On the class of potentials with trivial monodromy

  • Kh. K. Ishkin Bashkir State University
  • A. D. Akhmetshina Bashkir State University


We consider the problem of describing the class TM(Ω;A) potentials meromorphic in a simplyconnected domain Ω with a set of poles A satisfying the trivial monodromy condition: any solutionof the corresponding Sturm–Liouville equation for all values of the spectral parameter has nobranch points at any point in A.We have shown that in the case of a finite A the linear (with respectto the usual addition) space TM(Ω;A) has finite dimension modulo the subspace TM0(Ω;A) offunctions holomorphic in Ω and having at points A, zeros of a given multiplicity (its own for eachpoint). Thus, for a finite A, a complete description of TM(Ω; A;M) is obtained in terms of anyfinite set of functions – solutions of an interpolation problem with multiple nodes at points of theset A. The result obtained summarizes the well-known results on classes of potentials with trivialmonodromy on the C, decreasing at infinity (J.J. Duistermaat, F.A. Gr¨unbaum) or growing notfaster than the second (A. Oblomkov) or the sixth (J.Gibbons, A.P. Veselov) of degree. In the casewhen the set A is countable and has a unique limit point, a sufficiently wide class of functions thatsatisfy the condition of trivial monodromy is constructed.


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How to Cite
ISHKIN, Kh. K.; AKHMETSHINA, A. D.. On the class of potentials with trivial monodromy. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 99, n. 3, p. 43-52, dec. 2018. ISSN 1563-0277. Available at: <http://bm.kaznu.kz/index.php/kaznu/article/view/559>. Date accessed: 24 mar. 2019.
Keywords spectral instability, spectrum localization, Sturm–Liouville equation, trivial monodromy