Decay Law Regularity

Abstract

We conjecture that the non-decay probability in a model of decay through a δ barrier may be nowhere differentiable as a function of time if the initial state does not belong to the Hamiltonian domain. Mathematics Subject Classification (2010). 81Q05, 81Q80.Keywords. Non-decay probability, δ barrier.Let H α be a Hamiltonian on L 2 (R + ) with a δ interaction of strength α > 0 at a point a > 0 and Dirichlet condition at the origin, that is, H α ψ = −ψ with the domain of all ψ ∈ W 2,2 (R + \{a}) satisfying the boundary conditions ψ(0) = 0 and ψ(a+) = ψ(a−) =: ψ(a), ψ (a+) − ψ (a−) = αψ(a).Let P be the projection onto L 2 (0, a) in L 2 (R + ). Given ψ 0 ∈ L 2 (R + ) such that P ψ 0 = ψ 0 , consider the decay law, or survival probability at time t > 0,A numerical analysis of this and similar models [1] for a particular choice of ψ 0 gives a highly irregular function. A conjecture to be considered is that for ψ 0 (a−) = 0 the function P ψ0 (·) might be nowhere differentiable.