Integral transforms, spectral representation and the d-bar prolem

An alternative definition of the spectrum of a linear differential operator is introduced and a constructive procedure for finding the associated spectral representation is given. This construction is based on the solution of a Riemann-Hilbert or of a d-bar problem. The general theory is illustrated by analysing the following three examples: (a) the basic differential operator of order n, ∂nx⪖ 0; (b) the time-independent Schrödinger operator, ∂2x+ u(x)⪖ 0; (c) the differential operator with a complex non-constant leading coefficient (1 - il′∣(x))−1∂x, 0 ⪕ x ⪕ X. The general spectral representations associated with these operators are used to study several particular boundary conditions for them. We also discuss an extension of the spectral theory from ordinary differential operators to partial differential equations. In particular, we discuss a general linear dispersive evolution equation on x⪖0, 0⪕t⪕T.