We ask whether the only multiplicities in the spectrum of the clamped round plate are trivial, i.e., whether all existing multiplicities are due to the isometries of the sphere, or, equivalently, whether any eigenfunction is separated. We prove that any eigenfunction can be expressed as a sum of at most two separated ones, by showing that otherwise the corresponding eigenvalue is algebraic, contradicting the Siegel–Shidlovskii theory. In two dimensions, it follows that no eigenvalue is of multiplicity greater than four. The proof exploits a linear recursion of order two for cross-product Bessel functions with coefficients which are not even algebraic functions, though they do satisfy a non-linear algebraic recursion.

The symmetrized Asymptotic Mean Value Laplacian \tilde{\Delta} , obtained as limit of approximating operators \tilde{\Delta}_{r} , is an extension of the classical Euclidean Laplace operator to the realm of metric measure spaces. We show that, as r \downarrow 0 , the operators \tilde{\Delta}_{r} eventually admit isolated eigenvalues defined via min-max procedure on any compact uniformly locally doubling metric measure space. Then we prove L^{2} and spectral convergence of \tilde{\Delta}_{r} to the Laplace–Beltrami operator of a compact Riemannian manifold, imposing Neumann conditions when the manifold has a non-empty boundary.

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 on the dimension and the product \sqrt{\Lambda} |\Omega|^{1/d} , where \Lambda is the eigenvalue cut-off parameter in the definition of the Riesz mean. The same holds when |\Omega|^{1/d} is replaced by a generalized inradius of \Omega . Finally, we show similar inequalities in two dimensions in the presence of a constant magnetic field.

The main result of this paper are novel two-sided estimates of the essential resolvent norm for closed linear operators T . We prove that the growth of \|(T-\lambda)^{-1}\|_{\textup{e}} is governed by the distance of a point \lambda\in{}\rho(T){}\setminus{}W_{\textup{e}}(T) to the essential numerical range W_{\textup{e}}(T) . We extend these bounds even to points \lambda{}\in \mathbb{C}\setminus W_{\textup{e}}(T) outside the resolvent set \rho(T) with (T{}-{}\lambda)^{-1} replaced by the Moore–Penrose resolvent (T-\lambda)^{\dagger}{} . We use similar ideas to prove essential growth bounds in terms of the real part of the essential numerical range of generators of C_{0} -semigroups. Further, we study the essential approximate point spectrum \sigma_{{\textup{eap}}}(T) and the essential minimum modulus \gamma_{\textup{e}}(T) , in particular, their relations to the various essential spectra and the essential norm of the Moore–Penrose inverse, respectively. An important consequence of our results are new perturbation results for the spectra and essential spectra (of type 2) for accretive and sectorial T . Applications e.g. to Schrödinger operators with purely imaginary rapidly oscillating potentials in \mathbb{R}^{d} illustrate our results.

We consider certain non-integer base \beta -expansions of Parry’s type and we study various properties of the transfer (Perron–Frobenius) operator \mathcal{P}\colon L^{p}([0,1])\to L^{p}([0,1]) with p\geq 1 and its associated composition (Koopman) operator, which are induced by a discrete dynamical system on the unit interval related to these \beta -expansions.We show that if f is Lipschitz, then the iterated sequence \{\mathcal{P}^{N} f\}_{N\geq 1} converges exponentially fast (in the L^{1} norm) to an invariant state corresponding to the eigenvalue 1 of \mathcal{P} . This “attracting” eigenvalue is not isolated: for 1\leq p\leq 2 we show that the point spectrum of \mathcal{P} also contains the whole open complex unit disk and we explicitly construct an eigenfunction for every z with |z|<1 .

The original Calderón problem consists in recovering the potential (or the conductivity) from the knowledge of the related Neumann to Dirichlet map (or Dirichlet to Neumann map). Here, we first perturb the medium by injecting small-scaled and highly heterogeneous particles. Such particles can be bubbles or droplets in acoustics or nanoparticles in electromagnetism. They are distributed, periodically for instance, in the whole domain where we want to do reconstruction. Under critical scales between the size and contrast, these particles resonate at specific frequencies that can be well computed. Using incident frequencies that are close to such resonances, we show that (1) the corresponding Neumann to Dirichlet map of the composite converges to the one of the homogenised medium. In addition, the equivalent coefficient, which consists in the sum of the original potential and the effective coefficient, is negative valued with a controllable amplitude; (2) as the equivalent coefficient is negative valued, then we can linearise the corresponding Neumann to Dirichlet map using the effective coefficient’s amplitude; (3) from the linearised Neumann to Dirichlet map, we reconstruct the original potential using explicit complex geometrical optics solutions (CGOs).

In this paper, we explicitly provide expressions for a sequence of orthogonal polynomials associated with a weight matrix of size N , constructed from a collection of scalar weights w_{1}, \ldots, w_{N} of the form W(x) = T(x)\operatorname{diag}(w_{1}(x), \ldots, w_{N}(x))T(x)^{\ast} , where T(x) is a specific polynomial matrix. We provide sufficient conditions on the scalar weights to ensure that the weight matrix W is irreducible. Furthermore, we give sufficient conditions on the scalar weights to ensure that each term in the constructed sequence of matrix orthogonal polynomials is an eigenfunction of a differential operator. We also study the Darboux transformations and bispectrality of the orthogonal polynomials in the particular case where the scalar weights are the classical weights of Jacobi, Hermite, and Laguerre. With these results, we construct a wide variety of bispectral matrix-valued orthogonal polynomials of arbitrary size, which satisfy a second-order differential equation.

Inequalities between Dirichlet and Neumann eigenvalues of the Laplacian and of other differential operators have been intensively studied in the past decades. The aim of this paper is to introduce differential forms and the de Rham complex in the study of such inequalities. We show how differential forms lie hidden at the heart of the work of Rohleder on inequalities between Dirichlet and Neumann eigenvalues for the Laplacian on planar domains. Moreover, we extend the ideas of Rohleder to a new proof of Friedlander’s inequality for any bounded Lipschitz domain.

We obtain the spectral and resolvent estimates for semiclassical pseudodifferential operators with symbol of Gevrey- s regularity, near the boundary of the range of the principal symbol. We prove that the boundary spectrum free region is of size {\mathcal{O}}(h^{1-\frac{1}{s}}) where the resolvent is at most fractional exponentially large in h , as the semiclassical parameter h\to 0^{+} . This is a natural Gevrey analogue of a result by N. Dencker, J. Sjöstrand, and M. Zworski in the C^{\infty} and analytic cases.

We consider the graph whose edges connect all nearest neighbours of the lattice \mathbb{Z}^{d} . We prove several theorems establishing localization of eigenfunctions of the Schrödinger operator on this graph. The energies for which we establish the localization belong to specific intervals whose union is an unbounded subset of the real line.