Abstract. Let Ω be some domain in the hyperbolic space H n (with n ≥ 2) and S 1 the geodesic ball that has the same first Dirichlet eigenvalue as Ω. We prove the Payne-Pólya-Weinberger conjecture for H n , i.e., that the second Dirichlet eigenvalue on Ω is smaller or equal than the second Dirichlet eigenvalue on S 1 .We also prove that the ratio of the first two eigenvalues on geodesic balls is a decreasing function of the radius.

The purpose of this manuscript is to present a series of lecture notes on isoperimetric inequalities for the Laplacian, for the Schrödinger operator, and related problems.

Abstract. Let λ i (Ω, V ) be the ith eigenvalue of the Schrödinger operator with Dirichlet boundary conditions on a bounded domain Ω ⊂ R n and with the positive potential V . Following the spirit of the Payne-Pólya-Weinberger conjecture and under some convexity assumptions on the spherically rearranged potential V ⋆ , we prove that λ 2 (Ω, V ) ≤ λ 2 (S 1 , V ⋆ ). Here S 1 denotes the ball, centered at the origin, that satisfies the condition λ 1 (Ω, V ) = λ 1 (S 1 , V ⋆ ).Further we prove under the same convexity assumptions on a spherically symmetric potential V , that λ 2 (B R , V )/λ 1 (B R , V ) decreases when the radius R of the ball B R increases.We conclude with several results about the first two eigenvalues of the Laplace operator with respect to a measure of Gaussian or inverted Gaussian density.

We prove new inequalities of the Lieb-Thirring type on the eigenvalues of Schrödinger operators in wave guides with local perturbations. The estimates are optimal in the weak-coupling case. To illustrate their applications, we consider, in particular, a straight strip and a straight circular tube with either mixed boundary conditions or boundary deformations.