The inverse nodal problem for Hill's equation

Abstract

We study the inverse nodal problem for Hill's equation. In particular, we solve the uniqueness, reconstruction and stability problems using the nodal set of periodic (or anti-periodic) eigenfunctions. Furthermore, we show that the space of periodic potential functions q normalized by ∫10q = 0 is homeomorphic to the partition set of the space of quasinodal sequences. Our method is to make a translation so that the periodic (or anti-periodic) problem is reduced to a Dirichlet problem.

A nodal inverse problem for second order Sturm–Liouville operators with indefinite weights

“…Given q, the determination of a potential q for Àu 00 þ qðxÞu ¼ kqðxÞu; x 2 ½a; b in terms of nodal data has attracted a great deal of interest, see for example [4,9,12,18,14]. Usually, q is assumed constant, although an indefinite problem was studied in [22], with q > 0 in ½a; cÞ and q > 0 in ðc; b.…”

An inverse nodal problem for differential pencils with complex spectral parameter dependent boundary conditions

“…Independently, Shen studied the relations between the nodal points and the density function of string equation in 1988 [9]. Later, many authors have studied inverse nodal problem for different operators (see [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]). …”

An inverse problem for a differential pencil using nodal points as data

“…The problems of this type are related to some questions of mechanics and mathematical physics. Inverse nodal problems for Sturm-Liouville operators with separated boundary conditions on an interval are studied in sufficient detail in [1,3,4,5,6,7,8,13,14,15,18,21,22,23,26,27,31,32].…”

“…The inverse nodal problem for the differential pencil (1.1) with separated boundary conditions was studied in [13] (with an excessive assumption that the function p(x) was known a priori) and [23], in [4] for the case of the Robin boundary conditions, and in [5] for the Dirichlet boundary conditions. We note that Cheng and Law first studied the inverse nodal problem for the Sturm-Liouville operator with non-separated boundary conditions [7] (in fact, periodic or antiperiodic boundary conditions). However, the inverse nodal problem for the differential operator with non-separated boundary conditions (1.2) has not been studied yet.…”