Italo Guarneri

Formal Eigenfunctions
Sequences {u n (x, E)} that solve L n u n (x, E) = Eu n (x, E) for all
integer n ≥ 0 and satisfy the matching conditions. May be written
as
u n (x, E) = C(n, E) v n (x, E)
where v n (x) = normalized sol. of L n v n = Ev n , and constants
C(n, E) satisfy:
2nd order Difference Equation
h 2 (n, E)C(n + 2, E) + h 1 (n, E)C(n + 1, E) + h 0 (n, E)C(n, E) = 0 , (n
with ”initial” condition:
h 2 (−1, E)C(1, E) = −h 1 (−1, E)C(0, E) .