A Spectral Theorem for Sturm–Liouville Operators with Potentials That Are Finite Sums of Exponentials

Abstract

We study eigenfunction expansions for the differential operator L generated by the differential expression l(y) ≡ −y′′ + q(x)y in the space L2(−∞; +∞) under the assumption that the potential q has the form q(x) = ∑K l=1 cle l, where cl ∈ C and γl > 0. The operator L is not self-adjoint except in the trivial case q(x) ≡ 0. At present, nonself-adjoint differential operators with almost periodic coefficients are under intensive study. The spectrum of an operator with periodic potential representable by a sum of an absolutely convergent series in the exponentials e, as well as the corresponding eigenfunction expansions, was analyzed in [1]. The operator L generated in L2(−∞; +∞) by the differential expression