arXiv subjects
Publications and source records attributed to Luc Hillairet.
We use Krein formula and the S-matrix formalism to give formulas for the zeta-regularized determinant of non-Friedrichs extensions of the Laplacian on Euclidean surfaces with Conical Singularities. This formula involves S(0) and we show that the latter can be expressed using the Bergman projective connection on the underlying Riemann surface.
Let X be a compact Riemannian manifold of dimension two or three and let P be a point of X. We derive comparison formulas relating the zeta-regularized determinant of an arbitrary self-adjoint extension of (symmetric) Laplace operator with domain, consisting of smooth functions with compact supports which does not contain P, to the zeta-regularized determinant of the self-adjoint Laplacian on X.
In specific types of partially rectangular billiards we estimate the mass of an eigenfunction of energy $E$ in the region outside the rectangular set in the high-energy limit. We use the adiabatic ansatz to compare the Dirichlet energy form with a second quadratic form for which separation of variables applies. This allows us to use sharp one-dimensional control estimates and to derive the bound assuming that $E$ is not resonating with the Dirichlet spectrum of the rectangular part.
We give a proof that in settings where Von Neumann deficiency indices are finite the spectral counting functions of two different self-adjoint extensions of the same symmetric operator differ by a uniformly bounded term (see also Birman-Solomjak's 'Spectral Theory of Self-adjoint operators in Hilbert Space') >. We apply this result to quantum graphs, pseudo-laplacians and surfaces with conical singularities.
We describe a method for comparing the real analytic eigenbranches of two families of quadratic forms that degenerate as t tends to zero. One of the families is assumed to be amenable to `separation of variables' and the other one not. With certain additional assumptions, we show that if the families are asymptotic at first order as t tends to 0, then the generic spectral simplicity of the separable family implies that the eigenbranches of the second family are also generically one-dimensional. As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant curvature space form) each eigenspace of the Laplacian is one-dimensional. We also show that for all but countably many t, the geodesic triangle in the hyperbolic plane with interior angles 0, t, and t, has simple spectrum.
We show that the behaviour of analytic eigenbranches of a Schrödinger operator depends on the way eigenfunctions concentrate in the phase space.
Partially rectangular domains are compact two-dimensional Riemannian manifolds $X$, either closed or with boundary, that contain a flat rectangle or cylinder. In this paper we are interested in partially rectangular domains with ergodic billiard flow; examples are the Bunimovich stadium, the Sinai billiard or Donnelly surfaces. We consider a one-parameter family $X_t$ of such domains parametrized by the aspect ratio $t$ of their rectangular part. There is convincing theoretical and numerical evidence that the Laplacian on $X_t$ with Dirichlet or Neumann boundary conditions is not quantum unique ergodic (QUE). We prove that this is true for all $t \in [1,2]$ excluding, possibly, a set of Lebesgue measure zero. This yields the first examples of ergodic billiard systems proven to be non-QUE.
In this note, we extend the results on eigenfunction concentration in billiards as proved by the third author in \cite{M1}. There, the methods developed in Burq-Zworski \cite{BZ3} to study eigenfunctions for billiards which have rectangular components were applied. Here we take an arbitrary polygonal billiard $B$ and show that eigenfunction mass cannot concentrate away from the vertices; in other words, given any neighbourhood $U$ of the vertices, there is a lower bound $$ \int_U |u|^2 \geq c \int_B |u|^2 $$ for some $c = c(U) > 0$ and any eigenfunction $u$.
We introduce a small slit into a planar domain and study the resulting effect upon the eigenvalues of the Laplacian. In particular, we show that as the length of the slit tends to zero, each real-analytic eigenvalue branch tends to an eigenvalue of the original domain. By combining this with our earlier work (arXiv:math/0703616), we obtain the following application: The generic multiply connected polygon has simple spectrum.
We study the Laplace operator with Dirichlet or Neumann boundary condition on polygons in the Euclidean plane. We prove that almost every simply connected polygon with at least four vertices has simple spectrum. We also address the more general case of geodesic polygons in a constant curvature space form.