In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible invariant semiclassical measures obtained as limits of Wigner measures corresponding to eigenfunctions. These measures describe simultaneously the concentration...

Abstract. We study extreme values of desymmetrized eigenfunctions (so called Hecke eigenfunctions) for the quantized cat map, a quantization of a hyperbolic linear map of the torus. In a previous paper it was shown that for prime values of the inverse Planck’s constant N = 1/h, such that the map is diagonalizable (but not upper triangular) modulo N , the Hecke eigenfunctions are uniformly bound...

We determine the asymptotics of the joint eigenfunctions of the torus action on a toric Kähler variety. Such varieties are models of completely integrable systems in complex geometry. We first determine the pointwise asymptotics of the eigenfunctions, which show that they behave like Gaussians centered at the corresponding classical torus. We then show that there is a universal Gaussian scaling...

Essential information about the stationary and slow kinetic properties of macromolecules is contained in the eigenvalues and eigenfunctions of the dynamical operator of the molecular dynamics. A recent variational formulation allows to optimally approximate these eigenvalues and eigenfunctions when a basis set for the eigenfunctions is provided. In this study, we propose that a suitable choice ...

We study the spectrum of the linear operator L = −∂θ − ǫ∂θ(sin θ∂θ) subject to the periodic boundary conditions on θ ∈ [−π, π]. We prove that the operator is closed in L2([−π, π]) with the domain in H per([−π, π]) for |ǫ| < 2, its spectrum consists of an infinite sequence of isolated eigenvalues and the set of corresponding eigenfunctions is complete. By using numerical approximations of eigenv...

We study the small scale distribution of the L2 mass of eigenfunctions of the Laplacian on the flat torus Td . Given an orthonormal basis of eigenfunctions, we show the existence of a density one subsequencewhose L2 mass equidistributes at small scales. In dimension two our result holds all the way down to the Planck scale. For dimensions d = 3, 4 we can restrict to individual eigenspaces and s...

An accidental degeneracy of resonances gives rise to a double pole in the scattering matrix, a double zero in the Jost function and a Jordan chain of length two of generalized Gamow-Jordan eigenfunctions of the radial Schrödinger equation. The generalized Gamow-Jordan eigenfunctions are basis elements of an expansion in bound and resonant energy eigenfunctions plus a continuum of scattering wav...

We report the identification and construction of raising and lowering operators for the complete eigenfunctions of isotropic harmonic oscillators confined by dihedral angles, in circular cylindrical and spherical coordinates; as well as for the hydrogen atom in the same situation of confinement, in spherical, parabolic and prolate spheroidal coordinates. The actions of such operators on any eig...

We review some recent results on asymptotic properties of polynomials of large degree, of general holomorphic sections of high powers of positive line bundles over Kähler manifolds, and of Laplace eigenfunctions of large eigenvalue on compact Riemannian manifolds. We describe statistical patterns in the zeros, critical points and Lp norms of random polynomials and holomorphic sections, and the ...

We present a computational method for solving a class of boundary-value problems in Sturm-Liouville form. The algorithms are based on global polynomial collocation methods and produce discrete representations of the eigenfunctions. Error control is performed by evaluating the eigenvalue problem residuals generated when the eigenfunctions are interpolated to a finer discretization grid; eigenfun...