Two-spectra theorem with uncertainty

Abstract

The goal of this paper is to combine ideas from the theory of mixed spectral problems for differential operators with new results in the area of the Uncertainty Principle in Harmonic Analysis (UP). Using recent solutions of Gap and Type Problems of UP we prove a version of Borg's two-spectra theorem for Schrödinger operators, allowing uncertainty in the placement of the eigenvalues. We give a formula for the exact 'size of uncertainty', calculated from the lengths of the intervals where the eigenvalues may occur. Among other applications, we describe pairs of indeterminate operators in the three-interval case of the mixed spectral problem. At the end of the paper we discuss further questions and open problems.

Stability of Dirac resonances

“…In our paper we have used the connection between Jost and Hermite-Biehler functions. Similar connection in case of the Schrödinger operators was given by Baranov, Belov and Poltoratski in [3] (see also Makarov and Poltoratski [30]). In [26,27], the canonical systems associated with Dirac operators on the half-line and on the real line was considered.…”

Essentially isospectral transformations and their applications

“…Hald [22] Later, Gesztesy and Simon [19] and Ramm [40] showed that if the potential is known on more than half the interval, then only a finite density subset of eigenvalues is needed. See also [3,24,25,34,45] for further developments in this direction, and [4,44,46] for boundary conditions dependent on the eigenvalue parameter. Here, we generalize the Hochstadt-Lieberman theorem to the case of boundary value problems of the form (1.1)-(1.2).…”

Essentially isospectral transformations and their applications

“…Hald [22] proved that one of the boundary constants need not be assumed known, i.e., for a problem P(q, [19] and Ramm [40] showed that if the potential is known on more than half the interval then only a finite density subset of eigenvalues is needed. See also [3], [24], [25], [34], [45] for further developments in this direction, and [4], [44], [46] for boundary conditions dependent on the eigenvalue parameter.…”