum of the Laplace operator on manifolds with multi-cylindrical
ends.
|
math
|
3,033 |
Fredholm theory of the linearized ${\bar \partial}$-operator and additivity of its index
|
math.SP
|
In this paper, we established Fredholm theory of the linearized ${{\bar
\partial}}$-operator and studied the additivity of its index.
|
math
|
3,034 |
Discrete Nodal Domain Theorems
|
math.SP
|
We give a detailed proof for two discrete analogues of Courant's Nodal Domain
Theorem.
|
math
|
3,035 |
Perturbation of Domain: Ordinary Differential Equations
|
math.SP
|
We work with the Friedrichs extension of a one dimensional Schrodinger whose
potential has a certain type of regular singularity near one end point. We
study the effect on the eigenvalues of shrinking the region slightly near the
end point. In particular we calculate the rate at which the eigenvalues of the
smaller region converge to the eigenvalues of the larger region as the region
expands. A result of this type was needed by that author in another paper
concerned with a domain perturbation problem on Riemannian manifolds. The
problem here, however, is stated purely in terms of ordinary differential
equations.
|
math
|
3,036 |
Spectral Integration and Spectral Theory for non-Archimedean Banach spaces
|
math.SP
|
The non-Archimedean spectral theory and spectral integration is developed.
The analog of the Stone theorem is proved. Applications are considered for
algebras of operators.
|
math
|
3,037 |
Weyl-Titchmarsh M-Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators
|
math.SP
|
We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh
matrices associated with general Dirac-type operators on half-lines and on
$\bbR$. We also prove new local uniqueness results for Dirac-type operators in
terms of exponentially small differences of Weyl-Titchmarsh matrices. As
concrete applications of the asymptotic high-energy expansion we derive a trace
formula for Dirac operators and use it to prove a Borg-type theorem.
|
math
|
3,038 |
Commutators, Spectral Trace Identities, and Universal Estimates for Eigenvalues
|
math.SP
|
Using simple commutator relations, we obtain several trace identities
involving eigenvalues and eigenfunctions of an abstract self-adjoint operator
acting in a Hilbert space. Applications involve abstract universal estimates
for the eigenvalue gaps. As particular examples, we present simple proofs of
the classical universal estimates for eigenvalues of the Dirichlet Laplacian
(Payne-Polya-Weinberger, Hile-Protter, etc.), as well as of some known and new
results for other differential operators and systems. We also suggest an
extension of the methods to the case of non-self-adjoint operators.
|
math
|
3,039 |
Spectral behaviour of a simple non-self-adjoint operator
|
math.SP
|
We investigate the spectrum of a typical non-self-adjoint differential
operator $AD=-d^2/dx^2\otimes A$ acting on $\Lp(0,1)\otimes \mathbb{C}^2$,
where $A$ is a $2\times 2$ constant matrix. We impose Dirichlet and Neumann
boundary conditions in the first and second coordinate respectively at both
ends of $[0,1]\subset\mathbb{R}$. For $A\in \mathbb{R}^{2\times 2}$ we explore
in detail the connection between the entries of $A$ and the spectrum of $AD$,
we find necessary conditions to ensure similarity to a self-adjoint operator
and give numerical evidence that suggests a non-trivial spectral evolution.
|
math
|
3,040 |
Log--Sobolev Inequalities and Regions with Exterior Exponential Cusps
|
math.SP
|
We begin by studying semigroup estimates that are more singular than those
implied by a Sobolev embedding theorem but which are equivalent to certain
logarithmic Sobolev inequalities. We then give a method for showing such
log--Sobolev inequalities hold for Euclidean regions that satisfy a certain
Hardy-type inequality. Our main application is to show that domains with
exterior exponential cusps, and hence no Sobolev embedding theorem, satisfy
such heat kernel bounds provided the cusp is not too sharp. Finally we consider
a rotationally invariant domain with an exponentially sharp cusp and show that
ultracontractivity does break down after a certain point.
|
math
|
3,041 |
Eigenvalues of an elliptic system
|
math.SP
|
We describe the spectrum of a non-self-adjoint elliptic system on a finite
interval. Under certain conditions we find that the eigenvalues form a discrete
set and converge asymptotically at infinity to one of several straight lines.
The eigenfunctions need not generate a basis of the relevant Hilbert space, and
the larger eigenvalues are extremely sensitive to small perturbations of the
operator. We show that the leading term in the spectral asymptotics is closely
related to a certain convex polygon, and that the spectrum does not determine
the operator up to similarity. Two elliptic systems which only differ in their
boundary conditions may have entirely different spectral asymptotics. While our
study makes no claim to generality, the results obtained will have to be
incorporated into any future general theory.
|
math
|
3,042 |
Schrödinger operator on homogeneous metric trees: spectrum in gaps
|
math.SP
|
The paper studies the spectral properties of the Schr\"odinger operator
$A_{gV} = A_0 + gV$ on a homogeneous rooted metric tree, with a decaying
real-valued potential $V$ and a coupling constant $g\ge 0$. The spectrum of the
free Laplacian $A_0 = -\Delta$ has a band-gap structure with a single
eigenvalue of infinite multiplicity in the middle of each finite gap. The
perturbation $gV$ gives rise to extra eigenvalues in the gaps. These
eigenvalues are monotone functions of $g$ if the potential $V$ has a fixed
sign. Assuming that the latter condition is satisfied and that $V$ is
symmetric, i.e. depends on the distance to the root of the tree, we carry out a
detailed asymptotic analysis of the counting function of the discrete
eigenvalues in the limit $g\to\infty$. Depending on the sign and decay of $V$,
this asymptotics is either of the Weyl type or is completely determined by the
behaviour of $V$ at infinity.
|
math
|
3,043 |
Quasi-conformal mappings and periodic spectral problems in dimension two
|
math.SP
|
We study spectral properties of second order elliptic operators with periodic
coefficients in dimension two. These operators act in periodic simply-connected
waveguides, with either Dirichlet, or Neumann, or the third boundary condition.
The main result is the absolute continuity of the spectra of such operators.
The corner stone of the proof is an isothermal change of variables, reducing
the metric to a flat one and the waveguide to a straight strip. The main
technical tool is the quasi-conformal variant of the Riemann mapping theorem.
|
math
|
3,044 |
Birkhoff's theorem and multidimensional numerical range
|
math.SP
|
We study the relation between the spectrum of a self-adjoint operator and its
multidimensional numerical range. It turns out that the multidimensional
numerical range is a convex set whose extreme points are sequences of
eigenvalues of the operator. Every collection of eigenvalues which can be
obtained by the Rayleigh--Ritz formula generates an extreme point of the
multidimensional numerical range. However, it may also have other extreme
points.
|
math
|
3,045 |
A Prime Orbit Theorem for Self-Similar Flows and Diophantine Approximation
|
math.SP
|
Assuming some regularity of the dynamical zeta function, we establish an
explicit formula with an error term for the prime orbit counting function of a
suspended flow. We define the subclass of self-similar flows, for which we give
an extensive analysis of the error term in the corresponding prime orbit
theorem.
|
math
|
3,046 |
Eigenvalue asymptotics of perturbed periodic Dirac systems in the slow-decay limit
|
math.SP
|
A perturbation decaying to 0 at infinity and not too irregular at 0
introduces at most a discrete set of eigenvalues into the spectral gaps of a
one-dimensional Dirac operator on the half-line. We show that the number of
these eigenvalues in a compact subset of a gap in the essential spectrum is
given by a quasi-semiclassical asymptotic formula in the slow-decay limit,
which for power-decaying perturbations is equivalent to the large-coupling
limit. This asymptotic behaviour elucidates the origin of the dense point
spectrum observed in spherically symmetric, radially periodic three-dimensional
Dirac operators.
|
math
|
3,047 |
A Class of Matrix-Valued Schrödinger Operators with Prescribed Finite-Band Spectra
|
math.SP
|
We construct a class of matrix-valued Schr\"odinger operators with prescribed
finite-band spectra of maximum spectral multiplicity. The corresponding matrix
potentials are shown to be stationary solutions of the KdV hierarchy. The
methods employed in this paper rely on matrix-valued Herglotz functions,
Weyl--Titchmarsh theory, pencils of matrices, and basic inverse spectral theory
for matrix-valued Schr\"odinger operators.
|
math
|
3,048 |
Matrix-Valued Generalizations of the Theorems of Borg and Hochstadt
|
math.SP
|
We prove a generalization of the well-known theorems by Borg and Hochstadt
for periodic self-adjoint Schr\"odinger operators without a spectral gap,
respectively, one gap in their spectrum, in the matrix-valued context. Our
extension of the theorems of Borg and Hochstadt replaces the periodicity
condition of the potential by the more general property of being reflectionless
(the resulting potentials then automatically turn out to be periodic and we
recover Despr\'es' matrix version of Borg's result). In addition, we assume the
spectra to have uniform maximum multiplicity (a condition automatically
fulfilled in the scalar context considered by Borg and Hochstadt). Moreover,
the connection with the stationary matrix KdV hierarchy is established.
The methods employed in this paper rely on matrix-valued Herglotz functions,
Weyl--Titchmarsh theory, pencils of matrices, and basic inverse spectral theory
for matrix-valued Schr\"odinger operators.
|
math
|
3,049 |
Approximating Spectral invariants of Harper operators on graphs II
|
math.SP
|
We study Harper operators and the closely related discrete magnetic
Laplacians (DML) on a graph with a free action of a discrete group, as defined
by Sunada. The spectral density function of the DML is defined using the von
Neumann trace associated with the free action of a discrete group on a graph.
The main result in this paper states that when the group is amenable, the
spectral density function is equal to the integrated density of states of the
DML that is defined using either Dirichlet or Neumann boundary conditions. This
establishes the main conjecture in a paper by Mathai and Yates. The result is
generalized to other self adjoint operators with finite propagation.
|
math
|
3,050 |
Projection methods for discrete Schrodinger operators
|
math.SP
|
Let $H$ be the discrete Schr\"odinger operator
$Hu(n):=u(n-1)+u(n+1)+v(n)u(n)$, $u(0)=0$ acting on $l^2({\bf Z}^+)$ where the
potential $v$ is real-valued and $v(n)\to 0$ as $n\to \infty$. Let $P$ be the
orthogonal projection onto a closed linear subspace $L \subset l^2({\bf Z}^+)$.
In a recent paper E.B. Davies defines the second order spectrum ${\rm
Spec}_2(H,L)$ of $H$ relative to $L$ as the set of $z \in {\bf C}$ such that
the restriction to $L$ of the operator $P(H-z)^2P$ is not invertible within the
space $L$. The purpose of this article is to investigate properties of ${\rm
Spec}_2(H,L)$ when $L$ is large but finite dimensional. We explore in
particular the connection between this set and the spectrum of $H$. Our main
result provides sharp bounds in terms of the potential $v$ for the asymptotic
behaviour of ${\rm Spec}_2(H,L)$ as $L$ increases towards $l^2({\bf Z}^+)$.
|
math
|
3,051 |
The "Action" Variable is not an Invariant for the Uniqueness in the Inverse Scattering Problem
|
math.SP
|
We give a simple example of non-uniqueness in the inverse scattering for
Jacobi matrices: roughly speaking $S$-matrix is analytic. Then, multiplying a
reflection coefficient by an inner function, we repair this matrix in such a
way that it does uniquely determine a Jacobi matrix of Szeg\"o class; on the
other hand the transmission coefficient remains the same. This implies the
statement given in the title.
|
math
|
3,052 |
On Povzner--Wienholtz-type Self-Adjointness Results for Matrix-Valued Sturm--Liouville Operators
|
math.SP
|
We derive Povzner--Wienholtz-type self-adjointness results for $m\times m$
matrix-valued Sturm--Liouville operators
$T=R^{-1}\big[-\f{d}{dx}P\f{d}{dx}+Q\big]$ in $L^2((a,b);Rdx)^m$, $m\in\bbN$,
for $(a,b)$ a half-line or $\bbR$.
|
math
|
3,053 |
$\mathbf {SL_2(\bbR)}$, Exponential Herglotz Representations, and Spectral Averaging
|
math.SP
|
We revisit the concept of spectral averaging and point out its origin in
connection with one-parameter subgroups of $SL_2(\bbR)$ and the corresponding
M\"obius transformations. In particular, we identify exponential Herglotz
representations as the basic ingredient for the absolute continuity of average
spectral measures with respect to Lebesgue measure and the associated spectral
shift function as the corresponding density for the averaged measure. As a
by-product of our investigations we unify the treatment of rank-one
perturbations of self-adjoint operators and that of self-adjoint extensions of
symmetric operators with deficiency indices $(1,1)$. Moreover, we derive
separate averaging results for absolutely continuous, singularly continuous,
and pure point measures and conclude with an averaging result of the
$\kappa$-continuous part (with respect to the $\kappa$-dimensional Hausdorff
measure) of singularly continuous measures.
|
math
|
3,054 |
The Lie algebra f4(Os) (split f4) with Mathematica
|
math.SP
|
We present in this paper all the details for a complete description of the
Lie algebra a in the split case at any characteristic. We finish with the
determination of the expression of a generic element of this algebra. First of
all is necessary to implement its quadratic Jordan structure (see the O. Loss'
server http://mathematik.uibk.ac.at/jordan for our preprint). We write in this
file only the computational details.
|
math
|
3,055 |
Spectral distributions and isospectral sets of tridiagonal matrices
|
math.SP
|
We analyze the correspondence between finite sequences of finitely supported
probability distributions and finite-dimensional, real, symmetric, tridiagonal
matrices. In particular, we give an intrinsic description of the topology
induced on sequences of distributions by the usual Euclidean structure on
matrices. Our results provide an analytical tool with which to study ensembles
of tridiagonal matrices, important in certain inverse problems and integrable
systems. As an application, we prove that the Euler characteristic of any
generic isospectral set of symmetric, tridiagonal matrices is a tangent number.
|
math
|
3,056 |
Existence and uniqueness of solutions to the operator Riccati equation. A geometric approach
|
math.SP
|
We introduce a new concept of unbounded solutions to the operator Riccati
equation $A_1 X - X A_0 - X V X + V^\ast = 0$ and give a complete description
of its solutions associated with the spectral graph subspaces of the block
operator matrix $\mathbf{B} = \begin{pmatrix} A_0 & V V^\ast & A_1
\end{pmatrix}$. We also provide a new characterization of the set of all
contractive solutions under the assumption that the Riccati equation has a
contractive solution associated with a spectral subspace of the operator
$\mathbf{B}$. In this case we establish a criterion for the uniqueness of
contractive solutions.
|
math
|
3,057 |
L^p norms of eigenfunctions in the completely integrable case
|
math.SP
|
The eigenfunctions e^{i \lambda x} of the Laplacian on a flat torus have
uniformly bounded L^p norms. In this article, we prove that for every other
quantum integrable Laplacian, the L^p norms of the joint eigenfunctions must
blow up at a rate \gg \lambda^{p-2/4p - \epsilon} for every \epsilon >0 as
\lambda \to \infty.
|
math
|
3,058 |
Reconstructing Jacobi Matrices from Three Spectra
|
math.SP
|
Cut a Jacobi matrix into two pieces by removing the n-th column and n-th row.
We give neccessary and sufficient conditions for the spectra of the original
matrix plus the spectra of the two submatrices to uniqely determine the
original matrix. Our result contains Hostadt's original result as a special
case.
|
math
|
3,059 |
Inverse spectral problems for Sturm-Liouville operators with singular potentials
|
math.SP
|
The inverse spectral problem is solved for the class of Sturm-Liouville
operators with singular real-valued potentials from the space $W^{-1}_2(0,1)$.
The potential is recovered via the eigenvalues and the corresponding norming
constants. The reconstruction algorithm is presented and its stability proved.
Also, the set of all possible spectral data is explicitly described and the
isospectral sets are characterized.
|
math
|
3,060 |
Discrete Approximation of Non-Compact Operators Describing Continuum-of-Alleles Models
|
math.SP
|
We consider the eigenvalue equation for the largest eigenvalue of certain
kinds of non-compact linear operators given as the sum of a multiplication and
a kernel operator. It is shown that, under moderate conditions, such operators
can be approximated arbitrarily well by operators of finite rank, which
constitutes a discretization procedure. For this purpose, two standard methods
of approximation theory, the Nystr\"om and the Galerkin method, are
generalized. The operators considered describe models for mutation and
selection of an infinitely large population of individuals that are labeled by
real numbers, commonly called continuum-of-alleles (COA) models.
|
math
|
3,061 |
Inverse spectral problems for Sturm-Liouville operators with singular potentials, II. Reconstruction by two spectra
|
math.SP
|
We solve the inverse spectral problem of recovering the singular potentials
$q\in W^{-1}_{2}(0,1)$ of Sturm-Liouville operators by two spectra. The
reconstruction algorithm is presented and necessary and sufficient conditions
on two sequences to be spectral data for Sturm-Liouville operators under
consideration are given.
|
math
|
3,062 |
A generalization of the $tan 2Θ$ Theorem
|
math.SP
|
Let $\mathbf{A}$ be a bounded self-adjoint operator on a separable Hilbert
space $\mathfrak{H}$ and $\mathfrak{H}_0\subset\mathfrak{H}$ a closed invariant
subspace of $\mathbf{A}$. Assuming that $\sup\spec(A_0)\leq \inf\spec(A_1)$,
where $A_0$ and $A_1$ are restrictions of $\mathbf{A}$ onto the subspaces
$\mathfrak{H}_0$ and $\mathfrak{H}_1=\mathfrak{H}_0^\perp$, respectively, we
study the variation of the invariant subspace $\mathfrak{H}_0$ under bounded
self-adjoint perturbations $\mathbf{V}$ that are off-diagonal with respect to
the decomposition $\mathfrak{H} = \mathfrak{H}_0\oplus\mathfrak{H}_1$. We
obtain sharp two-sided estimates on the norm of the difference of the
orthogonal projections onto invariant subspaces of the operators $\mathbf{A}$
and $\mathbf{B}=\mathbf{A}+\mathbf{V}$. These results extend the celebrated
Davis-Kahan $\tan 2\Theta$ Theorem. On this basis we also prove new existence
and uniqueness theorems for contractive solutions to the operator Riccati
equation, thus, extending recent results of Adamyan, Langer, and Tretter.
|
math
|
3,063 |
Quasi-free resolutions of Hilbert modules
|
math.SP
|
The notion of a quasi-free Hilbert module over a function algebra
$\mathcal{A}$ consisting of holomorphic functions on a bounded domain $\Omega$
in complex $m$ space is introduced. It is shown that quasi-free Hilbert modules
correspond to the completion of the direct sum of a certain number of copies of
the algebra $\mathcal{A}$. A Hilbert module is said to be weakly regular
(respectively, regular) if there exists a module map from a quasi-free module
with dense range (respectively, onto). A Hilbert module $\mathcal{M}$ is said
to be compactly supported if there exists a constant $\beta$ satisfying $\|\phi
f \| \leq \beta \|\phi \|_X \|f\|$ for some compact subset $X$ of $\Omega$ and
$\phi$ in $\mathcal{A}$, $f$ in $\mathcal{M}$. It is shown that if a Hilbert
module is compactly supported then it is weakly regular. The paper identifies
several other classes of Hilbert modules which are weakly regular. In addition,
this result is extended to yield topologically exact resolutions of such
modules by quasi-free ones.
|
math
|
3,064 |
A new basis for eigenmodes on the Sphere
|
math.SP
|
The usual spherical harmonics $Y_{\ell m}$ form a basis of the vector space
${\cal V} ^{\ell}$ (of dimension $2\ell+1$) of the eigenfunctions of the
Laplacian on the sphere, with eigenvalue $\lambda_{\ell} = -\ell ~(\ell +1)$.
Here we show the existence of a different basis $\Phi ^{\ell}_j$ for ${\cal V}
^{\ell}$, where $\Phi ^{\ell}_j(X) \equiv (X \cdot N_j)^{\ell}$, the $\ell
^{th}$ power of the scalar product of the current point with a specific null
vector $N_j$. We give explicitly the transformation properties between the two
bases. The simplicity of calculations in the new basis allows easy
manipulations of the harmonic functions. In particular, we express the
transformation rules for the new basis, under any isometry of the sphere.
The development of the usual harmonics $Y_{\ell m}$ into thee new basis (and
back) allows to derive new properties for the $Y_{\ell m}$. In particular, this
leads to a new relation for the $Y_{\ell m}$, which is a finite version of the
well known integral representation formula. It provides also new development
formulae for the Legendre polynomials and for the special Legendre functions.
|
math
|
3,065 |
Connectedness of the Isospectral Manifold for One-Dimensional Half-Line Schrödinger Operators
|
math.SP
|
Let V_0 be a real-valued function on [0,\infty) and V\in L^1([0,R]) for all
R>0 so that H(V_0)= -\f{d^2}{dx^2}+V_0 in L^2([0,\infty)) with u(0)=0 boundary
conditions has discrete spectrum bounded from below. Let \calM (V_0) be the set
of V so that H(V) and H(V_0) have the same spectrum. We prove that \calM(V_0)
is connected.
|
math
|
3,066 |
Asymptotic behaviour of quasi-orthogonal polynomials
|
math.SP
|
We obtain explicit upper and lower bounds on the norms of the spectral
projections of the non-self-adjoint harmonic oscillator. Some of our results
apply to a variety of other families of orthogonal polynomials.
|
math
|
3,067 |
A trace formula and high energy spectral asymptotics for the perturbed Landau Hamiltonian
|
math.SP
|
A two-dimensional Schr\"odinger operator with a constant magnetic field
perturbed by a smooth compactly supported potential is considered. The spectrum
of this operator consists of eigenvalues which accumulate to the Landau levels.
We call the set of eigenvalues near the $n$'th Landau level an $n$'th
eigenvalue cluster, and study the distribution of eigenvalues in the $n$'th
cluster as $n\to\infty$. A complete asymptotic expansion for the eigenvalue
moments in the $n$'th cluster is obtained and some coefficients of this
expansion are computed. A trace formula involving the first eigenvalue moments
is obtained.
|
math
|
3,068 |
Generalized eigenfunctions of relativistic Schroedinger operators I
|
math.SP
|
Generalized eigenfunctions of the 3-dimensional relativistic Schr\"odinger
operator $\sqrt{\Delta} + V(x)$ with $|V(x)|\le C < x >^{{-\sigma}}$, $\sigma >
1$, are considered. We show that the generalized eigenfunctions can be
expressed as the sum of plane waves and solutions to the time-independent
relativistic Schr\"odinger equation with the radiation condition. If $\sigma
>3$, then we can give pointwise estimates of the differences between the sums
and the solutions.
|
math
|
3,069 |
On the spectrum of the Laplace-Beltrami operator for p-forms on asymptotically hyperbolic manifolds
|
math.SP
|
Under suitable conditions on the asymptotic decay of the metric, we compute
the essential spectrum of the Laplace-Beltrami operator acting on $p$-forms on
asymptotically hyperbolic manifolds.
|
math
|
3,070 |
On Weyl-Titchmarsh Theory for Singular Finite Difference Hamiltonian Systems
|
math.SP
|
We develop the basic theory of matrix-valued Weyl-Titchmarsh M-functions and
the associated Green's matrices for whole-line and half-line self-adjoint
Hamiltonian finite difference systems with separated boundary conditions.
|
math
|
3,071 |
Energy Decay of Damped Systems
|
math.SP
|
We present a new and simple bound for the exponential decay of second order
systems using the spectral shift. This result is applied to finite matrices as
well as to partial differential equations of Mathematical Physics. The type of
the generated semigroup is shown to be bounded by the upper real part of the
numerical range of the underlying quadratic operator pencil.
|
math
|
3,072 |
Kato's inequality and asymptotic spectral properties for discrete magnetic Laplacians
|
math.SP
|
In this paper, a discrete form of the Kato inequality for discrete magnetic
Laplacians on graphs is used to study asymptotic properties of the spectrum of
discrete magnetic Schrodinger operators. We use the existence of a ground state
with suitable properties for the ordinary combinatorial Laplacian and semigroup
domination to relate the combinatorial Laplacian with the discrete magnetic
Laplacian.
|
math
|
3,073 |
Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Vishik-Ljusternik theorem
|
math.SP
|
We extend the perturbation theory of Vishik, Ljusternik and Lidskii for
eigenvalues of matrices, using methods of min-plus algebra. We show that the
asymptotics of the eigenvalues of a perturbed matrix is governed by certain
discrete optimisation problems, from which we derive new perturbation formulae,
extending the classical ones and solving cases which where singular in previous
approaches. Our results include general weak majorisation inequalities,
relating leading exponents of eigenvalues of perturbed matrices and min-plus
analogues of eigenvalues.
|
math
|
3,074 |
The sharp form of the strong Szego theorem
|
math.SP
|
Let $f$ be a function on the unit circle and $D_n(f)$ be the determinant of
the $(n+1)\times (n+1)$ matrix with elements $\{c_{j-i}\}_{0\leq i,j\leq n}$
where $c_m =\hat f_m\equiv \int e^{-im\theta} f(\theta) \f{d\theta}{2\pi}$. The
sharp form of the strong Szeg\H{o} theorem says that for any real-valued $L$ on
the unit circle with $L,e^L$ in $L^1 (\f{d\theta}{2\pi})$, we have \[
\lim_{n\to\infty} D_n(e^L) e^{-(n+1)\hat L_0} = \exp \biggl(\sum_{k=1}^\infty
k\abs{\hat L_k}^2\biggr) \] where the right side may be finite or infinite. We
focus on two issues here: a new proof when $e^{i\theta}\to L(\theta)$ is
analytic and known simple arguments that go from the analytic case to the
general case. We add background material to make this article self-contained.
|
math
|
3,075 |
Semi-classical Analysis and Pseudospectra
|
math.SP
|
We prove an approximate spectral theorem for non-self-adjoint operators and
investigate its applications to second order differential operators in the
semi-classical limit. This leads to the construction of a twisted FBI
transform. We also investigate the connections between pseudospectra and
boundary conditions in the semi-classical limit.
|
math
|
3,076 |
On the absolutely continuous spectrum of the Laplace-Beltrami operator acting on p-forms for a class of warped product metrics
|
math.SP
|
We explicitely compute the absolutely continuous spectrum of the
Laplace-Beltrami operator for $p$-forms for the class of warped product metrics
$d\sigma^2= y^{2a}dy^2 + y^{2b}d\theta_{\Sphere^{N-1}}^2$, where $y$ is a
boundary defining function on the unit ball B(0,1) in $\Real^N$.
|
math
|
3,077 |
Perturbation of eigenvalues of matrix pencils and optimal assignment problem
|
math.SP
|
We consider a matrix pencil whose coefficients depend on a positive parameter
$\epsilon$, and have asymptotic equivalents of the form $a\epsilon^A$ when
$\epsilon$ goes to zero, where the leading coefficient $a$ is complex, and the
leading exponent $A$ is real. We show that the asymptotic equivalent of every
eigenvalue of the pencil can be determined generically from the asymptotic
equivalents of the coefficients of the pencil. The generic leading exponents of
the eigenvalues are the "eigenvalues" of a min-plus matrix pencil. The leading
coefficients of the eigenvalues are the eigenvalues of auxiliary matrix
pencils, constructed from certain optimal assignment problems.
|
math
|
3,078 |
The singularly continuous spectrum and non-closed invariant subspaces
|
math.SP
|
Let $\mathbf{A}$ be a bounded self-adjoint operator on a separable Hilbert
space $\mathfrak{H}$ and $\mathfrak{H}_0\subset\mathfrak{H}$ a closed invariant
subspace of $\mathbf{A}$. Assuming that $\mathfrak{H}_0$ is of codimension 1,
we study the variation of the invariant subspace $\mathfrak{H}_0$ under bounded
self-adjoint perturbations $\mathbf{V}$ of $\mathbf{A}$ that are off-diagonal
with respect to the decomposition $\mathfrak{H}=
\mathfrak{H}_0\oplus\mathfrak{H}_1$. In particular, we prove the existence of a
one-parameter family of dense non-closed invariant subspaces of the operator
$\mathbf{A}+\mathbf{V}$ provided that this operator has a nonempty singularly
continuous spectrum. We show that such subspaces are related to non-closable
densely defined solutions of the operator Riccati equation associated with
generalized eigenfunctions corresponding to the singularly continuous spectrum
of $\mathbf{B}$.
|
math
|
3,079 |
Nash type inequalities for fractional powers of non-negative self-adjoint operators
|
math.SP
|
Assuming that a Nash type inequality is satisfied by a non-negative
self-adjoint operator $A$, we prove a Nash type inequality for the fractional
powers $A^{\alpha}$ of $A$. Under some assumptions, we give ultracontractivity
bounds for the semigroup $(T_{t,{\alpha}})$ generated by $-A^{\alpha}$.
|
math
|
3,080 |
Lectures on scattering theory
|
math.SP
|
The first two lectures are devoted to describing the basic concepts of
scattering theory in a very compressed way. A detailed presentation of the
abstract part can be found in \cite{I} and numerous applications in \cite{RS}
and \cite{Y2}. The last two lectures are based on the recent research of the
author.
|
math
|
3,081 |
On the discrete spectrum of a family of differential operators
|
math.SP
|
A family $\BA_\a$ of differential operators depending on a real parameter
$\a$ is considered. The problem can be formulated in the language of
perturbation theory of quadratic forms. The perturbation is only relatively
bounded but not relatively compact with respect to the unperturbed form.
The spectral properties of the operator $\BA_\a$ strongly depend on $\a$. In
particular, for $\a<\sqrt2$ the spectrum of $\BA_\a$ below 1/2 is finite, while
for $\a>\sqrt2$ the operator has no eigenvalues at all. We study the asymptotic
behaviour of the number of eigenvalues as $\a\nearrow\sqrt2$. We reduce this
problem to the one on the spectral asymptotics for a certain Jacobi matrix.
|
math
|
3,082 |
Trace-class approach in scattering problems for perturbations of media
|
math.SP
|
We consider the operators $H_0=M_0^{-1}(x) P(D)$ and $H =M^{-1} (x) P(D)$
where $M_0 (x)$ and $M (x)$ are positively definite bounded matrix-valued
functions and $P(D)$ is an elliptic differential operator. Our main result is
that the wave operators for the pair $H_0$, $H$ exist and are complete if the
difference $ M(x)-M_0(x)=O(|x|^{- rho})$, $ rho>d$, as $|x| to infty$. Our
point is that no special assumptions on $M_0(x)$ are required. Similar results
are obtained in scattering theory for the wave equation.
|
math
|
3,083 |
Inverse problem for one class of nonselfadjoint operator's bunches with nonperiodic coefficients
|
math.SP
|
In this paper the complete spectral analysis of the operators is carried out
and also with help of generalized normalizing numbers the inverse problem is
solved.
|
math
|
3,084 |
Generalized Reflection Coefficients in Toeplitz-Block-Toeplitz Matrix Case and Fast Inverse 2D levinson Algorithm
|
math.SP
|
A factorization of the inverse of a Hermetian positive definite matrix based
on a diagonal by diagonal recurrence formulae permits the inversion of Toeplitz
Block Toeplitz matrices using minimized matrix-vector products, with a
complexity of ((n1)^3)((n2)^2), where n1 is the block size, and n2 is the block
matrix size. A 2D levinson algorithm is introduced that outperform Wittle,
Wiggins and Robinson Algorithm
|
math
|
3,085 |
Necessary and sufficient conditions for the inverse problem of one class ordinary differential operators with complex periodic coefficients
|
math.SP
|
The basic purpose of the present paper is the full solutions of the inverse
problem (i.e. a finding of necessary and sufficient conditions) for the
operator with complex periodic coefficients.
|
math
|
3,086 |
Limits of Zeros of Orthogonal Polynomials on the Circle
|
math.SP
|
We prove that there is a universal measure on the unit circle such that any
probability measure on the unit disk is the limit distribution of some
subsequence of the corresponding orthogonal polynomials. This follows from an
extension of a result of Alfaro and Vigil (which answered a question of
Tur\'an): namely, for $n<N$, one can freely prescribe the $n$-th polynomial and
$N-n$ zeros of the $N$-th one. We shall also describe all possible limit sets
of zeros within the unit disk.
|
math
|
3,087 |
Normalized Ricci flow on Riemann surfaces and determinants of Laplacian
|
math.SP
|
In this note we give a simple proof of the fact that the determinant of
Laplace operator in smooth metric over compact Riemann surfaces of arbitrary
genus $g$ monotonously grows under the normalized Ricci flow. Together with
results of Hamilton that under the action of the normalized Ricci flow the
smooth metric tends asymptotically to metric of constant curvature for $g\geq
1$, this leads to a simple proof of Osgood-Phillips-Sarnak theorem stating that
that within the class of smooth metrics with fixed conformal class and fixed
volume the determinant of Laplace operator is maximal on metric of constant
curvatute.
|
math
|
3,088 |
Tau-functions on spaces of Abelian differentials and higher genus generalizations of Ray-Singer formula
|
math.SP
|
Let $w$ be an Abelian differential on compact Riemann surface of genus $g\geq
1$. We obtain an explicit holomorphic factorization formula for
$\zeta$-regularized determinant of the Laplacian in flat conical metrics with
trivial holonomy $|w|^2$, generalizing the classical Ray-Singer result in
$g=1$.
|
math
|
3,089 |
On a theorem of Kac and Gilbert
|
math.SP
|
We prove a general operator theoretic result that asserts that many
multiplicity two selfadjoint operators have simple singular spectrum.
|
math
|
3,090 |
Orthogonal polynomials on the unit circle: New results
|
math.SP
|
We announce numerous new results in the theory of orthogonal polynomials on
the unit circle.
|
math
|
3,091 |
On location of discrete spectrum for complex Jacobi matrices
|
math.SP
|
We study spectrum inclusion regions for complex Jacobi matrices which are
compact perturbations of the discrete laplacian. The condition sufficient for
the lack of discrete spectrum for such matrices is given.
|
math
|
3,092 |
Geometric lower bounds for the spectrum of elliptic PDEs with Dirichlet conditions in part
|
math.SP
|
An extension of the lower-bound lemma of Boggio is given for the weak forms
of certain elliptic operators, which have partially Dirichlet and partially
Neumann boundary conditions, and are in general nonlinear. Its consequences and
those of an adapted Hardy inequality for the location of the bottom of the
spectrum are explored in corollaries wherein a variety of assumptions are
placed on the shape of the Dirichlet and Neumann boundaries.
|
math
|
3,093 |
Time-Dependent Solutions of a Discrete Schrodinger's Equation
|
math.SP
|
By substituting the diagonal and the other two adjacent diagonals terms with
two different functions depending on two parameters of the discrete Laplacian
operator, the nature of its spectrum changes from being purely continuous to
partially continuous. We present the existence of three isolated eigenvalues by
altering those two parameters.
|
math
|
3,094 |
Introduction to the spectral theory of self-adjoint differential vector-operators
|
math.SP
|
We study the spectral theory of operators, generated as direct sums of
self-adjoint extensions of quasi-differential minimal operators on a
multi-interval set (self-adjoint vector-operators), acting in a Hilbert space.
Spectral theorems for such operators are discussed, the structure of the
ordered spectral representation is investigated for the case of differential
coordinate operators. One of the main results is the construction of spectral
resolutions. Finally, we study the matters connected with analytical
decompositions of generalized eigenfunctions of such vector-operators and build
a matrix spectral measure leading to the matrix Hilbert space theory. Results,
connected with other spectral properties of self-adjoint vector-operators, such
as the introduction of the identity resolution and the spectral multiplicity
have also been obtained.
Vector-operators have been mainly studied by W.N. Everitt, L. Markus and A.
Zettl. Being a natural continuation of Everitt-Markus-Zettl theory, the
presented results reveal the internal structure of self-adjoint
vector-operators and are essential for the further study of their spectral
properties.
|
math
|
3,095 |
Spectral problems with mixed Dirichlet-Neumann boundary conditions: isospectrality and beyond
|
math.SP
|
Consider a bounded domain with the Dirichlet condition on a part of the
boundary and the Neumann condition on its complement. Does the spectrum of the
Laplacian determine uniquely which condition is imposed on which part? We
present some results, conjectures and problems related to this variation on the
isospectral theme.
|
math
|
3,096 |
Existence of eigenvalues of a linear operator pencil in a curved waveguide -- localized shelf waves on a curved coast
|
math.SP
|
The study of the possibility of existence of the non-propagating, trapped
continental shelf waves (CSWs)along curved coasts reduces mathematically to a
spectral problem for a self-adjoint operator pencil in a curved strip. Using
the methods developed in the setting of the waveguide trapped mode problem, we
show that such CSWs exist for a wide class of coast curvature and depth
profiles.
|
math
|
3,097 |
The characterization problem for one class high order ordinary differential operators with periodic coefficients
|
math.SP
|
The main purpose of the present work is solving the characterization problem
which consist of identification of necessary and sufficient conditions on the
scattering data ensuring that the reconstructed potential belongs to a
particular class
|
math
|
3,098 |
Aizenman's Theorem for Orthogonal Polynomials on the Unit Circle
|
math.SP
|
For suitable classes of random Verblunsky coefficients, including
independent, identically distributed, rotationally invariant ones, we prove
that if \[ \mathbb{E} \biggl(\int\frac{d\theta}{2\pi}
\biggl|\biggl(\frac{\mathcal{C} + e^{i\theta}}{\mathcal{C} -e^{i\theta}}
\biggr)_{k\ell}\biggr|^p \biggr) \leq C_1 e^{-\kappa_1 |k-\ell|} \] for some
$\kappa_1 >0$ and $p<1$, then for suitable $C_2$ and $\kappa_2 >0$, \[
\mathbb{E} \bigl(\sup_n |(\mathcal{C}^n)_{k\ell}|\bigr) \leq C_2 e^{-\kappa_2
|k-\ell|} \] Here $\mathcal{C}$ is the CMV matrix.
|
math
|
3,099 |
Fine Structure of the Zeros of Orthogonal Polynomials, I. A Tale of Two Pictures
|
math.SP
|
Mhaskar-Saff found a kind of universal behavior for the bulk structure of the
zeros of orthogonal polynomials for large $n$. Motivated by two plots, we look
at the finer structure for the case of random Verblunsky coefficients and for
what we call the BLS condition: $\alpha_n = Cb^n + O((b\Delta)^n)$. In the
former case, we describe results of Stoiciu. In the latter case, we prove
asymptotically equal spacing for the bulk of zeros.
|
math
|
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