- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 07:12:58
7
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JST&vol=4&iss=4&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
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
4
2014
4
Resolvents, Poisson operators and scattering matrices on asymptotically hyperbolic and de Sitter spaces
András
Vasy
Stanford University, STANFORD, UNITED STATES
Asymptotically hyperbolic spaces, asymptotically de Sitter spaces, asymptotically Minkowski spaces, scattering matrices, resolvents, backward and forward solution operator
We describe how the global operator induced on the boundary of an asymptotically Minkowski space links two even asymptotically hyperbolic spaces and an even asymptotically de Sitter space, and compute the scattering operator of the linked problem in terms of the scattering operator of the constituent pieces.
Global analysis, analysis on manifolds
Partial differential equations
643
673
10.4171/JST/82
http://www.ems-ph.org/doi/10.4171/JST/82
A graph discretization of the Laplace–Beltrami operator
Dmitri
Burago
The Pennsylvania State University, UNIVERSITY PARK, UNITED STATES
Sergei
Ivanov
Russian Academy of Sciences, ST. PETERSBURG, RUSSIAN FEDERATION
Yaroslav
Kurylev
University College London, LONDON, UNITED KINGDOM
Laplace, graph, discretization, Riemannian
We show that eigenvalues and eigenfunctions of the Laplace–Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.
Global analysis, analysis on manifolds
Combinatorics
Differential geometry
Numerical analysis
675
714
10.4171/JST/83
http://www.ems-ph.org/doi/10.4171/JST/83
Supersymmetry and Schrödinger-type operators with distributional matrix-valued potentials
Jonathan
Eckhardt
Universität Wien, WIEN, AUSTRIA
Fritz
Gesztesy
Baylor University, WACO, UNITED STATES
Roger
Nichols
The University of Tennessee at Chattanooga, CHATTANOOGA, UNITED STATES
Gerald
Teschl
Universität Wien, WIEN, AUSTRIA
Sturm–Liouville operators, distributional coefficients, Weyl–Titchmarsh theory, supersymmetry
Building on work on Miura's transformation by Kappeler, Perry, Shubin, and Topalov, we develop a detailed spectral theoretic treatment of Schrödinger operators with matrix-valued potentials, with special emphasis on distributional potential coefficients. Our principal method relies on a supersymmetric (factorization) formalism underlying Miura's transformation, which intimately connects the triple of operators $(D, H_1, H_2)$ of the form \[ D= \left(\begin{smallmatrix} 0 & A^* \\ A & 0 \end{smallmatrix}\right) \, \text{ in } \, L^2(\mathbb{R})^{2m} \, \text{ and } \, H_1 = A^* A, \;\; H_2 = A A^* \, \text{ in } L^2(\mathbb{R})^m. \] Here $A= I_m (d/dx) + \phi$ in $L^2(\mathbb{R})^m$, with a matrix-valued coefficient $\phi = \phi^* \in L^1_{\text{loc}}(\mathbb{R})^{m \times m}$, $m \in \mathbb{N}$, thus explicitly permitting distributional potential coefficients $V_j$ in $H_j$, $j=1,2$, where \[ H_j = - I_m \frac{d^2}{dx^2} + V_j(x), \quad V_j(x) = \phi(x)^2 + (-1)^{j} \phi'(x), \; j=1,2. \] Upon developing Weyl–Titchmarsh theory for these generalized Schrödinger operators $H_j$, with (possibly, distributional) matrix-valued potentials $V_j$, we provide some spectral theoretic applications, including a derivation of the corresponding spectral representations for $H_j$, $j=1,2$. Finally, we derive a local Borg–Marchenko uniqueness theorem for $H_j$, $j=1,2$, by employing the underlying supersymmetric structure and reducing it to the known local Borg–Marchenko uniqueness theorem for $D$.
Operator theory
Ordinary differential equations
715
768
10.4171/JST/84
http://www.ems-ph.org/doi/10.4171/JST/84
Point spectrum for quasi-periodic long range operators
Jiangong
You
Nanjing University, NANJING, CHINA
Shiwen
Zhang
Nanjing University, NANJING, CHINA
Qi
Zhou
Université Pierre et Marie Curie, PARIS CEDEX 05, FRANCE
Point spectrum, quasi-periodic, long-range operator, reducibility
We generalize Gordon type argument to quasi-periodic operators with finite range interaction and prove that these operators have no point spectrum when the rational approximation rate of the base frequency is relatively large. We also show that, for any irrational frequency, there are operators with in finite range interaction possessing point spectrum. is is a new phenomenon which can not happen in the finite range interaction case.
Quantum theory
Dynamical systems and ergodic theory
769
781
10.4171/JST/85
http://www.ems-ph.org/doi/10.4171/JST/85
Eigenvalue estimates for the resolvent of a non-normal matrix
Oleg
Szehr
TU München, GARCHING, GERMANY
Resolvent, non-normal matrix, Markov chain
We investigate the relation between the spectrum of a non-normal matrix and the norm of its resolvent. We provide spectral estimates for the resolvent of matrices whose largest singular value is bounded by 1 (so-called Hilbert space contractions) and for power-bounded matrices. In the rst case our estimate is optimal and we present explicit matrices that achieve equality in the bound. is result recovers and generalizes previous estimates obtained by E.B. Davies and B. Simon in the study of orthogonal polynomials on the unit circle. In case of power-bounded matrices we achieve the strongest estimate so far. Our result uni es previous approaches, where the resolvent was estimated in certain restricted regions of the complex plane. To achieve our estimates we relate the problem of bounding the norm of a function of a matrix to a Nevanlinna–Pick interpolation problem in a corresponding function space. In case of Hilbert space contractions this problem is connected to the theory of compressed shift operators to which we contribute by providing explicit matrix representations for such operators. Finally, we apply our results to study the sensitivity of the stationary states of a classical or quantumMarkov chainwith respect to perturbations of the transition matrix.
Linear and multilinear algebra; matrix theory
Numerical analysis
783
813
10.4171/JST/86
http://www.ems-ph.org/doi/10.4171/JST/86
Square-summable variation and absolutely continuous spectrum
Milivoje
Lukic
Rice University, HOUSTON, UNITED STATES
OPUC, CMV matrix, bounded variation, absolutely continuous spectrum
Recent results of Denisov [5] and Kaluzhny–Shamis [9] describe the absolutely continuous spectrum of Jacobi matrices with coe fficients that obey an $\mathcal l^2$ bounded variation condition with step $p$ and are asymptotically periodic. We extend these results to orthogonal polynomials on the unit circle. We also replace the asymptotic periodicity condition by the weaker condition of convergence to an isospectral torus and, for $p=1$ and $p=2$, we remove even that condition.
Operator theory
Difference and functional equations
Fourier analysis
815
840
10.4171/JST/87
http://www.ems-ph.org/doi/10.4171/JST/87
Uniqueness and stability of Lamé parameters in elastography
Ru-Yu
Lai
University of Washington, SEATTLE, UNITED STATES
Inverse Problems, elasticity system, Lamé parameters
Th is paper concerns an hybrid inverse problem involving elastic measurements called Transient Elastography (TE) which enables detection and characterization of tissue abnormalities. In this paper we assume that the displacements are modeled by linear isotropic elasticity system and the tissue displacement has been obtained by the rst step in hybrid methods. We reconstruct Lamé parameters of this system from knowledge of the tissue displacement. More precisely, we show that for a su ciently large number of solutions of the elasticity system and for an open set of the well-chosen boundary conditions, Lamé parameters can be uniquely and stably reconstructed. e set of well-chosen boundary conditions is characterized in terms of appropriate complex geometrical optics solutions.
Partial differential equations
841
877
10.4171/JST/88
http://www.ems-ph.org/doi/10.4171/JST/88