COMPLEX PATH-INTEGRAL REPRESENTATION FOR SEMICLASSICAL LINEAR FUNCTIONALS

Semiclassical linear Functionals are characterized by the distribution al equation D(phi L)+ psi L = 0 where phi and psi are arbitrary polyno mials With the condition deg(psi) greater than or equal to l. Two case s are considered: (A) deg(phi) > deg(psi) (B) deg(phi) less than or eq ual to deg(psi). In an earlier work by the authors (J. Comput. Appl. M ath. 57 (1995), 239-249) integral representations are given for semicl assical functionals in case (A). Here the problem is continued and cas e (B) is solved: it is always possible to choose some path y in the co mplex plane such that every solution, regular or not, of D(phi L) + ps i L = 0 can be represented in the form [L, p] = integral(y) w(z) p(z) dz where w(z) is a solution of the differential equation (phi w)' + ps i w = 0. In some cases, the expression for L is a singular integral an d a regularization process is given. (C) 1998 Academic Press.