In this paper we study the WKB-Langer asymptotic expansion of the eigenfunctions of a Schrödinger operator H = − 12 2 ∂2 ∂x2 + V (x). Then applying these asymptotic formulae we prove that the exact L eigenfunction ΨE(N, ) and its derivative ΨE(N, ) of the Schrödinger operator with a well-shaped analytic potential are approximated up to arbitrary order m by the semi-classical WKB-Langer approxim...

We build a one–parameter family of S−invariant metrics on the unit disc with fixed total area for which the second eigenvalue of the Laplace operator in the case of both Neumann and Dirichlet boundary conditions is simple and has an eigenfunction with a closed nodal line. In the case of Neumann boundary conditions, we also prove that this eigenfunction attains its maximum at an interior point, ...

Furthermore, the exponent of λ is sharp on every such manifold (see e.g., [15]). In the case of a sphere, the examples which prove sharpness are in fact eigenfunctions. For (1.2) the appropriate example is an eigenfunction which concentrates in a λ− 1 2 diameter tube about a geodesic. For (1.3), the example is a zonal eigenfunction of L norm λ n−1 2 which takes on value comparable to λ on a λ−1...