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Gaps and bands of one dimensional periodic Schrödinger operators. (English) Zbl 0554.34013
Let q(x) be the periodic extension to the whole line of a function in $$L^ 2_ R[0,1]$$, the Hilbert space of all real valued square integrable functions on the unit interval. The spectrum of the Schrödinger operator $$-(d^ 2/dx^ 2)+q(x)$$ acting on $$L^ 2(R^ 1)$$ is the union of purely absolutely continuous bands $$B_ n(q)$$, $$n\geq 1$$. The nth band $$B_ n$$ is the set $$\{v_ n(k,q);-1/2\leq k\leq 1/2\}$$. Here $$v_ n(k,q),n\geq 1$$, is the nth eigenvalue (counted with the multiplicities when $$k=0,\pm 1/2)$$ of the boundary value problem $$- y''+q(x)y=\lambda y$$, $$y(x+1)=e^{2\pi ik}y(x),-\infty <x<\infty$$. The eigenvalue $$v_ n(k)$$ is a continuous function of k so that $$B_ n$$ is a closed subinterval of $$R^ 1$$. The authors study the following question: When is a collection of closed subintervals of $$R^ 1$$ the set of bands corresponding to a function q in $$L^ 2_ R[0,1]?$$
Reviewer: S.L.Kalla

##### MSC:
 34L99 Ordinary differential operators
##### Keywords:
Hilbert space; spectrum of the Schrödinger operator
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