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In this article, we provide the spectral analysis of a Dirac-type operator on $\mathbb{Z}^2$ by describing the behavior of the spectral shift function associated with a sign-definite trace-class perturbation by a multiplication operator. We prove that it remains bounded outside a single threshold and obtain its main a... |
We obtain bounds on the complex eigenvalues of non-self-adjoint Schrödinger operators with complex potentials, and also on the complex resonances of self-adjoint Schrödinger operators. Our bounds are compared with numerical results, and are seen to provide useful information. |
Motivated by the work of Abreu and Freitas, we study the invariant spectrum of the Laplace operator associated to hermitian line bundles endowed with invariant metrics over $\mathbb{p}^1$. |
This is a comprehensive review of the uses of potential theory in studying the spectral theory of orthogonal polynomials. Much of the article focuses on the Stahl-Totik theory of regular measures, especially the case of OPRL and OPUC. |
A measure $\mu$ on the unit circle $\mathbb{T}$ belongs to Steklov class $\mathcal{S}$ if its density $w$ with respect to the Lebesgue measure on $\mathbb{T}$ is strictly positive: $\inf_{\mathbb{T}} w > 0$. Let $\mu$, $\mu_{-1}$ be measures on the unit circle $\mathbb{T}$ with real recurrence coefficients $\{\alph... |
We outline some recent proofs of quantum ergodicity on large graphs and give new applications in the context of irregular graphs. We also discuss some remaining questions. |
We give a mathematically rigorous construction of a magnetic Schrödinger operator corresponding to a field with flux through finitely many holes of the Sierpinski Gasket. The operator is shown to have discrete spectrum accumulating at $\infty$, and it is shown that the asymptotic distribution of eigenvalues is the sam... |
Superconductivity for Type II superconductors in external magnetic fields of magnitude between the second and third critical fields is known to be restricted to a narrow boundary region. The profile of the superconducting order parameter in the Ginzburg-Landau model is expected to be governed by an effective one-dimen... |
Using the method of similar operators we study an even order differential operator with periodic, semiperiodic, and Dirichlet boundary conditions. We obtain asymptotic formulas for eigenvalues of this operator and estimates for its spectral decompositions and spectral projections. |
The present work aims at obtaining estimates for transformation operators for one-dimensional perturbed radial Schrödinger operators. It provides more details and suitable extensions to already existing results, that are needed in other recent contributions dealing with these kinds of operators. |
We demonstrate criteria, purely based on finite subwords of the potential, to guarantee spectral inclusion as well as Hausdorff approximation of pseudospectra or even spectra of generalized Schrödinger operators on the discrete line or half-line. In fact, our results are neither limited to Schrödinger or self-adjoint ... |
The aim of the present paper is to introduce the notion of first order (supersymmetric) Dirac operators on discrete and metric (``quantum'') graphs. In order to cover all self-adjoint boundary conditions for the associated metric graph Laplacian, we develop systematically a new type of discrete graph operators... |
In this paper, we study eigenvalues and eigenfunctions of $p$-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. |
This paper is part of a series papers devoted to geometric and spectral theoretic applications of the hypoelliptic calculus on Heisenberg manifolds. More specifically, in this paper we make use of the Heisenberg calculus of Beals-Greiner and Taylor to analyze the spectral theory of hypoelliptic operators on Heisenberg... |
We prove Lieb-Thirring-type bounds on eigenvalues of non-selfadjoint Jacobi operators, which are nearly as strong as those proven previously for the case of selfadjoint operators by Hundertmark and Simon. We use a method based on determinants of operators and on complex function theory, extending and sharpening earlie... |
Our goal is to extend the theory of the spectral shift function to the case where only the difference of some powers of the resolvents of self-adjoint operators belongs to the trace class. As an example, we consider a couple of Dirac operators. |
We consider semiclassical Schroedinger operators on R^n, with C^\infty potentials decaying polynomially at infinity. The usual theories of resonances do not apply in such a non-analytic framework. |
In this paper, we reduce the general linear integral equation of the third kind in $L^2(Y,\mu)$, with largely arbitrary kernel and coefficient, to an equivalent integral equation either of the second kind or of the first kind in $L^2(\mathbb{R})$, with the kernel being the linear pencil of bounded infinitely differenti... |
We introduce a transfer matrix method for the spectral analysis of discrete Hermitian operators with locally finite hopping. Such operators can be associated with a locally finite graph structure and the method works in principle on any such graph. |
The first nonzero eigenvalue of the Neumann Laplacian is shown to be minimal for the degenerate acute isosceles triangle, among all triangles of given diameter. Hence an optimal Poincaré inequality for triangles is derived. |
Let $\fre\subset\bbR$ be a finite union of $\ell+1$ disjoint closed intervals and denote by $\omega_j$ the harmonic measure of the $j$ leftmost bands. The frequency module for $\fre$ is the set of all integral combinations of $\omega_1,..., \omega_\ell$. |
We investigate the eigenvalues of perturbed spherical Schrödinger operators under the assumption that the perturbation $q(x)$ satisfies $x q(x) \in L^1(0,1)$. We show that the square roots of eigenvalues are given by the square roots of the unperturbed eigenvalues up to an decaying error depending on the behavior of $... |
We consider a periodic magnetic Schrödinger operator on a noncompact Riemannian manifold $M$ such that $H^1(M, \RR)=0$ endowed with a properly discontinuous cocompact isometric action of a discrete group. We assume that there is no electric field and that the magnetic field has a periodic set of compact magnetic wells... |
We consider discrete Schrödinger operators with periodic potentials on periodic graphs. Their spectra consist of a finite number of bands. |
In this paper it is considered a spectral density for a class of Jacobi matrices with absolutely continuous spectrum that was examined first by Moszynski. It is shown that the corresponding spectral density is equivalent to the positive continuous function everywhere except the point $x=0$. |
We consider the two-dimensional Dirac operator with infinite mass boundary conditions posed in a tubular neighborhood of a $C^4$-planar curve. Under generic assumptions on its curvature $\kappa$, we prove that in the thin-width regime the splitting of the eigenvalues is driven by the one dimensional Schrödinger operat... |
Mhaskar-Saff found a kind of universal behavior for the bulk structure of the zeros of orthogonal polynomials for large $n$. Motivated by two plots, we look at the finer structure for the case of random Verblunsky coefficients and for what we call the BLS condition: $\alpha_n = Cb^n + O((b\Delta)^n)$. |
This paper deals with the massive three-dimensional Dirac operator coupled with a Lorentz scalar shell interaction supported on a compact smooth surface. The rigorous definition of the operator involves suitable transmission conditions along the surface. |
This paper addresses inverse spectral problems associated with Dirac-type operators with a constant delay, specifically when this delay is less than one-third of the interval length. Our research focuses on eigenvalue behavior and operator recovery from spectra. |
We consider Schrödinger operators on a bounded, smooth domain of dimension $d \ge 2$ with Dirichlet boundary conditions and a properly scaled potential, which depends only on the distance to the boundary of the domain. Our aim is to analyse the convergence of these operators as the scaling parameter tends to zero. |
We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic func... |
We consider semiclassical self-adjoint operators whose symbol, defined on a two-dimensional symplectic manifold, reaches a non-degenerate minimum $b_0$ on a closed curve. We derive a classical and quantum normal form which allows us, in addition to the complete integrability of the system, to obtain eigenvalue asympto... |
We introduce computational strategies for measuring the ``size'' of the spectrum of bounded self-adjoint operators using various metrics such as the Lebesgue measure, fractal dimensions, the number of connected components (or gaps), and other spectral characteristics. Our motivation comes from the study of alm... |
The main result of this paper is a sharp upper bound on the first positive eigenvalue of Dirac operators in two dimensional simply connected $C^3$-domains with infinite mass boundary conditions. This bound is given in terms of a conformal variation, explicit geometric quantities and of the first eigenvalue for the dis... |
We consider semi-infinite Jacobi matrices with discrete spectrum. We prove that the Jacobi operator can be uniquely recovered from one spectrum and subsets of another spectrum and norming constants corresponding to the first spectrum. |
We consider the question of whether the high-energy eigenfunctions of certain Schrödinger operators on the $d$-dimensional hyperbolic space of constant curvature $-\kappa^2$ are flexible enough to approximate an arbitrary solution of the Helmholtz equation $\Delta h+h=0$ on $\mathbf{R}^d$, over the natural length scale... |
We improve a specific method to obtain the dimension of the eigenspaces of the Laplace-Beltrami operator on lens spaces and establish some applications related to the explicit description of the dimension of the smallest positive eigenvalue and the parity of the dimensions of the eigenspaces. |
The isospectral reduction of matrix, which is closely related to its Schur complement, allows to reduce the size of a matrix while maintaining its eigenvalues up to a known set. Here we generalize this procedure by increasing the number of possible ways a matrix can be isospectrally reduced. |
We analyze the spectral properties and peculiar behavior of solutions of a damped wave equation on a finite interval with a singular damping of the form $\alpha/x$, $\alpha>0$. We establish the exponential stability of the semigroup for all positive $\alpha$, and determine conditions for the spectrum to consist of ... |
The so-called unified method expresses the solution of an initial-boundary value problem (IBVP) for an evolution PDE in the finite interval in terms of an integral in the complex Fourier (spectral) plane. Simple IBVP, which will be referred to as problems of type~I, can be solved via a classical transform pair. |
Let $(M,g)$ be a compact Riemannian manifold with a boundary of class $\mathscr{C}^{1}$. We are interested in the spectrum of the weighted Laplacian on $M$ with Neumann boundary conditions. |
McDonald and Paparella [Linear Algebra Appl. 498 (2016), 145--159] gave a necessary condition on the structure of Jordan chains of $h$-cyclic matrices. In this work, that necessary condition is shown to be sufficient. |
We explore the Steklov eigenvalue problem on convex polygons, focusing mainly on the inverse Steklov problem. Our primary finding reveals that, for almost all convex polygonal domains, there exist at most finitely many non-congruent domains with the same Steklov spectrum. |
The paper is devoted to optimization of resonances associated with 1-D wave equations in inhomogeneous media. The medium's structure is represented by a nonnegative function B. The problem is to design for a given $\alpha \in \R$ a medium that generates a resonance on the line $\alpha + ı\R$ with a minimal possibl... |
We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general matrix-valued Schrödinger operators on a half-line. |
We deal with a spectral problem for the Laplace-Beltrami operator posed on a stratified set $\Omega$ which is composed of smooth surfaces joined along a line $\gamma$, the junction. Through this junction we impose the Kirchhoff-type vertex conditions, which imply the continuity of the solutions and some balance for no... |
We prove that the resonances of long range perturbations of the (semiclassical) Laplacian are the zeroes of natural perturbation determinants. We more precisely obtain factorizations of these determinants of the form $ \prod_{w = {\rm resonances}}(z-w) \exp (\varphi_p(z,h)) $ and give semiclassical bounds on $ \partia... |
We study operators of Kramers-Fokker-Planck type in the semiclassical limit, assuming that the exponent of the associated Maxwellian is a Morse function with a finite number $n_0$ of local minima. Under suitable additional assumptions, we show that the first $n_0$ eigenvalues are real and exponentially small, and esta... |
We study the distribution of eigenvalues for selfadjoint $h$--pseudodifferential operators in dimension two, arising as perturbations of selfadjoint operators with a periodic classical flow. When the strength $\varepsilon$ of the perturbation is $\ll h$, the spectrum displays a cluster structure, and assuming that $\v... |
In the paper we consider singular spectral Sturm--Liouville problem $-(py')'+(q-\lambda r)y=0$, $(U-1)y^{\vee}+i(U+1)y^{\wedge}=0$, where function $p\in L_{\infty}[0,1]$ is uniformly positive, generalized function $q\in W_2^{-1}[0,1]$ is real-valued, generalized weight function $r\in W_2^{-1}[0,1]$ is positive ... |
We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem and its higher eigenvalue versions via the direct method of calculus of variations. |
For the discrete Schrödinger operator we obtain sharp estimates for the number of negative eigenvalues. |
Let M be a surface with conical singularities, and consider a degenerating family of surfaces obtained from M by removing disks of smaller and smaller radius around a subset of the conical singularities. Such families arise naturally in the study of the moduli space of flat metrics on higher-genus surfaces with bounda... |
Inverse problems for differential pencils with nonlocal conditions are investigated. Several uniqueness theorems of inverse problems from the Weyl-type function and spectra are proved, which are generalizations of the well-known Weyl function and Borg's inverse problem for the classical Sturm-Liouville operator. |
We study the Sturm--Liouville operator $$ <br>T(\varepsilon)y=-\frac{1}{\varepsilon}y''+ p(x)y, $$ with concrete $\mathcal{PT}$-- symmetric potential $p(x) = ix$ and Dirichlet boundary conditions on the segment $[-1,1]$. Here $\varepsilon \in (0, \infty)$ is a physical parameter. |
Given a compact Riemannian manifold $(M,g)$, Chazarain, Hörmander, Duistermaat, and Guillemin study the half-wave trace $\operatorname{HWT}_{M,g}(\tau) \in \mathscr{S}'(\mathbb{R}_\tau)$. From the asymptotics of the half-wave trace as $\tau\to 0$, Hörmander deduces the now standard remainder $\smash{O(\sigma^{d-1}... |
We prove quadratic estimates for complex perturbations of Dirac-type operators, and thereby show that such operators have a bounded functional calculus. As an application we show that spectral projections of the Hodge--Dirac operator on compact manifolds depend analytically on $L_\infty$ changes in the metric. |
In the present article we will give a new proof of the ground state asymptotics of the Dirichlet Laplacian with a Neumann window acting on functions which are defined on a two-dimensional infinite strip or a three-dimensional infinite layer. The proof is based on the analysis of the corresponding Dirichlet-to-Neumann ... |
We investigate the kernels of the transformation operators for one-dimensional Schroedinger operators with potentials, which are asymptotically close to Bohr almost periodic infinite-gap potentials. |
Let M be a compact manifold without boundary. Associated to a metric g on M there are various Laplace operators, for example the de Rham Laplacian on forms and the conformal Laplacian on functions. |
We exhibit limit-periodic Schrödinger operators that are uniformly localized in the strongest sense possible. That is, for these operators there are uniform exponential decay rates such that every element of the hull has a complete set of eigenvectors that decay exponentially off their centers of localization at least... |
The Berezin--Li--Yau and the Kröger inequalities show that Riesz means of order $\geq 1$ of the eigenvalues of the Laplacian on a domain $\Omega$ of finite measure are bounded in terms of their semiclassical limit expressions. We show that these inequalities can be improved by a multiplicative factor that depends only... |
We prove that a two-component mirror-symmetric analytic obstacle in the plane is determined by its resonance poles among such obstacles. The proof is essentially the same as in the interior case (part II of the series). |
A Hardy inequality of the form \[\int_{\tilde{\Omega}} |\nabla f({\bf{x}})|^p d {\bf{x}} \ge (\frac{p-1}{p})^p \int_{\tilde{\Omega}} \{1 + a(\delta, \partial \tilde{\Omega})(\x)\}\frac{|f({\bf{x}})|^p}{\delta({\bf{x}})^p} d{\bf{x}}, \] for all $f \in C_0^{\infty}({\tilde{\Omega}})$, is considered for $p\in (1,\infty)$,... |
We prove the meromorphic extension to C for the resolvent of the Laplacian on a class of geometrically finite hyperbolic manifolds with infinite volume and we give a polynomial bound on the number of resonances. This class notably contains the geometrically finite quotients with rational non-maximal rank cusps previou... |
In this article Dirac operators $A_{\eta, \tau}$ coupled with combinations of electrostatic and Lorentz scalar $\delta$-shell interactions of constant strength $\eta$ and $\tau$, respectively, supported on compact surfaces $\Sigma \subset \mathbb{R}^3$ are studied. In the rigorous definition of these operators the $\d... |
The paper presents estimates for the number of negative eigenvalues of a two-dimensional Schrödinger operator in terms of $L\log L$ type Orlicz norms of the potential and proves a conjecture by N.N. Khuri, A. Martin and T.T. Wu. |
It is proven following [18[ that Laplacians with standard vertex continuous on metric trees and with standard and Dirichlet conditions on arbitrary metric graphs possess an infinite sequence of simple eigenvalues with the eigenfunctions not equal to zero in any non-Dirichlet vertex. |
In this note, under a certain assumption on an affine space of operators, which admit embedded eigenvalues, it is shown that the singular part of the spectral shift function of any pair of operators from this space is an integer-valued function. The proof uses a natural decomposition of Pushnitski's $\mu$-invarian... |
We establish necessary and sufficient conditions for the discreteness of spectrum and strict positivity of magnetic Schrödinger operators with a positive scalar potential. They are expressed in terms of Wiener's capacity and the local energy of the magnetic field. |
Spectral asymptotics of the Neumann problem for the Sturm-Liouville equation with generalized derivative of a self-similar generalized Cantor type function as a weight are considered. The spectrum is shown to have a periodicity property for a wide class of Cantor type self-similar functions, and also the weaker "q... |
Time-frequency concentration operators restrict the integral analysis-synthesis formula for the short-time Fourier transform to a given compact domain. We estimate how much the corresponding eigenvalue counting function deviates from the Lebesgue measure of the time-frequency domain. |
The purpose of this paper is to give a systematic description of potentials decaying to zero at infinity, which generate eigenvalues at the edge of the absolutely continuous spectrum when combined with non-local operators defined by Bernstein functions of the Laplacian. By introducing suitable Hölder-Zygmund type spac... |
We classify all sets of the form $\bigcup_{t\in\mathbb{R}}\mathrm{spec}(A+tB)$ where $A$ and $B$ are self-adjoint operators and $B$ is bounded, non-negative, and non-zero. We show that these sets are exactly the complements of discrete subsets of $\mathbb{R}$, that is, of at most countable subsets of $\mathbb{R}$ that... |
In this paper, we study the direct and inverse spectral problems for the Schrodinger operator with two generalized Regge boundary conditions. For the direct problem, we give the properties of the spectrum, including the asymptotic distribution of the eigenvalues. |
We consider Fredholm determinants of the form identity minus product of spectral projections corresponding to isolated parts of the spectrum of a pair of self-adjoint operators. We show an identity relating such determinants to an integral over the spectral shift function in the case of a rank-one perturbation. |
Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A, B in the self-adjoint Jacobi operator H=AS^+ + A^-S^- + B (with S^\pm the right/left shift operators on the lattice Z) and the sp... |
The purpose of this article is to study pseudospectral properties of the one-dimensional Schrödinger operator perturbed by a complex steplike potential. By constructing the resolvent kernel, we show that the pseudospectrum of this operator is trivial if and only if the imaginary part of the potential is constant. |
We provide the full asymptotic description of the quasi-normal modes (resonances) in any strip of fixed width for Dirac fields in slowly rotating Kerr-Newman-de Sitter black holes. The resonances split in a way similar to the Zeeman effect. |
Let $(M,g)$ be a compact, boundaryless, Riemannian manifold whose geodesic flow on its unit sphere bundle is Anosov. Consider the (semiclassical) Laplace-Beltrami operator on $M$. |
A completion problem to recover a rational matrix function which is j-unitary on the line is treated. A Dirac type system with singularities on the semiaxis is recovered explicitly by its left reflection coefficient. |
This paper is devoted to a general min-max characterization of the eigenvalues in a gap of the essential spectrum of a self-adjoint unbounded operator. We prove an abstract theorem, then we apply it to the case of Dirac operators with a Coulomb-like potential. |
The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator valued Titchmarsh--Weyl $m$-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries, Schrödinge... |
The paper contains two results on the equivalence classes of block Jacobi matrices: first, that the Jacobi matrix of type 2 in the Nevai class has A_n coefficients converging to 1, and second, that under an L1-type condition on the Jacobi coefficients, equivalent Jacobi matrices of type 1, 2 and 3 are pairwise asymptot... |
In this paper we study the eigenvalue sums of Dirichlet Laplacians on bounded domains. Among our results we establish an improvement of the Li-Yau bound in the presence of a constant magnetic field. |
This work deals with the interior transmission eigenvalue problem: $y'' + {k^2}\eta \left( r \right)y = 0$ with boundary conditions ${y\left( 0 \right) = 0 = y'\left( 1 \right)\frac{\sin k}{k} - y\left( 1 \right)\cos k},$ where the function $\eta(r)$ is positive. We obtain the asymptotic distribution of no... |
We use a model operator approach and the spectral theorem for self-adjoint operators in a Hilbert space to derive the basic results of abstract left-definite theory in a straightforward manner. The theory is amply illustrated with a variety of concrete examples employing scales of Hilbert spaces, fractional Sobolev sp... |
We obtain the formula computing the number of isomorphic classes of ramification systems with characters over group $S_n$ ($n \not=6$) and their representatives. |
In this paper we study the self-adjoint Krein-von Neumann realization $A_K$ of the perturbed Laplacian $-\Delta+V$ in a bounded Lipschitz domain $\Omega\subset\mathbb{R}^n$. We provide an explicit and self-contained description of the domain of $A_K$ in terms of Dirichlet and Neumann boundary traces, and we establish ... |
We improve the classical discrete Hardy inequality \begin{equation*}\label{1} \sum _{n=1}^{\infty }a_{n}^{2}\geq \left({\frac {1}{2}}\right)^{2} \sum _{n=1}^{\infty }\left({\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right)^{2}, \end{equation*} where $\{a_n\}_{n=1}^\infty$ is any sequence of non-negative real numbers. |
We consider a linear finite spring mass system which is perturbed by modifying one mass and adding one spring. From knowledge of the natural frequencies of the original and the perturbed systems we study when masses and springs can be reconstructed. |
We study the minimum of the essential spectrum of canonical systems $Ju'=-zHu$. Our results can be described as a generalized and more quantitative version of the characterization of systems with purely discrete spectrum, which was recently obtained by Romanov and Woracek [6]. |
We give sufficient conditions for the presence of the absolutely continuous spectrum of a Schrödinger operator on a regular rooted tree without loops (also called regular Bethe lattice or Cayley tree). |
We consider a semiclassical asymptotics of local trace for the 3D-Schroedinger operator with self-generated magnetic field; it is given by Weyl expression with O(h^{-1}) error and under standard condition to Hamiltonian trajectories even o(h^{-1}). In comparison to v1,2 errors are corrected, new results are added and ... |
Given two compact Riemannian manifolds with boundary $M_1$ and $M_2$ such that their respective boundaries $\Sigma_1$ and $\Sigma_2$ admit neighborhoods $\Omega_1$ and $\Omega_2$ which are isometric, we prove the existence of a constant $C$, which depends only on the geometry of $\Omega_1\cong\Omega_2$, such that $|\si... |
We introduce a family of trees that interpolate between the Bethe lattice and $\bbZ$. We prove complete localization for the Anderson model on any member of that family. |
Consider an h-pseudodifferential operator P, whose symbol extends holomorphically to a tubular neighborhood of the real phase space and converges sufficiently fast to 1, so that the determinant of P is well-defined. We show that the modulus of this determinant is asymptotically bounded by an exponential of the integra... |
We consider off-diagonal Jacobi matrices $J$ with (faster-than-exponential) sparse perturbations. We prove (Theorem \ref{onehalf}) that the Fourier transform $\hat{\left\| f\right\| ^{2}d\rho}(t)$ of the spectral measure $\rho $ of $J$, whose sparse perturbations are at least separated by a distance $\exp \left(cj(\ln... |
We study the asymptotic distribution of the resonances near the Landau levels $\Lambda\_q =(2q+1)b$, $q \in \mathbb{N}$, of the Dirichlet (resp. Neumann, resp. Robin) realization in the exterior of a compact domain of $\mathbb{R}^3$ of the 3D Schr{ö}dinger operator with constant magnetic field of scalar intensity $b\te... |
We consider the Courant-sharp eigenvalues of the Robin Laplacian for bounded, connected, open sets in $\mathbb{R}^n$, $n \geq 2$, with Lipschitz boundary. We prove Pleijel's theorem which implies that there are only finitely many Courant-sharp eigenvalues in this setting as well as an improved version of Pleijel&#... |
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