De. Edmunds et Rm. Kauffman, CONTINUOUS-SPECTRUM EIGENFUNCTION-EXPANSIONS AND THE CAUCHY-PROBLEM IN L1, Proceedings - Royal Society. Mathematical and physical sciences, 441(1912), 1993, pp. 407-422
We consider the Cauchy problem a partial derivative PSI = -tauPSI, PSI
(x, 0) = f(x), where T is a ordinary differential operator in x (of or
der at least 2), x belongs to an unbounded interval I subset-of R, and
f is-an-element-of L1(I). The fact that f does not belong to L2(I) to
gether with the general nature of the differential expression tau prec
lude the application of classical methods in the case where the order
of tau is more than 2; instead we use a continuous spectrum eigenfunct
ion expansion, developed in the paper, to obtain a solution PSI(f) of
the problem which depends continuously on f in an appropriate sense an
d also converges to f in a natural topology as t --> 0. The solution d
epends upon a kernel function which may, in particular cases, be calcu
lated explicitly. Questions of approximation of the solution by finite
sums are also considered.