Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum

Abstract

Abstract. We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential q of a one-dimensional Schrödinger operator H = − d 2 dx 2 +q determine the potential completely. Included are theorems for finite intervals and for the whole line. In particular, we pose and solve a new type of inverse spectral problem involving fractions of the eigenvalues of H on a finite interval and knowledge of q over a corresponding fraction of the interval. The methods employed rest on Weyl m-function techniques and densities of zeros of a class of entire functions.

Inverse spectral problem with partial information given on the potential and norming constants

“…Later, refinements by Hald [17] and Suzuki [34] of Hochstadt and Lieberman's theorem showed that the boundary conditions at x = 1 for L 1 and L 2 need not be assumed a priori to be the same and that if q is continuous, then one only needs λ 1,n = λ 2,m(n) for all values of n but one, that is, one eigenvalue can be missing. However, this is no longer true if the boundary condition at x = 1 is different and q is discontinuous for all values of n but one (see [16]). The same boundary condition for L 1 and L 2 at x = 0, however, is crucial for Hochstadt and Lieberman's theorem to hold (see [33]).…”

“…Several important generalizations of Hochstadt and Lieberman's theorem are given by Gesztesy and Simon [16] who consider the case where the L 1 [0, 1] potential q is known on a larger interval [0, a] with a ∈ [1/2, 1) and assume that the set of common eigenvalues of L 1 and L 2 is sufficiently large. Another result in [16] is to assume that the potential q belongs to C 2k for some k ∈ N 0 near 1/2 so that C 2ksmoothness can replace the knowledge of some (k + 1) eigenvalues, that is, (k + 1) eigenvalues can be missing. In [10], del Rio, Gesztesy and Simon further study the case where a can be any number in the interval (0, 1).…”

Half Inverse Problem for the Sturm-Liouville Operator with Coulomb potential

“…In [9,10], Gesztesy, Simon and Malamud gave some new uniqueness results in inverse spectral analysis with partial information on the potential for scalar and matrix Sturm-Liouville equations, respectively. Gesztesy and Simon [9] showed that more information on the potential can compensate for less information on the spectrum. In 2001, Sakhnovich [11] studied the existence of solution to the half inverse problem.…”

An inverse problem for the Sturm-Liouville pencil with arbitrary entire functions in the boundary condition

“…The similar problems with the potential known on the interval (a, π) for arbitrary a ∈ (0, π/2) have been studied in [20,25] and other papers. Our results also can be applied to this case.…”