Mathematics

We prove that eigenfunctions of the Laplacian on a compact hyperbolic surface delocalise in terms of a geometric parameter dependent upon the number of short closed geodesics on the surface. In particular, we show that an L 2 normalised eigenfunction restricted to a measurable subset of the surface has squared L 2 -norm ε>0 , only if the set has a relatively large size -- exponential in the geometric parameter. For random surfaces with respect to the Weil-Petersson probability measure, we then show, with high probability as g→∞ , that the size of the set must be at least the genus of the surface to some power dependent upon the eigenvalue and ε .

We explicitly express the spectral determinant of Friederichs Dirichlet Laplacians on the 2-dimensional hyperbolic (Gaussian curvature -1) cones in terms of the cone angle and the geodesic radius of the boundary. The related results in the recent paper "Riemann-Roch isometries in the non-compact orbifold setting," J. Eur. Math. Soc. 22 (2020) by G. Freixas i Montplet and A. von Pippich turn out to be incorrect.

We prove a quantitative deterministic equivalence theorem for the logarithmic potentials of deterministic complex N×N matrices subject to small random perturbations. We show that with probability close to 1 this log-potential is, up to a small error, determined by the singular values of the unperturbed matrix which are larger than some small N -dependent cut-off parameter.

This paper is the first of a series where we study the spectral properties of Dirac operators with the Coulomb potential generated by any finite signed charge distribution μ . We show here that the operator has a unique distinguished self-adjoint extension under the sole condition that μ has no atom of weight larger than or equal to one. Then we discuss the case of a positive measure and characterize the domain using a quadratic form associated with the upper spinor, following earlier works by Esteban and Loss. This allows us to provide min-max formulas for the eigenvalues in the gap. In the event that some eigenvalues have dived into the negative continuum, the min-max formulas remain valid for the remaining ones. At the end of the paper we also discuss the case of multi-center Dirac-Coulomb operators corresponding to μ being a finite sum of deltas.

Consider the Coulomb potential −μ∗|x | −1 generated by a non-negative finite measure μ . It is well known that the lowest eigenvalue of the corresponding Schrödinger operator −Δ/2−μ∗|x | −1 is minimized, at fixed mass μ( R 3 )=ν , when μ is proportional to a delta. In this paper we investigate the conjecture that the same holds for the Dirac operator −iα⋅∇+β−μ∗|x | −1 . In a previous work on the subject we proved that this operator is self-adjoint when μ has no atom of mass larger than or equal to 1, and that its eigenvalues are given by min-max formulas. Here we consider the critical mass ν 1 , below which the lowest eigenvalue does not dive into the lower continuum spectrum for all μ≥0 with μ( R 3 )< ν 1 . We first show that ν 1 is related to the best constant in a new scaling-invariant Hardy-type inequality. Our main result is that for all 0≤ν< ν 1 , there exists an optimal measure μ≥0 giving the lowest possible eigenvalue at fixed mass μ( R 3 )=ν , which concentrates on a compact set of Lebesgue measure zero. The last property is shown using a new unique continuation principle for Dirac operators. The existence proof is based on the concentration-compactness principle.

The matrix Sturm-Liouville operator on a finite interval with the boundary conditions in the general self-adjoint form and with the singular potential from the class W −1 2 is studied. This operator generalizes Sturm-Liouville operators on geometrical graphs. We investigate structural and asymptotical properties of the spectral data (eigenvalues and weight matrices) of this operator. Furthermore, we prove the uniqueness of recovering the operator from its spectral data, by using the method of spectral mappings.

For a given self-adjoint operator A with discrete spectrum, we completely characterize possible eigenvalues of its rank-one perturbations~ B and discuss the inverse problem of reconstructing B from its spectrum.

We consider boundary conditions of self-adjoint banded Toeplitz matrices. We ask if boundary conditions exist for banded self-adjoint Toeplitz matrices which satisfy operator inequalities of Dirichlet-Neumann bracketing type. For a special class of banded Toeplitz matrices including integer powers of the discrete Laplacian we find such boundary conditions. Moreover, for this class we give a lower bound on the spectral gap above the lowest eigenvalue.

Under minimal regularity conditions on the photon dispersion and the coupling function, we prove that the spin-boson model with two massless photons in R d cannot have more than two bound state energies whenever the coupling strength is sufficiently strong.

All the known counterexamples to Kac' 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" isometries mapping the copies to each other. We prove and explore a complementary positive statement: if an operator intertwining two appropriate realisations of the Laplacian on a pair of domains preserves disjoint supports, then under additional assumptions on it generally far weaker than unitarity, the domains are congruent. We show this in particular for the Dirichlet, Neumann and Robin Laplacians on spaces of continuous functions and on L 2 -spaces.