Two-spectra theorem with uncertainty

  • Nikolai Makarov

    California Institute of Technology, Pasadena, USA
  • Alexei Poltoratski

    Texas A&M University, College Station, USA
Two-spectra theorem with uncertainty cover
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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.

Cite this article

Nikolai Makarov, Alexei Poltoratski, Two-spectra theorem with uncertainty. J. Spectr. Theory 9 (2019), no. 4, pp. 1249–1285

DOI 10.4171/JST/276