- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 08:40:54
8
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JST&vol=6&iss=1&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
6
2016
1
Universal measurability and the Hochschild class of the Chern character
Alan
Carey
The Australian National University, CANBERRA, AUSTRALIA
Adam
Rennie
University of Wollongong, WOLLONGONG, AUSTRALIA
Fedor
Sukochev
University of New South Wales, SYDNEY, AUSTRALIA
Dmitriy
Zanin
University of New South Wales, SYDNEY, AUSTRALIA
Singular trace, operator ideal, measurability, Chern character, spectral triple
We study notions of measurability for singular traces, and characterise universal measurability for operators in Dixmier ideals. Th is measurability result is then applied to improve on the various proofs of Connes’ identi cation of the Hochschild class of the Chern character of Dixmier summable spectral triples. The measurability results show that the identi cation of the Hochschild class is independent of the choice of singular trace. As a corollary we obtain strong information on the asymptotics of the eigenvalues of operators naturally associated to spectral triples $\mathcal A, H, D$ and Hochschild cycles for $\mathcal A$.
Functional analysis
1
41
10.4171/JST/116
http://www.ems-ph.org/doi/10.4171/JST/116
Isospectrality for graph Laplacians under the change of coupling at graph vertices
Yulia
Ershova
University of Bath, BATH, UNITED KINGDOM
Irina
Karpenko
Taurida National University, SIMFEROPOL, UKRAINE
Alexander
Kiselev
St.Petersburg State University, ST. PETERSBURG, RUSSIAN FEDERATION
Quantum graphs, Schrödinger operator, Laplace operator, inverse spectral problem, trace formulae, boundary triples, isospectral graphs
Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of $\delta$ and $\delta'$ types. An infinite series of trace formulae is obtained which link together two different quantum graphs under the assumption that their spectra coincide. The general case of graph Schrodinger operators is also considered, yielding the first trace formula. Tightness of results obtained under no additional restrictions on edge lengths is demonstrated by an example. Further examples are scrutinized when edge lengths are assumed to be rationally independent. In all but one of these impossibility of isospectral configurations is ascertained.
Operator theory
Ordinary differential equations
Quantum theory
43
66
10.4171/JST/117
http://www.ems-ph.org/doi/10.4171/JST/117
Titchmarsh–Weyl theory for Schrödinger operators on unbounded domains
Jussi
Behrndt
TU Graz, GRAZ, AUSTRIA
Jonathan
Rohleder
TU Graz, GRAZ, AUSTRIA
Dirichlet-to-Neumann map, Schrödinger operator, spectrum, Titchmarsh–Weyl function
In this paper it is proved that the complete spectral data of selfadjoint Schrödinger operators on unbounded domains can be described with an associated Dirichlet-to-Neumann map. In particular, a characterization of the isolated and embedded eigenvalues, the corresponding eigenspaces, as well as the continuous and absolutely continuous spectrum in terms of the limiting behaviour of the Dirichlet-to-Neumann map is obtained. Furthermore, a sufficient criterion for the absence of singular continuous spectrum is provided. The results are natural multidimensional analogs of classical facts from singular Sturm-Liouville theory.
Partial differential equations
Operator theory
Quantum theory
67
87
10.4171/JST/118
http://www.ems-ph.org/doi/10.4171/JST/118
On the spectra of the Sturm–Liouville operator with the fast oscillating potential
Rais
Ismagilov
Bauman Moscow State Technical University, MOSCOW, RUSSIAN FEDERATION
Sturm–Liouville operator, discrete spectra, oscillating potential
A spectral Problem for a Sturm–Liouville operator is considered in the functional Hilbert space on the half axis. The potential is "fast oscillating". The conditions giving a discrete spectra are indicated. The spectral asymptotic is determined.
Operator theory
Ordinary differential equations
89
97
10.4171/JST/119
http://www.ems-ph.org/doi/10.4171/JST/119
Hadamard type asymptotics for eigenvalues of the Neumann problem for elliptic operators
Vladimir
Kozlov
Linköping University, LINKÖPING, SWEDEN
Johan
Thim
Linköping University, LINKÖPING, SWEDEN
Hadamard formula, domain variation, asymptotics of eigenvalues, Neumann problem
This paper considers how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain. The proximity of two domains is measured in terms of the norm of the difference between the two resolvents corresponding to the reference domain and the perturbed domain, and the size of eigenfunctions outside the intersection of the two domains. This construction enables the possibility of comparing both nonsmooth domains and domains with different topology. An abstract framework is presented, where the main result is an asymptotic formula where the remainder is expressed in terms of the proximity quantity described above when this is relatively small. As an application, we develop a theory for the Laplacian in Lipschitz domains. In particular, if the domains are assumed to be $C^{1,\alpha}$ regular, an asymptotic result for the eigenvalues is given together with estimates for the remainder, and we also provide an example which demonstrates the sharpness of our obtained result.
Partial differential equations
Operator theory
Calculus of variations and optimal control; optimization
99
135
10.4171/JST/120
http://www.ems-ph.org/doi/10.4171/JST/120
A note on the resonance counting function for surfaces with cusps
Yannick
Bonthonneau
Université du Québec à Montréal, MONTREAL, CANADA
Resonances, surfaces with cusps, Weyl law
We prove sharp upper bounds for the number of resonances in boxes of size 1 at high frequency for the Laplacian on finite volume surfaces with hyperbolic cusps. As a corollary, we obtain a Weyl asymptotic for the number of resonances in balls of size $T \to \infty$ with remainder $O(T^{3/2})$.
Partial differential equations
137
144
10.4171/JST/121
http://www.ems-ph.org/doi/10.4171/JST/121
Inverse boundary problems for polyharmonic operators with unbounded potentials
Katsiaryna
Krupchyk
University of California at Irvine, IRVINE, UNITED STATES
Gunther
Uhlmann
University of Washington, SEATTLE, UNITED STATES
Inverse boundary problem, polyharmonic operator, unbounded potential, Carleman estimate, Green function
We show that the knowledge of the Dirichlet–to–Neumann map on the boundary of a bounded open set in $\mathbb R^n$ for the perturbed polyharmonic operator $(-\Delta)^m +q$ with $q\in L^{\frac{n}{2m}}$, $n>2m$, determines the potential $q$ in the set uniquely. In the course of the proof, we construct a special Green function for the polyharmonic operator and establish its mapping properties in suitable weighted $L^2$ and $L^p$ spaces. The $L^p$ estimates for the special Green function are derived from $L^p$ Carleman estimates with linear weights for the polyharmonic operator.
Partial differential equations
Potential theory
145
183
10.4171/JST/122
http://www.ems-ph.org/doi/10.4171/JST/122
Evolution PDEs and augmented eigenfunctions. Half-line
Beatrice
Pelloni
University of Reading, READING, UNITED KINGDOM
David
Smith
University of Michigan, ANN ARBOR, UNITED STATES
Spectral theory of non-self-adjoint differential operator, initial-boundary value problem, generalised eigenfunction expansion
Th e solution of an initial-boundary value problem for a linear evolution partial di fferential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. is representation is obtained by the unifi ed transform introduced by Fokas in the 90’s. On the other hand, it is known that many initial-boundary value problems can be solved via a classical transform pair, constructed via the spectral analysis of the associated spatial operator. For example, the Dirichlet problem for the heat equation can be solved by applying the Fourier sine transform pair. However, for many other initial-boundary value problems there is no suitable transform pair in the classical literature. Here we pose and answer two related questions: given any well-posed initial-boundary value problem, does there exist a (non-classical) transform pair suitable for solving that problem? If so, can this transform pair be constructed via the spectral analysis of a diff erential operator? Th e answer to both of these questions is positive and given in terms of augmented eigenfunctions, a novel class of spectral functionals. ese are eigenfunctions of a suitable di erential operator in a certain generalised sense, they provide an e ective spectral representation of the operator, and are associated with a transform pair suitable to solve the given initial-boundary value problem.
Partial differential equations
Operator theory
185
213
10.4171/JST/123
http://www.ems-ph.org/doi/10.4171/JST/123