General theory of the factorization of ordinary linear differential operators

Abstract

The problem of factoring the general ordinary linear differential operator L y = y ( n ) + p n − 1 y ( n − 1 ) + ⋯ + p 0 y Ly = {y^{(n)}} + {p_{n - 1}}{y^{(n - 1)}} + \cdots + {p_0}y into products of lower order factors is studied. The factors are characterized completely in terms of solutions of the equation L y = 0 Ly = 0 and its adjoint equation L ∗ y = 0 {L^ \ast }y = 0 . The special case when L is formally selfadjoint of order n = 2 k n = 2k and the factors are of order k and adjoint to each other reduces to a well-known result of Rellich and Heinz: L = Q ∗ Q L = {Q^ \ast }Q if and only if there exist solutions y 1 , ⋯ , y k {y_1}, \cdots ,{y_k} of L y = 0 Ly = 0 satisfying W ( y 1 , ⋯ , y k ) ≠ 0 W({y_1}, \cdots ,{y_k}) \ne 0 and [ y i ; y j ] = 0 [{y_i};{y_j}] = 0 for i , j = 1 , ⋯ , k i,j = 1, \cdots ,k ; where [ ; ] is the Lagrange bilinear form of L.