On the eigenvalues in problems with spherical symmetry. Ill
Edward Charles Titchmarsh · Proceedings of the Royal Society of London A Mathematical and Physical Sciences · 1959
Abstract Let N(λ) denote the number of eigenvalues not exceeding λ of the three-dimensional equation ∇2Ψ + {λ-q(r)} Ψ = 0 over the whole space. The problem of the behaviour of N(λ) as λ → ∞ is considered in the case where q(r — rc, c being a constant. It is shown that if c = 4 or 6 N(λ) = aμ3 + bμ2 + O(μ5/3), where μ =λ1/2+1/c and a and b are constants. This result is derived from a theorem due to van der Corput on the lattice-points in a region of a general type. It does not hold in the case c = 2, which is exceptional.