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European Mathematical Society Publishing House
2024-03-28 09:49:04
4
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JST&vol=2&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
2
2012
1
Resolvent conditions for the control of unitary groups and their approximations
Luc
Miller
Université Paris-Ouest La Défense, NANTERRE, FRANCE
Observability of PDEs, infinite dimensional Hautus test, Galerkin approximation of control problems.
A self-adjoint operator $\mathcal A$ and an operator $\mathcal C$ bounded from the domain $\mathcal D(\mathcal A)$ with the graph norm to another Hilbert space are considered. The admissibility or the exact observability in finite time of the unitary group generated by $i\mathcal A$ with respect to the observation operator $\mathcal C$ are characterized by some spectral inequalities on $\mathcal A$ and $\mathcal C$. E.g. both properties hold if and only if $x\mapsto\ \|(\mathcal A-\lambda)x \|+\|\mathcal C x \|$ is a norm on $\mathcal D(\mathcal A)$ equivalent to $x\mapsto\|(\mathcal A -\lambda)x\|+\|x\|$ uniformly with respect to $\lambda\in\mathbb R$. This paper generalizes and simplifies some results on the control of unitary groups obtained using these so-called resolvent conditions, also known as Hautus tests. It proves new theorems on the equivalence (with respect to admissibility and observability) between first and second order equations, between groups generated by $i\mathcal A$ and $if(\mathcal A)$ for positive $\mathcal A$ and convex $f$, and between a group and its Galerkin approximations. E.g. they apply to the control of linear Schrödinger, wave and plates equations and to the uniform control of their finite element semi-discretization.
Systems theory; control
Operator theory
General
1
55
10.4171/JST/20
http://www.ems-ph.org/doi/10.4171/JST/20
Lieb–Thirring inequalities on some manifolds
Alexei
Ilyin
Keldysh Institute of Applied Mathematics, Moscow, RUSSIAN FEDERATION
Lieb–Thirring inequalities, Schrödinger operators
We prove Lieb–Thirring inequalities with improved constants on the two-dimensional $\mathbb S^2$ sphere and the two-dimensional torus $\mathbb T^2$. In the one-dimensional periodic case we obtain a simultaneous bound for the negative trace and the number of negative eigenvalues.
Partial differential equations
Real functions
General
57
78
10.4171/JST/21
http://www.ems-ph.org/doi/10.4171/JST/21
Spectral inequalities for operators on H-type groups
James
Inglis
Imperial College London, London, UNITED KINGDOM
H-type groups, super-Poincaré inequality, spectral gap inequality, spectrum, logarithmic Sobolev inequality
In this paper the spectra of certain Schrödinger-type operators defined in the sub-elliptic setting of H-type groups are investigated. Two approaches are taken: the first one makes use of a convenient unitary transformation on the Heisenberg group, while the second proceeds via functional inequalities. The paper concludes by highlighting a major difference between the spectrum of the considered operators depending on which natural sub-Riemannian metric is chosen.
Global analysis, analysis on manifolds
Topological groups, Lie groups
General
79
105
10.4171/JST/22
http://www.ems-ph.org/doi/10.4171/JST/22
Almost all eigenfunctions of a rational polygon are uniformly distributed
Jens
Marklof
University of Bristol, Bristol, UNITED KINGDOM
Zeév
Rudnick
Tel Aviv University, Tel Aviv, ISRAEL
Billiards in rational polygons, quantum ergodicity, pseudo-integrable systems
We consider an orthonormal basis of eigenfunctions of the Dirichlet Laplacian for a rational polygon. The modulus squared of the eigenfunctions defines a sequence of probability measures. We prove that this sequence contains a density-one subsequence that converges to Lebesgue measure.
Partial differential equations
Global analysis, analysis on manifolds
Quantum theory
General
107
113
10.4171/JST/23
http://www.ems-ph.org/doi/10.4171/JST/23