Numerical reconstruction of the first band(s) in an inverse Hill’s problem

This paper concerns an inverse band structure problem for one dimensional periodic Schrödinger operators (Hill’s operators). Our goal is to find a potential for the Hill’s operator in order to reproduce as best as possible some given target bands, which may not be realisable. We recast the problem as an optimisation problem, and prove that this problem is well-posed when considering singular potentials (Borel measures).

On the Estimations of the Small Periodic Eigenvalues

We estimate the small periodic and semiperiodic eigenvalues of Hill's operator with sufficiently differentiable potential by two different methods. Then using it we give the high precision approximations for the length of th gap in the spectrum of Hill-Sehrodinger operator and for the length of th instability interval of Hill's equation for small values of Finally we illustrate and compare the results obtained by two different ways for some examples.