On the limit of the coefficients of the eigenfunction series associated with a certain non-self-adjoint differential system

«»» — q(x)u = uxt — p(x)ut where u(x, t) must satisfy the conditions u(a, t)=u(b, /)=0 and u(x, 0) =F(x). In attempting to solve equation (1) by separation of variables, one is led to the consideration of expanding an arbitrary function F(x) in terms of the eigenfunctions, or nonzero solutions, un(x) of the equation: (2) (A + \B)u = 0 satisfying the conditions u(a)=u(b)=0, where A is the operator d2/dx2+q(x) and B is the operator — d/dx-\-p(x). The system adjoint to (2) is: (3) (A* + \B*)v « 0, v(a) = v(b) = 0 where A =A* and B*=d/dx+p(x). Conditions have been established [2], under which a function F(x) of bounded variation on (a, b) can be expanded in terms of un(x). However, in the expansion F(jc)=E-« anun(x) there are certain properties of the coefficients, an, which differ quite radically from the corresponding properties of the coefficients of certain well-known selfadjoint eigenfunction expansions. For example, if Bn are the Fourier coefficients of a function g(x), it is well known that limn,«, Bn = 0. However, in the expansion i(x)=E-» a„u»(x), it is found that limn_M an is not in general equal to zero;. Consequently, the series 22? °n> unlike the corresponding series of Fourier coefficients, does not in general converge.