Matrix-valued generalizations of the theorems of Borg and Hochstadt. (English) Zbl 1070.47037
Ruiz Goldstein, Gisèle (ed.) et al., Evolution equations. Proceedings of the conference, Blaubeuren, Germany, June 11–17, 2001 in honor of the 60th birthdays of Philippe Bénilan, Jerome A. Goldstein and Rainer Nagel. New York, NY: Marcel Dekker (ISBN 0-8247-0975-6/pbk). Lect. Notes Pure Appl. Math. 234, 1-34 (2003).
Consider a Schrödinger operator \(h = -d^2/dx^2 + q\) on the real line with a real-valued potential \(q \in L_{1,\text{loc}}(\mathbb R)\). G. Borg’s theorem [Acta Math. 78, 1–96 (1946; Zbl 0063.00523)] states that if \(h\) is periodic and has spectrum equal to \([E_0, \infty)\), then \(q(x)=E_0\) a.e. H. Hochstadt’s theorem [Arch. Rat. Mech. Anal. 19, 353–362 (1965; Zbl 0128.31201)] is the analogue for a Schrödinger operator having exactly one gap. In the latter case, the potential can be expressed in terms of an elliptic Weierstrass function.
The authors generalise both results to matrix-valued (not necessarily periodic) Schrödinger operators \(H := -\text{Id}2/dx^2 + Q\) where \(Q=Q^* \in L_{1,\text{loc}}(\mathbb R)^{m \times m}\). As has already been recognised in [S. Clark, F. Gesztesy, H. Holden and B. M. Levitan, J. Differ. Eq. 167, 181–210 (2000; Zbl 0970.34073)], the key ingredient (besides \(Q=Q^*\) and the spectrum of \(H\) having no resp.exactly one gap) is not the periodicity but the fact that \(Q\) is reflectionless. This condition is more general than periodicity and can be expressed via Weyl-Titchmarsh matrices [cf., e.g., F. Gesztesy and E. Tsekanovskii, Math. Nachr. 218, 61–138 (2000; Zbl 0961.30027) or S. Kotani and B. Simon, Commun. Math. Phys. 119, 403–429 (1988; Zbl 0656.60068)]. The proof of the main result consists of two steps: First the authors show that a special class of matrix-valued Schrödinger operators \(H_\Sigma\) with prescribed spectrum \(\Sigma\) (having no gap resp.one gap) satisfies the conclusion of the main result. Then, in a second step, they prove that any reflectionless Schrödinger operator \(H\) is of the form \(H_\Sigma\).
For the entire collection see [Zbl 1027.00022].
The authors generalise both results to matrix-valued (not necessarily periodic) Schrödinger operators \(H := -\text{Id}2/dx^2 + Q\) where \(Q=Q^* \in L_{1,\text{loc}}(\mathbb R)^{m \times m}\). As has already been recognised in [S. Clark, F. Gesztesy, H. Holden and B. M. Levitan, J. Differ. Eq. 167, 181–210 (2000; Zbl 0970.34073)], the key ingredient (besides \(Q=Q^*\) and the spectrum of \(H\) having no resp.exactly one gap) is not the periodicity but the fact that \(Q\) is reflectionless. This condition is more general than periodicity and can be expressed via Weyl-Titchmarsh matrices [cf., e.g., F. Gesztesy and E. Tsekanovskii, Math. Nachr. 218, 61–138 (2000; Zbl 0961.30027) or S. Kotani and B. Simon, Commun. Math. Phys. 119, 403–429 (1988; Zbl 0656.60068)]. The proof of the main result consists of two steps: First the authors show that a special class of matrix-valued Schrödinger operators \(H_\Sigma\) with prescribed spectrum \(\Sigma\) (having no gap resp.one gap) satisfies the conclusion of the main result. Then, in a second step, they prove that any reflectionless Schrödinger operator \(H\) is of the form \(H_\Sigma\).
For the entire collection see [Zbl 1027.00022].
Reviewer: Olaf Post (Aachen)
MSC:
| 47E05 | General theory of ordinary differential operators |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 34A55 | Inverse problems involving ordinary differential equations |
