On eigenfunctions corresponding to a small resurgent eigenvalue

Resurgent Analysis of the Witten Laplacian in One Dimension

“…It is important, however, that locations of singularities of majors Ψ r (E r , ξ) of ψ r (E r , h) = ψ(E r h, h) do not depend on E r , and it is expected from the general theory that majors Ψ r (E r , ξ) are analytic with respect to E r . In [15] we have proven that for a small resurgent function φ(h) the composite function ψ r (φ(h), h) is again resurgent and has a hyperasymptotic expansion that one would expect from formal manipulation with hyperasymptotic expansions of φ and ψ r .…”

“…The outcome of the section IX is that the quantization condition that we set up treating E r as a complex number, can be solved for E r and the solution will be a resurgent function in h if all the ingredients µ j and τ j of the quantization condition are resurgent (plus some additional technical conditions). Note that we can substitute small resurgent functions for E r in the equation ( 6) for E = hE r , as we discussed in detail in [15].…”

“…In these cases we expect that the major of ψ(E, q, h), resp., ψ(E r , q, h) can be chosen to holomorphically depend on E, respectively, on E r , and on q. We proved in [15] that this would imply that the equation ( 4) also has resurgent solutions when E = hϕ(h) and ϕ(h) is a small resurgent function.…”

“…We refer to [15] for a proof that if g(t) is a holomorphic function in the neighborhood of the origin and ϕ(h) is a small resurgent function, then g(ϕ(h)) is again resurgent. In particular, in this case the function 1/(c + ϕ(h)) is resurgent for 0 = c ∈ C, and therefore functions representable by resurgent asymptotic expansions B e −c/h (a 0 + a 1 h + a 2 h 2 + ...) with a 0 = 0 have a resurgent inverse.…”

“…In fact, we will presume that E r is a number to set up this equation, then solve it and find its resurgent solutions, then by [15] one can substitute E r into the original equation and obtain its resurgent solutions satisfying the periodic boundary conditions.…”