New Characterizations for the Eigenvalues of the Prolate Spheroidal Wave Equation

Abstract

In this paper, we give new characterizations for the eigenvalues of the prolate wave equation as limits of the zeros of some families of polynomials: the coefficients of the formal power series appearing in the solutions near 0, 1, or ∞ (in the variables x,x−1, or 1/x, respectively). The result, which seems to be true for all values of the parameter τ, according to our numerical experiments, is here proved for small values of the parameter τ.

The UV prolate spectrum matches the zeros of zeta

“…ref. 12 ) together with the fact that imply that the logarithmic singularities of ξ on the left and the right of have to match. This reduces the number of parameters to four.…”

Prolate spheroidal operator and Zeta

“…In fact as shown in Proposition 14 of [13], the coefficients of this expansion are directly related to the coefficients of the expansion of the finite solution at λ and taking for simplicity λ = 1, if the latter is of the form…”

“…Any ξ ∈ Dom(W max ) satisfying W ξ = ±iξ is a piecewise real analytic function and is uniquely specified by six parameters in the complement of the two regular singular points ±λ. The known form of the solutions (cf.. [13]) together with the fact that W ξ ∈ L 2 (R) imply that the logarithmic singularities of ξ on the left and the right of ±λ have to match. This reduces the number of parameters to 4.…”

“…This follows from the above characterization ( 16), ( 17), (18) of the domain of W sa combined with the known bases of formal solutions for the equation W ξ = µξ, µ ∈ R, around ±λ and ±∞ (cf. [13]).…”

“…We can then find the asymptotic expansion at ∞ using the boundary condition that the leading term there is sin(2πλy) y . We take for simplicity λ = 1 and use [13] to get for the tentative eigenvector for eigenvalue µ the expansion at ∞…”