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On the periodic spectrum of the 1-dimensional Schrödinger operator. (English) Zbl 0703.34085
Consider Hill’s equation $$-y''(x)+q(x)y(x)=\lambda y(x)$$ (x$$\in {\mathbb{R}})$$ with q a potential in $$L^ 2[0,1]$$. Denote by $$\lambda_ k=\lambda_ k(q)$$ (k$$\geq 0)$$ the union of the periodic and antiperiodic spectrum of $$(-d^ 2/dx^ 2)+q$$ on the interval [0,1], written in nondecreasing order and with multiplicities. Introduce the set f potentials ISO(q), having the same periodic spectrum as q and the set G(q) of potential p having the same gaps as q, i.e. $$\lambda_ 0(q)=\lambda_ 0(p)$$ and $$\lambda_{2k}(q)-\lambda_{2k-1}(q)=\lambda_{2k}(p)-\lambda_{2k- 1}(p)$$ (k$$\geq 1)$$. This paper presents an elementary proof of the following result, due to J. Garnett and E. Trubowitz: Theorem. For all q in $$L^ 2[0,1]$$, $$ISO(q)=G(q)$$.
Reviewer: Th.Kappeler
##### MSC:
 34L05 General spectral theory of ordinary differential operators
##### Keywords:
Hill’s equation; antiperiodic spectrum; gaps
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