On the class of potentials with trivial monodromy

Abstract

We consider the problem of describing the class TM(Ω;A) potentials meromorphic in a simply
connected domain Ω with a set of poles A satisfying the trivial monodromy condition: any solution
of the corresponding Sturm–Liouville equation for all values of the spectral parameter has no
branch points at any point in A.We have shown that in the case of a finite A the linear (with respect
to the usual addition) space TM(Ω;A) has finite dimension modulo the subspace TM0(Ω;A) of
functions holomorphic in Ω and having at points A, zeros of a given multiplicity (its own for each
point). Thus, for a finite A, a complete description of TM(Ω; A;M) is obtained in terms of any
finite set of functions – solutions of an interpolation problem with multiple nodes at points of the
set A. The result obtained summarizes the well-known results on classes of potentials with trivial
monodromy on the C, decreasing at infinity (J.J. Duistermaat, F.A. Gr¨unbaum) or growing not
faster than the second (A. Oblomkov) or the sixth (J.Gibbons, A.P. Veselov) of degree. In the case
when the set A is countable and has a unique limit point, a sufficiently wide class of functions that
satisfy the condition of trivial monodromy is constructed.