On Hill's Equation with a Singular Complex-valued Potential

1. Introduction The differential equation y00 ‡ qy ˆ Ey is called Hill's equation if q is a realvalued, periodic, locally integrable function of a real variable. It is very well known that the spectrum of the operator associated with this equation consists of a countably in®nite number of compact intervals of the real line. These intervals are called (spectral) bands and are generally separated by open intervals called gaps. In some instances, however, all but a ®nite number of gaps are empty resulting in a spectrum consisting of a ®nite number of compact bands and one closed interval extending to negative in®nity. In this case one calls q a ®nite-band or ®nite-gap potential. The relationship of ®nite-band potentials with the hierarchy of Korteweg±de Vries equations has been the reason for a heightened interest in them during the past twenty years. For recent accounts see, for instance, the monographs by Dickey [8] and Belokolos et al. [3]. The ®rst non-trivial example of a ®nite-band potential was given by Ince in 1940 [18]: consider the Lame potentials q…x† ˆ ÿg…g‡ 1†`…x‡ c jq;q0† where g 2N and `… ́ jq;q0† denotes Weierstrass's elliptic function with fundamental half-periods q and q0. If q is real, q0 is pure imaginary, and c ˆ q0, then q is realvalued, periodic and continuous. Ince's result, when translated to the present language, shows that q then is a ®nite-band potential with g non-empty gaps between the bands. This, however, raises immediately the question of what happens when c 6ˆ q0 or when the half-periods are more general. In these cases one encounters potentials which are complex-valued (but continuous) as well as potentials which have singularities of the form ÿg…g‡ 1†=x (and are realor complex-valued). Before giving an overview of the content of the paper I want to give a brief account of the history of the subject. Let q be a real, locally integrable, periodic function with period p, L the differential expression d=dx ‡ q, and H the self-adjoint operator in L…R† associated with L. Then consider the following four statements. (A) There are precisely 2g‡ 1 real numbers E2g < : : : < E0 such that the spectrum of H is given by