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On the spectrum of critical almost Mathieu operators in the rational case J. Spectr. Theory (IF 1.0) Pub Date : 2022-03-24 Svetlana Jitomirskaya, Lyuben Konstantinov, Igor Krasovsky
We derive a new Chambers-type formula and prove sharper upper bounds on the measure of the spectrum of critical almost Mathieu operators with rational frequencies.
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Eigenfunction asymptotics and spectral rigidity of the ellipse J. Spectr. Theory (IF 1.0) Pub Date : 2022-03-24 Hamid Hezari, Steve Zelditch
Microlocal defect measures for Cauchy data of Dirichlet, resp. Neumann, eigenfunctions of an ellipse $E$ are determined. We prove that, for any invariant curve for the billiard map on the boundary phase space $B^* E$ of an ellipse, there exists a sequence of eigenfunctions whose Cauchy data concentrates on the invariant curve. We use this result to give a new proof that ellipses are infinitesimally
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Semiclassical Gevrey operators and magnetic translations J. Spectr. Theory (IF 1.0) Pub Date : 2022-03-24 Michael Hitrik, Richard Lascar, Johannes Sjöstrand, Maher Zerzeri
We study semiclassical Gevrey pseudodifferential operators acting on the Bargmann space of entire functions with quadratic exponential weights. Using some ideas from time frequency analysis, we show that such operators are uniformly bounded on a natural scale of exponentially weighted spaces of holomorphic functions, provided that the Gevrey index is greater than or equal to 2.
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Spectral shift via “lateral” perturbation J. Spectr. Theory (IF 1.0) Pub Date : 2022-03-24 Gregory Berkolaiko, Peter Kuchment
We consider a compact perturbation $H_0 = S + K_0^* K_0$ of a self-adjoint operator $S$ with an eigenvalue $\lambda^\circ$ below its essential spectrum and the corresponding eigenfunction $f$. The perturbation is assumed to be “along” the eigenfunction $f$, namely $K_0f=0$. The eigenvalue $\lambda^\circ$ belongs to the spectra of both $H_0$ and $S$. Let $S$ have $\sigma$ more eigenvalues below $\lambda^\circ$
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Complete asymptotic expansions of the spectral function for symbolic perturbations of almost periodic Schrödinger operators in dimension one J. Spectr. Theory (IF 1.0) Pub Date : 2022-03-24 Jeffrey Galkowski
In this article we consider asymptotics for the spectral function of Schrödinger operators on the real line. Let $P\colon L^2(\mathbb{R})\to L^2(\mathbb{R})$ have the form $$ P:=-\frac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order differential operator with certain modified almost periodic structure. We show that the kernel of the spectral projector, $\mathbf{1}_{(-\infty,\lambda^2]}(P)$
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Berezin–Toeplitz quantization associated with higher Landau levels of the Bochner Laplacian J. Spectr. Theory (IF 1.0) Pub Date : 2022-03-24 Yuri A. Kordyukov
In this paper, we construct a family of Berezin–Toeplitz type quantizations of a compact symplectic manifold. For this, we choose a Riemannian metric on the manifold such that the associated Bochner Laplacian has the same local model at each point (this is slightly more general than in almost-Kähler quantization). Then the spectrum of the Bochner Laplacian on high tensor powers $L^p$ of the prequantum
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On the Benjamin–Ono equation on $\mathbb{T}$ and its periodic and quasiperiodic solutions J. Spectr. Theory (IF 1.0) Pub Date : 2022-03-24 Patrick Gérard, Thomas Kappeler, Petar Topalov
In this paper, we survey our recent results on the Benjamin–Ono equation on the torus. As an application of the methods developed we construct large families of periodic or quasiperiodic solutions, which are not $C^\infty$-smooth
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The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander’s rediscovered manuscript J. Spectr. Theory (IF 1.0) Pub Date : 2022-03-24 Alexandre Girouard, Mikhail Karpukhin, Michael Levitin, Iosif Polterovich
How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat surprisingly, a lot of techniques involved can be traced back to a newly rediscovered manuscript of Hörmander from the 1950s. We present Hörmander’s approach and its applications, with an emphasis on eigenvalue estimates and
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Integral representations of isotropic semiclassical functions and applications J. Spectr. Theory (IF 1.0) Pub Date : 2022-03-24 Victor W. Guillemin, Alejandro Uribe, Zuoqin Wang
In a previous paper, we introduced a class of “semiclassical functions of isotropic type,” starting with a model case and applying Fourier integral operators associated with canonical transformations. These functions are a substantial generalization of the “oscillatory functions of Lagrangian type” that have played major role in semiclassical and microlocal analysis. In this paper we exhibit more clearly
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Eigenvalues of singular measures and Connes’ noncommutative integration J. Spectr. Theory (IF 1.0) Pub Date : 2022-03-24 Grigori Rozenblum
In a domain $\Omega\subset \mathbb{R}^{\operatorname{N}}$ we consider compact, Birman–Schwinger type operators of the form $\operatorname{T}_{P,\mathfrak{A}}=\mathfrak{A}^* P \mathfrak{A}$ with $P$ being a Borel measure in $\Omega,$ containing a singular part, and $\mathfrak{A}$ being an order $-\operatorname{N}/2$ pseudodifferential operator. Operators are defined by means of quadratic forms. For
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Invariant subspaces of elliptic systems II: Spectral theory J. Spectr. Theory (IF 1.0) Pub Date : 2022-03-24 Matteo Capoferri, Dmitri Vassiliev
Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$ M, whose principal symbol is assumed to have simple eigenvalues.We show that the spectrum of $A$ decomposes, up to an error with superpolynomial decay, into $m$ distinct series, each associated with one of the eigenvalues of the principal symbol of $A$. These spectral results
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The anisotropic Calderón problem on 3-dimensional conformally Stäckel manifolds J. Spectr. Theory (IF 1.0) Pub Date : 2021-12-02 Thierry Daudé, Niky Kamran, François Nicoleau
Conformally Stäckel manifolds can be characterized as the class of $n$-dimensional pseudo-Riemannian manifolds $(M,G)$ on which the Hamilton–Jacobi equation $$ G(\nabla u, \nabla u) = 0 $$ for null geodesics and the Laplace equation $-\Delta_G \, \psi = 0$ are solvable by R-separation of variables. In the particular case in which the metric has Riemannian signature, they provide explicit examples of
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The Hartree functional in a double well J. Spectr. Theory (IF 1.0) Pub Date : 2021-12-02 Alessandro Olgiati, Nicolas Rougerie
We consider a non-linear Hartree energy for bosonic particles in a symmetric double-well potential. In the limit where the wells are far apart and the potential barrier is high, we prove that the ground state and first excited state are given to leading order by an even, respectively odd, superposition of ground states in single wells. The corresponding energies are separated by a small tunneling term
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Spectrum of the semi-relativistic Pauli–Fierz model II J. Spectr. Theory (IF 1.0) Pub Date : 2021-12-02 Takeru Hidaka, Fumio Hiroshima, Itaru Sasaki
We consider the ground state of the semi-relativistic Pauli–Fierz Hamiltonian $$ H = |\textbf{p} - \textbf{A(x)}| + H_f + V\textbf{(x)}. $$ Here $\textbf{A(x)}$ denotes the quantized radiation field with an ultraviolet cutoff function and $H_f$ the free field Hamiltonian with dispersion relation $|\textbf{k}|$. The Hamiltonian $H$ describes the dynamics of a massless and semi-relativistic charged particle
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Uniform resolvent estimates for the discrete Schrödinger operator in dimension three J. Spectr. Theory (IF 1.0) Pub Date : 2021-12-02 Kouichi Taira
In this note, we prove the uniform resolvent estimate of the discrete Schrödinger operator with dimension three. To do this, we show a Fourier decay of the surface measure on the Fermi surface.
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Upper eigenvalue bounds for the Kirchhoff Laplacian on embedded metric graphs J. Spectr. Theory (IF 1.0) Pub Date : 2021-12-02 Marvin Plümer
We derive upper bounds for the eigenvalues of the Kirchhoff Laplacian on a compact metric graph depending on the graph’s genus $g$. These bounds can be further improved if $g=0$, i.e., if the metric graph is planar. Our results are based on a spectral correspondence between the Kirchhoff Laplacian and a particular combinatorial weighted Laplacian. In order to take advantage of this correspondence,
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Eigenfunctions growth of R-limits on graphs J. Spectr. Theory (IF 1.0) Pub Date : 2021-12-02 Siegfried Beckus, Latif Eliaz
A characterization of the essential spectrum of Schrödinger operators on infinite graphs is derived involving the concept of $\mathcal{R}$-limits. This concept, which was introduced previously for operators on $\mathbb{N}$ and $\mathbb{Z}^d$ as “right-limits,” captures the behaviour of the operator at infinity. For graphs with sub-exponential growth rate, we show that each point in $\sigma_{\textup{ess}}(H)$
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A Faber–Krahn inequality for the Riesz potential operator for triangles and quadrilaterals J. Spectr. Theory (IF 1.0) Pub Date : 2021-12-02 Rajesh Mahadevan, Franco Olivares-Contador
We prove an analog of the Faber–Krahn inequality for the Riesz potential operator. The proof is based on Riesz’s inequality under Steiner symmetrization and the continuity of the first eigenvalue of the Riesz potential operator with respect to the convergence, in the complementary Hausdorff distance, of a family of uniformly bounded non-empty convex open sets.
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A theorem on the multiplicity of the singular spectrum of a general Anderson-type Hamiltonian J. Spectr. Theory (IF 1.0) Pub Date : 2021-11-02 Dhriti Ranjan Dolai, Anish Mallick
In this work, we study the multiplicity of the singular spectrum for operators of the form $A^\omega=A+\sum_{n}\omega_n C_n$ on a separable Hilbert space $\mathcal{H}$, where $A$ is a self-adjoint operator and $\{C_n\}_{n}$ is a countable collection of non-negative finite-rank operators. When $\{\omega_n\}_n$ are independent real random variables with absolutely continuous distributions, we show that
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Spectral theory of the thermal Hamiltonian: 1D case J. Spectr. Theory (IF 1.0) Pub Date : 2021-09-24 Giuseppe De Nittis, Vicente Lenz
In 1964 J. M. Luttinger introduced a model for the quantum thermal transport. In this paper we study the spectral theory of the Hamiltonian operator associated with Luttinger’s model, with a special focus at the one-dimensional case. It is shown that the (so called) thermal Hamiltonian has a one-parameter family of self-adjoint extensions and the spectrum, the time-propagator group and the Green function
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Disjointness-preserving operators and isospectral Laplacians J. Spectr. Theory (IF 1.0) Pub Date : 2021-09-29 Wolfgang Arendt, James B. Kennedy
The most commonly considered counterexamples to Kac’s famous question “can one hear the shape of a drum?” – i.e., does isospectrality of two Laplacians on domains imply that the domains are congruent? – consist of pairs of domains composed of copies of isometric building blocks arranged in different ways, such that the unitary operator intertwining the Laplacians acts as a sum of overlapping “local”
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Jacobi matrices on trees generated by Angelesco systems: asymptotics of coefficients and essential spectrum J. Spectr. Theory (IF 1.0) Pub Date : 2021-10-27 Alexander I. Aptekarev, Sergey A. Denisov, Maxim L. Yattselev
We continue studying the connection between Jacobi matrices defined on a tree and multiple orthogonal polynomials (MOPs) that was recently discovered. In this paper, we consider Angelesco systems formed by two analytic weights and obtain asymptotics of the recurrence coefficients and strong asymptotics of MOPs along all directions (including the marginal ones). These results are then applied to show
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Quantum ergodicity for pseudo-Laplacians J. Spectr. Theory (IF 1.0) Pub Date : 2021-11-02 Elie Studnia
We prove quantum ergodicity for the eigenfunctions of the pseudo-Laplacian on surfaces with hyperbolic cusps and ergodic geodesic flows.
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Spectral invariants of Dirichlet-to-Neumann operators on surfaces J. Spectr. Theory (IF 1.0) Pub Date : 2021-11-03 Jean Lagacé, Simon St-Amant
We obtain a complete asymptotic expansion for the eigenvalues of the Dirichlet-to-Neumann maps associated with Schrödinger operators on Riemannian surfaces with boundary. For the zero potential, we recover the well-known spectral asymptotics for the Steklov problem. For non-zero potentials, we obtain new geometric invariants determined by the spectrum. In particular, for constant potentials, which
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Erratum to “Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains” J. Spectr. Theory (IF 1.0) Pub Date : 2021-11-02 Simon Larson
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Different completions of $A + CX$ J. Spectr. Theory (IF 1.0) Pub Date : 2021-07-30 Dragana S. Cvetković-Ilić, Qing Wen Wang, Yimin Wei
In this paper we use Takahasi’s idea of using properties of spectral measures in addressing the question of the existence of an operator $X$ such that $A+CX$ is of appropriate types. In particular, we consider the class of right semi-Fredholm operators. Also, in the case of right invertibility we will show how the results of this type can be used to address the appropriate completion problem of the
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Inequalities between Neumann and Dirichlet eigenvalues of Schrödinger operators J. Spectr. Theory (IF 1.0) Pub Date : 2021-07-30 Jonathan Rohleder
Given a Schrödinger operator with a real-valued potential on a bounded, convex domain or a bounded interval we prove inequalities between the eigenvalues corresponding to Neumann and Dirichlet boundary conditions, respectively. The obtained inequalities depend partially on monotonicity and convexity properties of the potential. The results are counterparts of classical inequalities for the Laplacian
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Efficiency and localisation for the first Dirichlet eigenfunction J. Spectr. Theory (IF 1.0) Pub Date : 2021-07-30 Michiel van den Berg, Francesco Della Pietra, Giuseppina di Blasio, Nunzia Gavitone
Bounds are obtained for the efficiency or mean to max ratio $E(\Omega)$ for the first Dirichlet eigenfunction (positive) for open, connected sets $\Omega$ with finite measure in Euclidean space $\mathbb{R}^m$. It is shown that (i) localisation implies vanishing efficiency, (ii) a vanishing upper bound for the efficiency implies localisation, (iii) localisation occurs for the first Dirichlet eigenfunctions
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Recovery of time-dependent coefficients from boundary data for hyperbolic equations J. Spectr. Theory (IF 1.0) Pub Date : 2021-08-10 Ali Feizmohammadi, Joonas Ilmavirta, Yavar Kian, Lauri Oksanen
We study uniqueness of the recovery of a time-dependent magnetic vector valued potential and an electric scalar-valued potential on a Riemannian manifold from the knowledge of the Dirichlet-to-Neumann map of a hyperbolic equation. The Cauchy data is observed on time-like parts of the space-time boundary and uniqueness is proved up to the natural gauge for the problem. The proof is based on Gaussian
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Szegő’s theorem for canonical systems: the Arov gauge and a sum rule J. Spectr. Theory (IF 1.0) Pub Date : 2021-07-30 David Damanik, Benjamin Eichinger, Peter Yuditskii
We consider canonical systems and investigate the Szegő class, which is defined via the finiteness of the associated entropy functional. Noting that the canonical system may be studied in a variety of gauges, we choose to work in the Arov gauge, in which we prove that the entropy integral is equal to an integral involving the coefficients of the canonical system. This sum rule provides a spectral theory
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Existence of metrics maximizing the first eigenvalue on non-orientable surfaces J. Spectr. Theory (IF 1.0) Pub Date : 2021-09-13 Henrik Matthiesen, Anna Siffert
We prove the existence of metrics maximizing the first eigenvalue normalized by area on closed, non-orientable surfaces assuming two spectral gap conditions. These spectral gap conditions are proved by the authors in [21].
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Anderson localization for a generalized Maryland model with potentials given by skew shifts J. Spectr. Theory (IF 1.0) Pub Date : 2021-09-20 Jia Shi, Xiaoping Yuan
In this paper, we proved Anderson localization for the following long-range operator $$ H=\tan\pi\Big(x_0+my_0+\frac{m(m-1)}{2}\omega\Big) \delta_{mn}+\epsilon S_\phi, $$ which generalized the Maryland model to potentials given by skew shifts.
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Quantum evolution and sub-Laplacian operators on groups of Heisenberg type J. Spectr. Theory (IF 1.0) Pub Date : 2021-09-24 Clotilde Fermanian-Kammerer, Véronique Fischer
In this paper we analyze the evolution of the time averaged energy densities associated with a family of solutions to a Schrödinger equation on a Lie group of Heisenberg type. We use a semi-classical approach adapted to the stratified structure of the group and describe the semi-classical measures (also called quantum limits) that are associated with this family. This allows us to prove an Egorov’s
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A Szegő type theorem and distribution of symplectic eigenvalues J. Spectr. Theory (IF 1.0) Pub Date : 2021-09-28 Rajendra Bhatia, Tanvi Jain, Ritabrata Sengupta
We study the properties of stationary G-chains in terms of their generating functions. In particular, we prove an analogue of the Szegő limit theorem for symplectic eigenvalues, derive an expression for the entropy rate of stationary quantum Gaussian processes, and study the distribution of symplectic eigenvalues of truncated block Toeplitz matrices. We also introduce a concept of symplectic numerical
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On Lieb–Thirring inequalities for one-dimensional non-self-adjoint Jacobi and Schrödinger operators J. Spectr. Theory (IF 1.0) Pub Date : 2021-09-29 Sabine Bögli, František Štampach
We study to what extent Lieb–Thirring inequalities are extendable from self-adjoint to general (possibly non-self-adjoint) Jacobi and Schrödinger operators. Namely, we prove the conjecture of Hansmann and Katriel from [12] and answer another open question raised therein. The results are obtained by means of asymptotic analysis of eigenvalues of discrete Schrödinger operators with rectangular barrier
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Trace class properties of the non homogeneous linear Vlasov–Poisson equation in dimension 1+1 J. Spectr. Theory (IF 1.0) Pub Date : 2021-07-14 Bruno Després
We consider the abstract scattering structure of the non homogeneous linearized Vlasov–Poisson equations from the viewpoint of trace class properties which are emblematic of the abstract scattering theory [13, 14, 15, 19]. In dimension 1+1, we derive an original reformulation which is trace class. It yields the existence of the Moller wave operators. The non homogeneous background electric field is
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Maximizing the ratio of eigenvalues of non-homogeneous partially hinged plates J. Spectr. Theory (IF 1.0) Pub Date : 2021-07-14 Elvise Berchio, Alessio Falocchi
We study the spectrum of non-homogeneous partially hinged plates having structural engineering applications. A possible way to prevent instability phenomena is to maximize the ratio between the frequencies of certain oscillating modes with respect to the density function of the plate; we prove existence of optimal densities and we investigate their analytic expression. This analysis suggests where
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Periodic Jacobi operators with complex coefficients J. Spectr. Theory (IF 1.0) Pub Date : 2021-07-14 Vassilis G. Papanicolaou
We present certain results on the direct and inverse spectral theory of the Jacobi operator with complex periodic coefficients. For instance, we show that any $N$-th degree polynomial whose leading coefficient is $(-1)^N$ is the Hill discriminant of finitely many discrete $N$-periodic Schrödinger operators (Theorem 1). Also, in the case where the spectrum is a closed interval we prove a result (Theorem
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Spectral analysis of 2D outlier layout J. Spectr. Theory (IF 1.0) Pub Date : 2021-07-14 Mihai Putinar
Thompson’s partition of a cyclic subnormal operator into normal and completely non-normal components is combined with a non-commutative calculus for hyponormal operators for separating outliers from the cloud, in rather general point distributions in the plane. The main result provides exact transformation formulas from the power moments of the prescribed point distribution into the moments of the
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Invertibility issues for a class of Wiener–Hopf plus Hankel operators J. Spectr. Theory (IF 1.0) Pub Date : 2021-07-14 Victor D. Didenko, Bernd Silbermann
The invertibility of Wiener–Hopf plus Hankel operators $W(a)+H(b)$ acting on the spaces $L^p(\mathbb{R}^+)$, $1 \leq p<\infty$ is studied. If $a$ and $b$ belong to a subalgebra of $L^\infty(\mathbb{R})$ and satisfy the condition $$ a(t) a(-t)=b(t) b(-t),\quad t\in\mathbb{R}, $$ we establish necessary and also sufficient conditions for the operators $W(a)+H(b)$ to be one-sided invertible, invertible
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Ergodic Schrödinger operators in the infinite measure setting J. Spectr. Theory (IF 1.0) Pub Date : 2021-07-14 Michael Boshernitzan, David Damanik, Jake Fillman, Milivoje Lukic
We develop the basic theory of ergodic Schrödinger operators, which is well known for ergodic probability measures, in the case of a base dynamics on an infinite measure space. This includes the almost sure constancy of the spectrum and the spectral type, the definition and discussion of the density of states measure and the Lyapunov exponent, as well as a version of the Pastur–Ishii theorem. We also
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Partial retraction of “Two-term, asymptotically sharp estimates for eigenvalue means of the Laplacian” J. Spectr. Theory (IF 1.0) Pub Date : 2021-05-20 Evans M. Harrell II, Joachim Stubbe
We regret that we have to retract portions of the article “Two-term, asymptotically sharp estimates for eigenvalue means of the Laplacian” [J. Spectral Theory 8 (2018), 1529–1550] due to an essential error in the proof of Theorem 1.2, which is used in other places in the paper.
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Lieb–Thirring inequalities for an effective Hamiltonian of bilayer graphene J. Spectr. Theory (IF 1.0) Pub Date : 2021-07-19 Philippe Briet, Jean-Claude Cuenin, Leonid B. Golinskii, Stanislas Kupin
Combining the methods of Cuenin [7] and Borichev, Golinskii, and Kupin [4] and [5], we obtain the so-called Lieb–Thirring inequalities for non-selfadjoint perturbations of an effective Hamiltonian for bilayer graphene.
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Maximal estimates for the Fokker–Planck operator with magnetic field J. Spectr. Theory (IF 1.0) Pub Date : 2021-07-19 Zeinab Karaki
We consider the Fokker–Planck operator with a strong external magnetic field. We show a maximal type estimate on this operator using a nilpotent approach on vector field polynomial operators and including the notion of representation of a Lie algebra. This estimate makes it possible to give an optimal characterization of the domain of the closure of the considered operator.
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Coexistence of absolutely continuous and pure point spectrum for kicked quasiperiodic potentials J. Spectr. Theory (IF 1.0) Pub Date : 2021-07-19 Kristian Bjerklöv, Raphaël Krikorian
We introduce a class of real analytic “peaky” potentials for which the corresponding quasiperiodic 1D-Schrödinger operators exhibit, for quasiperiodic frequencies in a set of positive Lebesgue measure, both absolutely continuous and pure point spectrum.
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The massless Dirac equation in two dimensions: zero-energy obstructions and dispersive estimates J. Spectr. Theory (IF 1.0) Pub Date : 2021-07-14 M. Burak Erdoğan, Michael Goldberg, William R. Green
We investigate $L^1\to L^\infty$ dispersive estimates for the massless two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies the natural $t^{-\frac{1}{2}}$ decay rate, which may be improved to $t^{-\frac{1}{2} - \gamma}$ for any $0\leq \gamma<\frac{3}{2}$ at the cost of spatial weights. We classify the structure of threshold obstructions as being
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Range characterizations and Singular Value Decomposition of the geodesic X-ray transform on disks of constant curvature J. Spectr. Theory (IF 1.0) Pub Date : 2021-07-14 Rohit Kumar Mishra, François Monard
For a one-parameter family of simple metrics of constant curvature ($4\kappa$ for $\kappa\in (-1,1)$) on the unit disk $M$, we first make explicit the Pestov–Uhlmann range characterization of the geodesic X-ray transform, by constructing a basis of functions making up its range and co-kernel. Such a range characterization also translates into moment conditions $à$ $la$ Helgason–Ludwig or Gel'fand–Graev
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Self-adjointness of two-dimensional Dirac operators on corner domains J. Spectr. Theory (IF 1.0) Pub Date : 2021-07-14 Fabio Pizzichillo, Hanne Van Den Bosch
We investigate the self-adjointness of the two-dimensional Dirac operator $D$, with $quantum$-$dot$ and $Lorentz$-$scalar$ $\delta$-$shell$ boundary conditions, on piecewise $C^2$ domains (with finitely many corners). For both models, we prove the existence of a unique self-adjoint realization whose domain is included in the Sobolev space $H^{1/2}$, the formal form domain of the free Dirac operator
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Jordan chains of elliptic partial differential operators and Dirichlet-to-Neumann maps J. Spectr. Theory (IF 1.0) Pub Date : 2021-07-14 Jussi Behrndt, A. F. M. ter Elst
Let $\Omega \subset \mathbb{R}^d$ be a bounded open set with Lipschitz boundary $\Gamma$. It will be shown that the Jordan chains of m-sectorial second-order elliptic partial differential operators with measurable coefficients and (local or non-local) Robin boundary conditions in $L_2(\Omega)$ can be characterized with the help of Jordan chains of the Dirichlet-to-Neumann map and the boundary operator
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Semiclassical study of shape resonances in the Stark effect J. Spectr. Theory (IF 1.0) Pub Date : 2021-04-12 Kentaro Kameoka
Semiclassical behavior of Stark resonances is studied. The complex distortion outside a cone is introduced to study resonances in any energy region for the Stark Hamiltonians with non-globally analytic potentials. The non-trapping resolvent estimate is proved by the escape function method. The Weyl law and the resonance expansion of the propagator are proved in the shape resonance model. To prove the
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On operator error estimates for homogenization of hyperbolic systems with periodic coefficients J. Spectr. Theory (IF 1.0) Pub Date : 2021-03-18 Yulia M. Meshkova
In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a selfadjoint matrix strongly elliptic second order differential operator $\mathcal{A}_\varepsilon$, $\varepsilon > 0$. The coefficients of the operator $\mathcal{A}_\varepsilon$ are periodic and depend on $\mathbf{x}/\varepsilon$. We study the asymptotic behavior of the operator $\mathcal{A}_\varepsilon ^{-1/2}\sin (\tau \mathcal{A}_\varepsilon ^{1/2})$
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A universality law for sign correlations of eigenfunctions of differential operators J. Spectr. Theory (IF 1.0) Pub Date : 2021-03-18 Felipe Gonçalves, Diogo Oliveira e Silva, Stefan Steinerberger
We establish a sign correlation universality law for sequences of functions $\{w_n\}_{n \in \mathbb{N}}$ satisfying a trigonometric asymptotic law. Our results are inspired by the classical WKB asymptotic approximation for Sturm–Liouville operators, and in particular we obtain non-trivial sign correlations for eigenfunctions of generic Schrödinger operators (including the harmonic oscillator), as well
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Steklov and Robin isospectral manifolds J. Spectr. Theory (IF 1.0) Pub Date : 2021-02-11 Carolyn Gordon, Peter Herbrich, David Webb
We use two of the most fruitful methods for constructing isospectral manifolds, the Sunada method and the torus action method, to construct manifolds whose Dirichlet-to-Neumann operators are isospectral at all frequencies. The manifolds are also isospectral for the Robin boundary value problem for all choices of Robin parameter. As in the sloshing problem, we can also impose mixed Dirichlet–Neumann
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Decorrelation estimates for random Schrödinger operators with non rank one perturbations J. Spectr. Theory (IF 1.0) Pub Date : 2021-02-11 Peter D. Hislop, Maddaly Krishna, Christopher Shirley
We prove decorrelation estimates for generalized lattice Anderson models on $\mathbb Z^d$ constructed with finite-rank perturbations in the spirit of Klopp [12]. These are applied to prove that the local eigenvalue statistics $\xi^\omega_{E}$ and $\xi^\omega_{E'}$, associated with two energies $E$ and $E'$ in the localization region and satisfying $|E - E'| > 4d$, are independent. That is, if $I,J$
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Spectral analysis on Barlow and Evans’ projective limit fractals J. Spectr. Theory (IF 1.0) Pub Date : 2021-02-24 Benjamin Steinhurst, Alexander Teplyaev
We review the projective limit construction of a state space for a Markov process use by Barlow and Evans. On this state space we construct a projective limit Dirichlet form in a process analogous to Barlow and Evan’s construction of a Markov process. Then we study the spectral properties of the corresponding Laplacian using the projective limit construction. For some examples, such as the Laakso spaces
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The density of states of 1D random band matrices via a supersymmetric transfer operator J. Spectr. Theory (IF 1.0) Pub Date : 2021-02-24 Margherita Disertori, Martin Lohmann, Sasha Sodin
Recently,M. and T. Shcherbina proved a pointwise semicircle law for the density of states of one-dimensional Gaussian band matrices of large bandwidth. The main step of their proof is a new method to study the spectral properties of non-self-adjoint operators in the semiclassical regime. The method is applied to a transfer operator constructed from the supersymmetric integral representation for the
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On open scattering channels for a branched covering of the Euclidean plane J. Spectr. Theory (IF 1.0) Pub Date : 2021-03-01 Rainer Hempel, Olaf Post
We study the interaction of two scattering channels for a simple geometric model consisting in a double covering of the plane with two branch points, equipped with the Euclidean metric. We show that the scattering channels are open in the sense of [11] and that this property is stable under suitable perturbations of the metric.
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Wildly perturbed manifolds: norm resolvent and spectral convergence J. Spectr. Theory (IF 1.0) Pub Date : 2021-03-01 Colette Anné, Olaf Post
The publication of the important work of Rauch and Taylor [J. Funct. Anal. 18 (1975)] started a hole branch of research on wild perturbations of the Laplace–Beltrami operator. Here, we extend certain results and show norm convergence of the resolvent. We consider a (not necessarily compact) manifold with many small balls removed, the number of balls can increase as the radius is shrinking, the number
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Mixed data in inverse spectral problems for the Schrödinger operators J. Spectr. Theory (IF 1.0) Pub Date : 2021-03-10 Burak Hatinoğlu
We consider the Schrödinger operator on a finite interval with an $L^1$-potential. We prove that the potential can be uniquely recovered from one spectrum and subsets of another spectrum and point masses of the spectral measure (or norming constants) corresponding to the first spectrum. We also solve this Borg–Marchenko-type problem under some conditions on two spectra, when missing part of the second
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Nodal line estimates for the second Dirichlet eigenfunction J. Spectr. Theory (IF 1.0) Pub Date : 2021-03-10 Thomas Beck, Yaiza Canzani, Jeremy L. Marzuola
We study the nodal curves of low energy Dirichlet eigenfunctions in generalized curvilinear quadrilaterals. The techniques can be seen as a generalization of the tools developed by Grieser–Jerison in a series of works on convex planar domains and rectangles with one curved edge and a large aspect ratio. Here, we study the structure of the nodal curve in greater detail, in that we find precise bounds