We prove delocalization properties for eigenvectors of vertex-transitive graphs via elementary estimates of the spectral projector. We recover in this way known results which were formerly proved using representation theory. Similar techniques show that for general symmetric matrices, most approximate eigenvectors spectrally localized in a given window containing sufficiently many eigenvalues are delocalized in L^{q} norms. Building upon this observation, we prove a delocalization result for approximate eigenvectors of large graphs containing few short loops, under an assumption on the resolvent which is verified in some standard cases, for instance, random lifts of a fixed base graph.

We consider planar elastic beam Hamiltonians defined on hexagonal lattices. These quantum graphs are constructed from Euler–Bernoulli beams, each governed by the fourth-order Schrödinger operator with a real periodic symmetric potential function. In contrast to the second-order Schrödinger operator commonly studied in the quantum graph literature, here vertex matching conditions encode the geometry of the underlying graph by their dependence on angles at which the edges meet.We show that on the hexagonal lattice, the dispersion relation has a structure similar to that reported for the periodic second-order Schrödinger operator, known as the “graphene Hamiltonian.” This property is then utilized to prove the existence of Dirac points (conical singularities). We further discuss the (ir)reducibility of Fermi surfaces. Moreover, we obtain the point spectrum, the absolutely continuous spectrum, and the singular continuous spectrum.Applying perturbation analysis, we derive the dispersion relation for the planar elastic beam Hamiltonians on angle-perturbed irregular hexagonal lattices, defined in a geometric neighborhood of the hexagonal lattice. On these graphs, we find that, unlike the hexagonal lattice, the dispersion relation is not split into purely energy- and quasimomentum-dependent terms; however, Dirac points exist similar to the hexagonal-lattice case.

Eigenvalue interlacing is a useful tool in linear algebra and spectral analysis. In its simplest form, the interlacing inequality states that a rank-one positive perturbation shifts each eigenvalue up, but not further than the next unperturbed eigenvalue. For different types of perturbations, this idea is known as Weyl interlacing, Cauchy interlacing, Dirichlet–Neumann bracketing, and so on.We prove a sharp version of the interlacing inequalities for “finite-dimensional perturbations in boundary conditions,” expressed as bounds on the spectral shift between two self-adjoint extensions of a fixed semibounded symmetric operator with finite and equal defect numbers. The bounds are given in terms of the Duistermaat index, a topological invariant describing the relative position of three Lagrangian planes in a symplectic space. Two of the Lagrangian planes describe the self-adjoint extensions being compared, while the third corresponds to the Friedrichs extension, which acts as a reference point.Along the way, numerous auxiliary results are established, including one-sided continuity properties of the Duistermaat index, smoothness of the Cauchy data space without unique continuation-type assumptions, and a formula for the Morse index of an extension of a non-negative symmetric operator.

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.