A characterization of some even vector-valued Sturm--Liouville problems

Анотація

We call `even` a Sturm-Liouville problem

-y″+Q(x)y=ly, 0≤x≤ p,(1)

y`(0)-hy(0)=0, (2)

y`(p)+Hy(p)=0 (3)

in which H=h and Q(p-x)≡Q(x) on [0,p]. In this paper we study the vector-valued case, where the Q(x) is a real symmetric d ´ d matrix for each x in [0,p], and the entries of Q and their first derivatives (in the distribution sense) are all in L²[0,p]. We assume that h and H are real symmetric d ´ d matrices.

We prove that a vector-valued Sturm-Liouville problem (1)-(3) is even if and only if, for each eigenvalue l, whose multiplicity is r = r_l (where 1≤ r≤d, and where j₁(x,l),...,j_r(x,l) denote orthonormal eigenfunctions belonging to l), there exists an r ´ r matrix A=(a_ij) (which may depend on l and on the choice of basis { j_i(x, l)}^r_i=1, but does not depend on x) such that

(1) A is orthogonal and symmetric,

and

(2) for 1 ≤ i ≤ r, j_i(p,l)=∑ ^r_j=1a_ijj_j(0,l). To some extent our theorem can be considered a generalization of N. Levinson`s results in [2].