- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 06:12:39
23
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JST&vol=3&update_since=2024-03-29
Journal of Spectral Theory
J. Spectr. Theory
JST
1664-039X
1664-0403
Quantum theory
10.4171/JST
http://www.ems-ph.org/doi/10.4171/JST
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
3
2013
1
An estimate on the number of eigenvalues of a quasiperiodic Jacobi matrix of size $n$ contained in an interval of size $n^{–C}$
Ilia
Binder
University of Toronto, TORONTO, ONTARIO, CANADA
Mircea
Voda
University of Toronto, TORONTO, ONTARIO, CANADA
Eigenvalues, eigenfunctions, quasiperiodic Jacobi matrix, avalanche principle, large deviations
We consider infinite quasi-periodic Jacobi self-adjoint matrices for which the three main diagonals are given via values of real analytic functions on the trajectory of the shift $x\rightarrow x+\omega$. We assume that the Lyapunov exponent $L(E_{0})$ of the corresponding Jacobi cocycle satisfies $L(E_{0})\ge\gamma>0$. In this setting we prove that the number of eigenvalues $E_{j}^{(n)}(x)$ of a submatrix of size $n$ contained in an interval $I$ centered at $E_{0}$ with $|I|=n^{-C_{1}}$ does not exceed $\left(\log n\right)^{C_{0}}$ for any $x$. Here $n\ge n_{0}$, and $n_{0}$, $C_{0}$, $C_{1}$ are constants depending on $\gamma$ (and the other parameters of the problem).
Quantum theory
Operator theory
Statistical mechanics, structure of matter
General
1
45
10.4171/JST/36
http://www.ems-ph.org/doi/10.4171/JST/36
Scattering zippers and their spectral theory
Laurent
Marin
Universität Erlangen-Nürnberg, ERLANGEN, GERMANY
Hermann
Schulz-Baldes
Universität Erlangen-Nürnberg, ERLANGEN, GERMANY
Weyl theory, Sturm–Liouville oscillation theory, orthogonal polynomials on the unit circle
A scattering zipper is a system obtained by concatenation of scattering events with equal even number of incoming and out going channels. The associated scattering zipper operator is the unitary equivalent of Jacobi matrices with matrix entries and generalizes Blatter–Browne and Chalker–Coddington models and CMV matrices. Weyl discs are analyzed and used to prove a bijection between the set of semi-infinite scattering zipper operators and matrix valued probability measures on the unit circle. Sturm–Liouville oscillation theory is developed as a tool to calculate the spectra of finite and periodic scattering zipper operators.
Ordinary differential equations
General
47
82
10.4171/JST/37
http://www.ems-ph.org/doi/10.4171/JST/37
The Tan 2$\Theta$ Theorem for indefinite quadratic forms
Luba
Grubišić
University of Zagreb, ZAGREB, CROATIA
Vadim
Kostrykin
Johannes Gutenberg-Universität Mainz, MAINZ, GERMANY
Konstantin
Makarov
University of Missouri, COLUMBIA, UNITED STATES
Krešimir
Veselić
FernUniversität Hagen, HAGEN, GERMANY
Perturbation theory, quadratic forms, invariant subspaces
A version of the Davis–Kahan Tan 2$\Theta$ theorem [3] for not necessarily semibounded linear operators defined by quadratic forms is proven. This theorem generalizes a result by Motovilov and Selin [13].
Operator theory
Ordinary differential equations
General
83
100
10.4171/JST/38
http://www.ems-ph.org/doi/10.4171/JST/38
Spectral analysis of tridiagonal Fibonacci Hamiltonians
William
Yessen
Rice University, HOUSTON, UNITED STATES
Spectral theory, quasiperiodicity, Jacobi operators, Fibonacci Hamiltonians, trace maps
We consider a family of discrete Jacobi operators on the one-dimensional integer lattice, with the diagonal and the off-diagonal entries given by two sequences generated by the Fibonacci substitution on two letters. We show that the spectrum is a Cantor set of zero Lebesgue measure, and discuss its fractal structure and Hausdorff dimension. We also extend some known results on the diagonal and the off-diagonal Fibonacci Hamiltonians.
Operator theory
Statistical mechanics, structure of matter
General
101
128
10.4171/JST/39
http://www.ems-ph.org/doi/10.4171/JST/39
2
A spectral alternative for continuous families of self-adjoint operators
Alexander
Gordon
University of North Carolina at Charlotte, CHARLOTTE, UNITED STATES
Self-adjoint operator, operator family, eigenvalue, continuous branch, Cantor spectrum, ergodic operator
We consider a continuous family of self-adjoint operators $A_s$ in a separable Hilbert space, the parameter $s$ being a point of a complete metric space $S$. It is well known that isolated simple eigenvalues (assuming that the operators are bounded and the mapping $s\mapsto A_s$ is continuous in the norm sense) behave "well'': under small changes of the parameter they do not disappear and change continuously. Unlike this, the eigenvalues embedded in the essential spectrum can display a very "bad'' behavior. It turns out, nevertheless, that if the set of eigenvalues is non-empty for a topologically rich (e.g., open) set of values of the parameter, then the (multi-valued) eigenvalue function has continuous branches. One application is as follows. Suppose a one-dimen\-sional quasi-periodic Schr\"{o}\-dinger operator has Cantor spectrum; then a Baire generic operator in its hull does not have eigenvalues.
Operator theory
Ordinary differential equations
General
129
145
10.4171/JST/40
http://www.ems-ph.org/doi/10.4171/JST/40
Schrödinger operators with slowly decaying Wigner–von Neumann type potentials
Milivoje
Lukic
Rice University, HOUSTON, UNITED STATES
Schrödinger operator, bounded variation,Wigner–von Neumann potential
We consider Schr\"odinger operators with potentials satisfying a generalized bounded variation condition at infinity and an $L^p$ decay condition. This class of potentials includes slowly decaying Wigner--von~Neumann type potentials $\sin(ax)/x^b$ with $b>0$. We prove absence of singular continuous spectrum and show that embedded eigenvalues in the continuous spectrum can only take values from an explicit finite set. Conversely, we construct examples where such embedded eigenvalues are present, with exact asymptotics for the corresponding eigensolutions.
Ordinary differential equations
Partial differential equations
General
147
169
10.4171/JST/41
http://www.ems-ph.org/doi/10.4171/JST/41
Fredholm and invertibility theory for a special class of Toeplitz + Hankel operators
Estelle
Basor
American Institute of Mathematics, PALO ALTO, UNITED STATES
Torsten
Ehrhardt
University of California at Santa Cruz, SANTA CRUZ, UNITED STATES
Toeplitz operator, Hankel operator, Toeplitz plus Hankel operator
We develop a complete Fredholm and invertibility theory for Toeplitz+Hankel operators $T(a)+H(b)$ on the Hardy space $\Hp$, $1
Operator theory
General
171
214
10.4171/JST/42
http://www.ems-ph.org/doi/10.4171/JST/42
On a class of spectral problems on the half-line and their applications to multi-dimensional problems
Michael
Solomyak
Weizmann Institute of Science, REHOVOT, ISRAEL
Sturm–Liouville operator on $\mathbb R_+$, estimates on the number of bound states
\begin{abstract} A survey of estimates on the number $N_-(\mathbf M_{\alpha G})$ of negative eigenvalues (bound states) of the Sturm--Liouville operator $\mathbf M_{\alpha G}u=-u''-\alpha G$ on the half-line, as depending on the properties of the function $G$ and the value of the coupling parameter $\alpha>0$. The central result is Theorem 5.1, giving a sharp sufficient condition for the semi-classical behavior $N_-(\mathbf M_{\alpha G})=O(\alpha^{1/2})$, and the necessary and sufficient conditions for a ``super-classical'' growth rate $N_-(\mathbf M_{\alpha G})=O(\alpha^q)$ with any given $q>1/2$. Similar results for the problem on the whole $\mathbb R$ are also presented. Applications to the multi-dimensional spectral problems are discussed.
Ordinary differential equations
Partial differential equations
General
215
235
10.4171/JST/43
http://www.ems-ph.org/doi/10.4171/JST/43
3
Difference Sturm–Liouville problems in the imaginary direction
Yury
Neretin
Universität Wien, WIEN, AUSTRIA
Difference operators, Sturm--Liouville problem, Kontorovich--Lebedev transform, Wimp transform, index hypergeometric transform, defect indices, self-a
We consider difference operators in $L^2$ on $\mathbb R$ of the form $$\mathcal L f(s)=p(s)f(s+i)+q(s) f(s)+r(s) f(s-i),$$ where $i$ is the imaginary unit. The domain of definiteness are functions holomorphic in a strip with some conditions of decreasing at infinity. Problems of such type with discrete spectra are well known (Meixner--Pollaczek, continuous Hahn, continuous dual Hahn, and Wilson hypergeometric orthogonal polynomials). We write explicit spectral decompositions for several operators $\mathcal L$ with continuous spectra. We also discuss analogs of 'boundary conditions' for such operators.
Ordinary differential equations
Special functions
Integral transforms, operational calculus
Operator theory
237
269
10.4171/JST/44
http://www.ems-ph.org/doi/10.4171/JST/44
Commutator methods for unitary operators
Claudio
Fernández
Pontificia Universidad Católica de Chile, SANTIAGO DE CHILE, CHILE
Serge
Richard
Université Claude Bernard Lyon 1, VILLEURBANNE CEDEX, FRANCE
Rafael
Tiedra de Aldecoa
Pontificia Universidad Católica de Chile, SANTIAGO DE CHILE, CHILE
Unitary operators, spectral analysis, Mourre theory, limiting absorption principle, cocycles over rotations
We present an improved version of commutatormethods for unitary operators under a weak regularity condition. Once applied to a unitary operator, the method typically leads to the absence of singularly continuous spectrum and to the local finiteness of point spectrum. Large families of locally smooth operators are also exhibited. Half of the paper is dedicated to applications, and a special emphasis is put on the study of cocycles over irrational rotations. It is apparently the first time that commutator methods are applied in the context of rotation algebras, for the study of their generators.
Quantum theory
Operator theory
General
271
292
10.4171/JST/45
http://www.ems-ph.org/doi/10.4171/JST/45
Convergence of Dirichlet eigenvalues for elliptic systems on perturbed domains
Justin
Taylor
Murray State University, MURRAY, UNITED STATES
Eigenvalues, elliptic systems, perturbed domains
We consider the eigenvalues of an elliptic operator \[(Lu)^{\beta}=-\frac{\partial}{\partial x_j} (a^{\alpha \beta}_{ij}\frac{\partial u^{\alpha}}{\partial x_i}),\quad \beta=1,\ldots,m,\] where{\verone} $u=(u^1\ldots,u^m)^t$ is a vector valued function and $a^{\alpha \beta}(x)$ are $(n \times n)$ matrices whose elements $a^{\alpha \beta}_{ij}(x)$ are at least uniformly bounded measurable real-valued functions such that $$a^{\alpha \beta}_{ij}(x)=a^{\beta \alpha}_{ji}(x)$$ for any combination of $\alpha, \beta, i,$ and $j$. We assume we have two non-empty, open, disjoint, and bounded sets, $\Omega$ and $\tilde{\Omega}$, in $\mathbb{R}^n$, and add a set $T_{\varepsilon}$ of small measure to form the domain $\Omega_{\varepsilon}$. Then we show that as $\varepsilon \rightarrow 0^+$, the Dirichlet eigenvalues corresponding to the family{\verone} of domains $\{\Omega_{\varepsilon}\}_{\varepsilon>0}$ converge to the Dirichlet eigenvalues corresponding to $\Omega_0=\Omega \cup \tilde{\Omega}$. 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.
Partial differential equations
Abstract harmonic analysis
General
293
316
10.4171/JST/46
http://www.ems-ph.org/doi/10.4171/JST/46
The spectral function of a first order elliptic system
Olga
Chervova
University College London, LONDON, UNITED KINGDOM
Robert
Downes
University College London, LONDON, UNITED KINGDOM
Dmitri
Vassiliev
University College London, LONDON, UNITED KINGDOM
Spectral theory, asymptotic distribution of eigenvalues
We consider an elliptic self-adjoint first order pseudodifferential operator acting on columns of complex-valued half-densities over a connected compact manifold without boundary. The eigenvalues of the principal symbol are assumed to be simple but no assumptions are made on their sign, so the operator is not necessarily semi-bounded. We study the following objects: the propagator (time-dependent operator which solves the Cauchy problem for the dynamic equation), the spectral function (sum of squares of Euclidean norms of eigenfunctions evaluated at a given point of the manifold, with summation carried out over all eigenvalues between zero and a positive $\lambda$), and the counting function (number of eigenvalues between zero and a positive $\lambda$). We derive explicit two-term asymptotic formulae for all three. For the propagator "asymptotic'' is understood as asymptotic in terms of smoothness, whereas for the spectral and counting functions "asymptotic'' is understood as asymptotic with respect to $\lambda\to+\infty$.
Partial differential equations
General
317
360
10.4171/JST/47
http://www.ems-ph.org/doi/10.4171/JST/47
Spectral theory of higher order differential operators by examples
Horst
Behncke
Universität Osnabrück, OSNABRÜCK, GERMANY
Don
Hinton
Universität Osnabrück, OSNABRÜCK, GERMANY
Spectral theory, asymptotic solutions, absolutely continuous spectrum, singular spectrum, Titchmarsh–Weyl $m$-functions
Asymptotic integration has turned out to be a powerful method to determine the deficiency indices and spectra of higher order differential operators. Since the general method is by now well established we shall only outline this method and illustrate typical results and properties via examples. In addition to the calculation of deficiency indices, the location and multiplicity of the absolutely continuous spectrumwill be found as well as showing the absence of singular continuous spectrum. Finite singular pointswill also be considered. For unbounded coefficients new results arise from competing terms of the operators.
Ordinary differential equations
General
361
398
10.4171/JST/48
http://www.ems-ph.org/doi/10.4171/JST/48
Semiclassical estimates of the cut-off resolvent for trapping perturbations
Jean-François
Bony
Université Bordeaux I, TALENCE CEDEX, FRANCE
Vesselin
Petkov
Université Bordeaux I, TALENCE CEDEX, FRANCE
Resolvent estimate, quantum resonances, semiclassical analysis, resonant states
This paper is devoted to the study of the cut-off resolvent of a semiclassical "black box'' operator $P$. We estimate the norm of $\varphi ( P - z )^{- 1} \varphi$, for any $\varphi \in C^{\infty}_{0} ( \mathbb R^{n} )$, by the norm of ${\mathds{1}}_{\mathcal C_{a,b}} ( P - z )^{- 1} {\mathds{1}}_{\mathcal C_{a,b}}$ where $\mathcal C_{a,b} = \{ x \in \mathbb R^{n} ; \ a < \vert x \vert < b \}$ and $a \gg 1$. For $z$ in the unphysical sheet with $- M h \vert \ln h \vert \leq \operatorname{Im} z \leq 0$, we prove that this estimate holds with a constant $\frac{h}{\vert \operatorname{Im} z \vert} e^{C \vert \operatorname{Im} z \vert / h}$. We also study the resonant states $u$ of the operator $P$ and we obtain bounds for $\Vert \varphi u \Vert$ by $\Vert \mathds{1}_{\mathcal C_{a,b}} u \Vert$. These results hold without any assumption on the trapped set nor any assumption on the multiplicity of the resonances.
Linear and multilinear algebra; matrix theory
Partial differential equations
Operator theory
Quantum theory
399
422
10.4171/JST/49
http://www.ems-ph.org/doi/10.4171/JST/49
Semiclassical analysis with vanishing magnetic fields
Nicolas
Dombrowski
Université de Nice, NICE CEDEX 02, FRANCE
Nicolas
Raymond
Université de Rennes 1, RENNES CEDEX, FRANCE
Semi-classical limit, microlocal analysis, quasimode, Agmon estimates, Born–Oppenheimer approximation
We analyze the 2D magnetic Laplacian in the semiclassical limit in the case when the magnetic field vanishes along a smooth curve. In particular, we prove local and micro-local estimates for the eigenfunctions and a complete asymptotic expansion of the eigenpairs in powers of $h^{1/6}$.
Partial differential equations
Quantum theory
General
423
464
10.4171/JST/50
http://www.ems-ph.org/doi/10.4171/JST/50
4
Spectral estimates for Dirichlet Laplacians and Schrödinger operators on geometrically nontrivial cusps
Pavel
Exner
Mathematical Physics and Applied Mathematics, PRAGUE, CZECH REPUBLIC
Diana
Barseghyan
Mathematical Physics and Applied Mathematics, PRAGUE, CZECH REPUBLIC
Dirichlet Laplacian, cusp-shaped region, Lieb–Thirring inequalities, bending and twisting
The goal of this paper is to derive estimates of eigenvalue moments for Dirichlet Laplacians and Schrödinger operators in regions having infinite cusps which are geometrically nontrivial being either curved or twisted; we are going to show how those geometric properties enter the eigenvalue bounds. The obtained inequalities reflect the essentially one-dimensional character of the cusps and we give an example showing that in an intermediate energy region they can be much stronger than the usual semiclassical bounds.
Partial differential equations
Operator theory
Quantum theory
465
484
10.4171/JST/51
http://www.ems-ph.org/doi/10.4171/JST/51
On nonlinear wave equations with parabolic potentials
Alexander
Komech
Universität Wien, WIEN, AUSTRIA
Elena
Kopylova
Universität Wien, WIEN, AUSTRIA
Sergey
Kopylov
Russian State University of Tourism and Service, CHERKIZOVO, RUSSIAN FEDERATION
Relativistic invariant nonlinearwave equations, soliton, kink, spectrum, resonance, eigenvalue, Fermi golden rule
We introduce a new class of piece-wise quadratic potentials for nonlinear wave equations with a kink solutions. The potentials allow an exact description of the spectral properties for the linearized equation at the kink. This description is necessary for the study of the stability properties of the kinks. In particular, we construct examples of the potentials of Ginzburg–Landau type providing the asymptotic stability of the kinks [6] and [7].
Partial differential equations
Dynamical systems and ergodic theory
485
503
10.4171/JST/52
http://www.ems-ph.org/doi/10.4171/JST/52
On spectral estimates for two-dimensional Schrödinger operators
Ari
Laptev
Imperial College London, LONDON, UNITED KINGDOM
Michael
Solomyak
Weizmann Institute of Science, REHOVOT, ISRAEL
Schrödinger operator on $\mathbb R^2$, bound states, spectral estimates
For the two-dimensional Schrödinger operator $\mathrm H_{\alpha V}=-\Delta-\alpha V,\ V\ge 0$, we study the behavior of the number $N_-(\mathrm H_{\alpha V})$ of its negative eigenvalues (bound states), as the coupling parameter $\alpha$ tends to infinity. A wide class of potentials is described, for which $N_-(\mathrm H_{\alpha V})$ has the semi-classical behavior, i.e. $N_-(\mathrm H_{\alpha V})=O(\alpha)$. For the potentials from this class, the necessary and sufficient condition is found for the validity of the Weyl asymptotic law.
Partial differential equations
505
515
10.4171/JST/53
http://www.ems-ph.org/doi/10.4171/JST/53
The spectral density of the scattering matrix of the magnetic Schrödinger operator for high energies
Daniel
Bulger
King's College London, London, UNITED KINGDOM
Alexander
Pushnitski
King's College London, LONDON, UNITED KINGDOM
Scattering matrix, scattering phase, Schrödinger operator, magnetic field, spectral density
The scattering matrix of the Schrödinger operator with smooth short-range electric and magnetic potentials is considered. The asymptotic density of the eigenvalues of this scattering matrix in the high energy regime is determined. An explicit formula for this density is given. This formula involves only the magnetic vector-potential.
Quantum theory
Partial differential equations
517
534
10.4171/JST/54
http://www.ems-ph.org/doi/10.4171/JST/54
On the limit behaviour of second order relative spectra of self-adjoint operators
Eugene
Shargorodsky
King's College London, London, UNITED KINGDOM
Self-adjoint operators, second order relative spectra, projection methods
It is well known that the standard projectionmethods allow one to recover the whole spectrum of a bounded self-adjoint operator but they often lead to spectral pollution, i.e. to spurious eigenvalues lying in the gaps of the essential spectrum. Methods using second order relative spectra are free from spectral pollution, but they have not been proven to approximate the whole spectrum. L. Boulton ([3] and [4]) has shown that second order relative spectra approximate all isolated eigenvalues of finite multiplicity. The main result of the present paper is that second order relative spectra do not in general approximate the whole of the essential spectrum of a bounded self-adjoint operator.
Operator theory
Numerical analysis
535
552
10.4171/JST/55
http://www.ems-ph.org/doi/10.4171/JST/55
Spectral asymptotics for magnetic Schrödinger operators in domains with corners
Ayman
Kachmar
Lebanese University, HADATH, LEBANON
Abdallah
Khochman
Lebanese University, HADATH, LEBANON
Spectral Theory, Schrödinger Operator, semiclassical analysis, piecewise smooth domains
This paper is on magnetic Schrodinger operators in two dimensional domains with corners. Semiclassical formulas are obtained for the sum and number of eigenvalues. The obtained results extend former formulas for smooth domains in [11] and [10] to piecewise smooth domains.
Quantum theory
Partial differential equations
553
574
10.4171/JST/56
http://www.ems-ph.org/doi/10.4171/JST/56
Uniform stability of the Dirichlet spectrum for rough outer perturbations
Bruno
Colbois
Université de Neuchâtel, NEUCHÂTEL, SWITZERLAND
Alexandre
Girouard
Université Laval, QUEBEC (QUEBEC), CANADA
Mette
Iversen
University of Bristol, BRISTOL, UNITED KINGDOM
Spectral stability, Dirichlet spectrum
The goal of this paper is to study the Dirichlet eigenvalues of bounded domains $\Omega\subset\Omega'$. With a local spectral stability requirement on $\Omega$, we show that the difference of the Dirichlet eigenvalues of $\Omega'$ and $\Omega$ is explicitly controlled from above in terms of the first eigenvalue of $\Omega'\setminus\bar{\Omega}$ and of geometric constants depending on the inner domain $\Omega$. In particular, $\Omega'$ can be an arbitrary bounded domain.
Partial differential equations
Global analysis, analysis on manifolds
575
599
10.4171/JST/57
http://www.ems-ph.org/doi/10.4171/JST/57
A multichannel scheme in smooth scattering theory
Alexander
Pushnitski
King's College London, LONDON, UNITED KINGDOM
Dmitri
Yafaev
Université de Rennes I, RENNES CEDEX, FRANCE
Multichannel problem, Fredholm resolvent equations, smoothness, absolutely continuous and singular spectra, wave operators, scattering matrix
In this paper we develop the scattering theory for a pair of self-adjoint operators \mbox{$A_{0}=A_{1}\oplus\dots \oplus A_{N}$} and $A=A_{1}+\dots +A_{N}$ under the assumption that all pair products $A_{j}A_{k}$ with $j\neq k$ satisfy certain regularity conditions. Roughly speaking, these conditions mean that the products $A_{j}A_{k}$, $j\neq k$, can be represented as integral operators with smooth kernels in the spectral representation of the operator $A_{0}$. We show that the absolutely continuous parts of the operators $A_{0}$ and $A$ are unitarily equivalent. This yields a smooth version of Ismagilov's theorem known earlier in the trace class framework. We also prove that the singular continuous spectrum of the operator $A$ is empty and that its eigenvalues may accumulate only to "thresholds'' of the absolutely continuous spectra of the operators $A_{j}$. Our approach relies on a system of resolvent equations which can be considered as a generalization of Faddeev's equations for three particle quantum systems.
Operator theory
601
634
10.4171/JST/58
http://www.ems-ph.org/doi/10.4171/JST/58