- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 16:54:28
8
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JST&vol=4&iss=1&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
4
2014
1
Cwikel's theorem and the CLR inequality
Rupert
Frank
Caltech, PASADENA, UNITED STATES
Cwikel inequality, Eigenvalue inequalities, trace ideals, Schrödinger operators
We give a short proof of the CLR bound on the number of negative eigenvalues of Schrödinger operators. The argument, which is based on work of Rumin, leads to remarkably good constants and applies to the case of operator-valued potentials as well. Moreover, we obtain the general form of Cwikel's estimate about the singular values of operators of the form $f(X) g(-i\nabla)$.
Operator theory
Functional analysis
Quantum theory
1
21
10.4171/JST/59
http://www.ems-ph.org/doi/10.4171/JST/59
Spectral sets of periodic matrices related to the strong moment problem
Ionela
Moale
Johannes Kepler University Linz, Linz, AUSTRIA
Peter
Yuditskii
Johannes Kepler University Linz, LINZ, AUSTRIA
Strong Moment Problem, periodic CMV matrices, Hardy spaces on Riemann surfaces, conformal mappings, comb domains, reproducing kernels
The main result of this work is a parametric description of the spectral surfaces of a class of periodic 5-diagonal matrices, related to the strong moment problem. This class is a self-adjoint twin of the class of CMV matrices. Jointly they form the simplest possible classes of 5-diagonal matrices.
Functions of a complex variable
Functional analysis
Operator theory
23
52
10.4171/JST/60
http://www.ems-ph.org/doi/10.4171/JST/60
A priori bounds and existence of non-real eigenvalues of indefinite Sturm–Liouville problems
Jiangang
Qi
Shandong University (Weihai), WEIHAI, SHANDONG PROVINCE, CHINA
Shaozhu
Chen
Shandong University (Weihai), WEIHAI, SHANDONG PROVINCE, CHINA
A priori bound, non-real eigenvalue, indefinite Sturm-Liouville problem, PT-symmetry
The present paper gives a priori bounds on the moduli of the possible non-real eigenvalues to regular indefinite Sturm-Liouville problems and obtains sufficient conditions for such problems to admit non-real eigenvalues.
Ordinary differential equations
Operator theory
53
63
10.4171/JST/61
http://www.ems-ph.org/doi/10.4171/JST/61
Quantum ergodicity for a class of mixed systems
Jeffrey
Galkowski
University of California Berkeley, BERKELEY, CA, UNITED STATES
quantum ergodicity, mixed dynamics, semiclassical
We examine high energy eigenfunctions for the Dirichlet Laplacian on domains where the billiard flow exhibits mixed dynamical behavior. (More generally, we consider semiclassical Schrödinger operators with mixed assumptions on the Hamiltonian flow.) Specifically, we assume that the billiard flow has an invariant ergodic component, $U$, and study defect measures, $\mu$, of positive density subsequences of eigenfunctions (and, more generally, of almost orthogonal quasimodes). We show that any defect measure associated to such a subsequence satisfies $\mu|_{U}=c\mu_L|_{U}$, where $\mu_L$ is the Liouville measure. This proves part of a conjecture of Percival [18].
Global analysis, analysis on manifolds
Dynamical systems and ergodic theory
Quantum theory
65
85
10.4171/JST/62
http://www.ems-ph.org/doi/10.4171/JST/62
Spectral properties of bipolar surfaces to Otsuki tori
Mikhail
Karpukhin
Moscow State University, MOSCOW, RUSSIAN FEDERATION
Otsuki tori, extremal metric, bipolar surface
The $i$-th eigenvalue $\lambda_i$ of the Laplace–Beltrami operator on a surface can be considered as a functional on the space of all Riemannian metrics of unit volume on this surface. Surprisingly only few examples of extremal metrics for these functionals are known. In the present paper a new countable family of extremal metrics on the torus is provided.
Global analysis, analysis on manifolds
87
111
10.4171/JST/63
http://www.ems-ph.org/doi/10.4171/JST/63
The Galerkin method for perturbed self-adjoint operators and applications
Michael
Strauss
Flat 0/2, GLASGOW, UNITED KINGDOM
Eigenvalue problem, spectral pollution, Galerkin method, finite-section method
We consider the Galerkin method for approximating the spectrum of an operator $T+A$ where $T$ is semi-bounded self-adjoint and $A$ satisfies a relative compactness condition. We show that the method is reliable in all regions where it is reliable for the unperturbed problem - which always contains $\mathbb{C}\backslash\mathbb{R}$. The results lead to a new technique for identifying eigenvalues of $T$, and for identifying spectral pollution which arises from applying the Galerkin method directly to $T$. The new technique benefits from being applicable on the form domain.
Operator theory
113
151
10.4171/JST/64
http://www.ems-ph.org/doi/10.4171/JST/64
Best constants in Lieb-Thirring inequalities: a numerical investigation
Antoine
Levitt
Université Paris-Dauphine, PARIS, FRANCE
Lieb-Thirring inequalities, finite elements
We investigate numerically the optimal constants in Lieb–Thirring inequalities by studying the associated maximization problem. Using a monotonic fixed-point algorithm and a finite element discretization, we obtain radial trial potentials which provide lower bounds on the optimal constants. These results confirm existing conjectures, and provide insight into the behavior of the maximizers. Based on our numerical results, we formulate a complete conjecture about the best constants for all possible values of the parameters.
Partial differential equations
Quantum theory
153
175
10.4171/JST/65
http://www.ems-ph.org/doi/10.4171/JST/65
Scattering theory of the $p$-form Laplacian on manifolds with generalized cusps
Eugenie
Hunsicker
Loughborough University, LOUGHBOROUGH, UNITED KINGDOM
Nikolaos
Roidos
Universität Hannover, HANNOVER, GERMANY
Alexander
Strohmaier
Loughborough University, LOUGHBOROUGH, UNITED KINGDOM
Scattering matrix, spectral resolution, generalized cusps, Laplacian, resolvent
In this paper we consider scattering theory on manifolds with special cusp-like metric singularities of warped product type $g=dx^2 + x^{-2 a}h$, where $a>0$. These metrics form a natural subset in the class of metrics with warped product singularities and they can be thought of as interpolating between hyperbolic and cylindrical metrics. We prove that the resolvent of the Laplace operator acting on $p$-forms on such a manifold extends to a meromorphic function defined on the logarithmic cover of the complex plane with values in the bounded operators between weighted $L^2$-spaces. This allows for a construction of generalized eigenforms for the Laplace operator as well as for a meromorphic continuation of the scattering matrix. We give a precise description of the asymptotic expansion of generalized eigenforms on the cusp and find that the scattering matrix satisfies a functional equation.
Global analysis, analysis on manifolds
Special functions
Partial differential equations
177
209
10.4171/JST/66
http://www.ems-ph.org/doi/10.4171/JST/66