- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 14:13:59
4
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JST&vol=1&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
1
2011
4
Eigenvalue bounds for two-dimensional magnetic Schrödinger operators
Hynek
Kovařík
Università degli Studi di Brescia, BRESCIA, ITALY
Eigenvalue estimates, magnetic Schrödinger operator, Hardy inequalities
We prove that the number of negative eigenvalues of two-dimensional magnetic Schrödinger operators is bounded from above by the strength of the corresponding electric potential. Such estimates fail in the absence of a magnetic field. We also show how the corresponding upper bounds depend on the properties of the magnetic field and discuss their connection with Hardy-type inequalities.
Partial differential equations
General
363
387
10.4171/JST/16
http://www.ems-ph.org/doi/10.4171/JST/16
How opening a hole affects the sound of a flute
Romain
Joly
Université de Grenoble I, SAINT-MARTIN D'HERES, FRANCE
Thin domains, convergence of operators, resonance, mathematics for music and acoustic
In this paper, we consider an open tube of diameter $\varepsilon>0$, on the side of which a small hole of size $\varepsilon^2$ is pierced. The resonances of this tube correspond to the eigenvalues of the Laplacian operator with homogeneous Neumann condition on the inner surface of the tube and Dirichlet one on the open parts of the tube. We show that this spectrum converges when $\varepsilon$ goes to $0$ to the spectrum of an explicit one-dimensional operator. At a first order of approximation, the limit spectrum describes the note produced by a flute, for which one of its holes is open.
Partial differential equations
General
389
408
10.4171/JST/17
http://www.ems-ph.org/doi/10.4171/JST/17
Ballistic behavior for random Schrödinger operators on the Bethe strip
Abel
Klein
University of California, Irvine, IRVINE, UNITED STATES
Christian
Sadel
Pontificia Universidad Católica de Chile, SANTIAGO DE CHILE, CHILE
Random Schrödinger operators, Anderson model, spreading of wave packets, ballistic behavior, Bethe strip
The Bethe strip of width $m$ is the Cartesian product $\mathbb B\times\{1,\ldots,m\}$, where $\mathbb B$ is the Bethe lattice (Cayley tree). We consider Anderson-like Hamiltonians $\;H_\lambda=\frac12 \Delta \otimes 1 + {1 \otimes A}+\lambda \mathcal{V}$ on a Bethe strip with connectivity $K \ge 2$, where $A$ is an $m\times m$ symmetric matrix, $\mathcal{V}$ is a random matrix potential, and $\lambda$ is the disorder parameter. Under certain conditions on $A$ and $K$, for which we previously proved the existence of absolutely continuous spectrum for small $\lambda$, we now obtain ballistic behavior for the spreading of wave packets evolving under $H_\lambda$ for small $\lambda$.
Statistical mechanics, structure of matter
Operator theory
Probability theory and stochastic processes
General
409
442
10.4171/JST/18
http://www.ems-ph.org/doi/10.4171/JST/18
A family of anisotropic integral operators and behavior of its maximal eigenvalue
Boris
Mityagin
Ohio State University, COLUMBUS, UNITED STATES
Alexander
Sobolev
University College London, LONDON, UNITED KINGDOM
Eigenvalues, asymptotics, positivity improving integral operators, pseudo-differential operators, superconductivity
We study the family of compact integral operators ${\bf K}_\beta$ in $L2(\mathbb R)$ with the kernel \begin{equation*} K_\beta(x, y) = \frac{1}{\pi}\frac{1}{1 + (x-y)^2 + \beta^2\Theta(x, y)}, \end{equation*} depending on the parameter $\beta >0$, where $\Theta(x, y)$ is a symmetric non-\hspace{0pt}negative homogeneous function of degree $\gamma\ge 1$. The main result is the following asymptotic formula for the maximal eigenvalue $M_\beta$ of $\bf K_\beta$: \begin{equation*} M_\beta = 1 - \lambda_1 \beta^{\frac{2}{\gamma+1}} + o(\beta^{\frac{2}{\gamma+1}}),\quad \beta\to 0, \end{equation*} where $\lambda_1$ is the lowest eigenvalue of the operator $\bf A = |d/dx| + \Theta(x, x)/2$. A central role in the proof is played by the fact that $\bf K_\beta, \beta>0,$ is positivity improving. The case $\Theta(x, y) = (x^2 + y^2)^2$ has been studied earlier in the literature as a simplified model of high-temperature superconductivity.
Integral equations
Operator theory
General
443
460
10.4171/JST/19
http://www.ems-ph.org/doi/10.4171/JST/19