We solve the Regge-Wheeler equation for axial perturbations of the Schwarzschild metric in the black hole interior in terms of Heun’s functions and give a description of the spectrum and the eigenfunctions of the interior problem. The phenomenon of attraction and repulsion of the discrete eigenvalues of gravitational waves is discovered.

Various examples of exactly solvable ‘discrete’ quantum mechanics are explored explicitly with emphasis on shape invariance, Heisenberg operator solutions, annihilationcreation operators, the dynamical symmetry algebras and coherent states. The eigenfunctions are the (q-)Askey-scheme of hypergeometric orthogonal polynomials satisfying difference equation versions of the Schrödinger equation. Va...

We study the one-dimensional discrete quasi-periodic Schrödinger equation −ϕ(n + 1) − ϕ(n − 1) + λV (x + nω)ϕ(n) = Eϕ(n), n ∈ Z We show that for " typical " C 3 potential V , if the coupling constant λ is large, then for most frequencies ω, the Lyapunov exponent is positive for all energies E, and the corresponding eigenfunctions ϕ decay exponentially.