Gaps and bands of one dimensional periodic Schrödinger operators (Q759884)

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 \(L^ 2_ R[0,1]?\)