The isospectrality problem for some vector-valued Sturm-Liouville boundaryproblems

In this paper we characterize and partially construct the family of all vec tor-valued Sturm-Liouville problems that are isospectral, in a certain sens e, to a given vector-valued Sturm-Liouville problem -y'' + Qo(x)y = lambda y, 0 less than or equal to x less than or equal to p i (1) y'(0) - h(o)y(0) = 0, (2) y'(pi) + H(o)y(pi) = 0, (3) where h(o) and H-o are real symmetric constant dxd matrices and Q(o)(x) is a matrix with continuously differentiable real entries that is symmetric fo r each x. The special sense of isospectrality involves a condition on the s ets of all possible initial values y(0) of each of the problems: these sets must be equal for each lambda. This condition implies (but is not equivale nt to) the condition that the spectra are equal and have the same multiplic ities. We show that two vector-valued Sturm-Liouville problems are isospect ral in this sense if and only if there exists a "transmutation" between the two problems that has the form I+K, where K is a Volterra operator whose k ernel K(x, y) solves a certain overdetermined Goursat problem.