Asymptotics of solutions of the Sturm–Liouville equation with meromorphic potential

Abstract

In the present paper, we study the asymptotics for large spectral parameter of the solutions of the
Sturm–Liouville equation with a meromorphic potential. It is shown that the asymptotic behavior
of solutions depends entirely on the location of the poles and on the fulfillment of the condition
of trivial monodromy. So, if a smooth curve γ, free from poles, and a solution φ with Cauchy
data at one of the ends of γ are given, then provided that the chord contracting γ also does not
contain poles, and all the poles lying inside the domain g bounded by γ and chorda satisfy the
trivial monodromy condition, the φ and its derivative at the other end of γ obtained by analytic
continuation along γ have the same asymptotics as in the case of a holomorphic potential. The
situation is greatly complicated if the domain g contains at least one pole that does not satisfy
the condition of trivial monodromy. In this case, the asymptotics of φ and its derivative will be
determined by the monodromy matrices of the part of the poles lying inside g. Based on the
estimates obtained, we found the spectrum asymptotics of the Sturm–Liouville operator on some
smooth curve γ, with a potential having inside the convex hull γ a unique second-order pole that
does not satisfy the trivial monodromy condition.