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European Mathematical Society Publishing House
2024-03-28 15:30:47
4
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JST&vol=1&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
1
2011
1
Trace formulae for perturbations of class $\boldsymbol{{\boldsymbol S}_m}$
Alexei
Aleksandrov
Steklov Institute of Mathematics, St-Petersburg, RUSSIAN FEDERATION
Vladimir
Peller
Michigan State University, EAST LANSING, UNITED STATES
Trace formulae, Schatten–von Neumann classes, multiple operator integrals, Besov spaces, spectral shift
We offer a new approach to trace formulae for functions of perturbed self-adjoint operators. We establish most general trace formulae in the case of perturbation of Schatten–von Neumann class ${\boldsymbol S}_m$, where $m$ is a positive integer. We consider several special cases of our general trace formulae. In particular, as a special case we obtain the trace formula for the $m$th operator Taylor polynomial. In the case $m=1$ this corresponds to the Livshits–Krein trace formula, while in the case $m=2$ this corresponds to the Koplienko trace formula. In the case of an arbitrary positive integer $m$, the trace formula for the $m$th operator Taylor polynomial was obtained recently in [25]. Our results allow us to essentially enlarge the class of functions, for which the trace formula obtained in [25] holds. Namely, we prove that it holds for functions in the Besov space $B_{\infty1}^m(\Bbb R)$.
Operator theory
Real functions
Fourier analysis
Functional analysis
1
26
10.4171/JST/1
http://www.ems-ph.org/doi/10.4171/JST/1
Approximate Quantum and Acoustic Cloaking
Allan
Greenleaf
University of Rochester, ROCHESTER, UNITED STATES
Yaroslav
Kurylev
University College London, LONDON, UNITED KINGDOM
Matti
Lassas
University of Helsinki, Helsinki, FINLAND
Gunther
Uhlmann
University of Washington, SEATTLE, UNITED STATES
Isotropic transformation optics, quantum cloaking, acoustic cloaking
For any $E\ge 0$, we construct a sequence of bounded potentials $V^E_{n},\, n\in\Bbb N$, supported in {an annular region $B_{out}\setminus B_{inn}\subset\Bbb R ^3$,} which act as approximate cloaks for solutions of Schrödinger's equation at energy $E$: For any potential $V_0\in L^\infty(B_{inn})$ {such that $E$ is not a Neumann eigenvalue of $-\Delta+V_0$ in $B_{inn}$}, the scattering amplitudes $a_{V_0+V_n^E}(E,\theta,\omega)\to 0$ as $n\to\infty$. The $V^E_{ n}$ thus not only form a family of approximately transparent potentials, but also function as approximate invisibility cloaks in quantum mechanics. {On the other hand, for $E$ close to interior eigenvalues, resonances develop and there exist almost trapped states concentrated in $B_{inn}$.} We derive the $V_n^E$ from singular, anisotropic transformation optics-based cloaks by a de-anisotropization procedure, which we call \emph{isotropic transformation optics}. This technique uses truncation, inverse homogenization and spectral theory to produce nonsingular, isotropic approximate cloaks. As an intermediate step, we also obtain approximate cloaking for a general class of equations including the acoustic equation.
Partial differential equations
Optics, electromagnetic theory
Quantum theory
General
27
80
10.4171/JST/2
http://www.ems-ph.org/doi/10.4171/JST/2
On the Removal of Finite Discrete Spectrum by Coefficient Stripping
Barry
Simon
California Institute of Technology, PASADENA, UNITED STATES
Coefficient stripping, block Jacobi matrices
We prove for a large class of operators, $J$, including block Jacobi matrices, if $\sigma(J) \setminus [\alpha,\beta]$ is a finite set, each point of which is an eigenvalue of finite multiplicity, then a finite coefficient stripped, $J_N$, has $\sigma(J_N)\subset [\alpha,\beta]$. We use an abstract Dirichlet decoupling.
Operator theory
Ordinary differential equations
General
81
85
10.4171/JST/3
http://www.ems-ph.org/doi/10.4171/JST/3
Geometrical Versions of improved Berezin–Li–Yau Inequalities
Leander
Geisinger
Universität Stuttgart, STUTTGART, GERMANY
Ari
Laptev
Imperial College London, LONDON, UNITED KINGDOM
Timo
Weidl
Universität Stuttgart, STUTTGART, GERMANY
Dirichlet Laplace operator, semiclassical estimates, Berezin–Li–Yau inequality
We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in $\mathbb{R}^d$, $d \geq 2$. In particular, we derive upper bounds on Riesz means of order $\sigma \geq 3/2$, that improve the sharp Berezin inequality by a negative second term. This remainder term depends on geometric properties of the boundary of the set and reflects the correct order of growth in the semi-classical limit. Under certain geometric conditions these results imply new lower bounds on individual eigenvalues, which improve the Li–Yau inequality.
Partial differential equations
Operator theory
General
87
109
10.4171/JST/4
http://www.ems-ph.org/doi/10.4171/JST/4