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...

In his comment [1] on our paper [2], Pikovsky contrasts two definitions for the phase of a stochastic oscillator by way of an analytically solvable model system. In [3] the phase is defined in terms of a system of isochrons Σθ, analogous to Poincaré sections, with the property that for any initial condition on one isochron Σa, the mean first passage time (MFPT) to a second isochron Σb, b > a, w...

We consider initial value/boundary value problems for fractional diffusion-wave equation: ∂ α t u(x, t) = Lu(x, t), where 0 < α ≤ 2, where L is a symmetric uniformly elliptic operator with t-independent smooth coefficients. First we establish the unique existence of ths weak solutions and the asymptotic behaviour as the time t goes to ∞ and the proofs are based on the eigenfunction expansions. ...

We prove an uncertainty principle for certain eigenfunction expansions on L2(R+,w(r)dr) and use it to analogues of theorems Chernoff Ingham Laplace-Beltrami operators compact symmetric spaces, special Hermite operator Cn Rn.

is completely integrable and hence L(k) is in a commuting system of differential operators with n algebraically independent operators. Then we have the following fundamental result (cf. [1]). Theorem [Heckman, Opdam]. When kα are generic, the function F (λ, k;x) has an analytic extension on R and defines a unique simultaneous eigenfunction of the commuting system of differential operators with ...