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European Mathematical Society Publishing House
2024-03-28 22:41:14
7
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JST&vol=3&iss=3&update_since=2024-03-28
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
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
3
2013
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