ar X iv : q ua nt - p h / 02 10 11 8 v 1 1 5 O ct 2 00 2 Toward the spectral zone control

The desired shifts of the boundaries of spectral allowed zones of periodical systems are demonstrated. In particular, the phenomenon of merging neighbor allowed zones is exhibited and its simple explanation is given. It is also shown how to change the additional fundamental spectral parameter, the degree of exponential solution growth, at arbitrary given energy points inside the forbidden zones. This allows one to control tunneling through fragments of periodic structures at energies belonging to spectral gap. All the results are based on the finite interval inverse eigenvalue problem which provides us with complete sets of exactly solvable models. This is a radical extension (continuous ! multiplicity) in comparison to the famous finite-gap models. The periodical structures represent an area of intensive research and diverse practical applications, e.g., in microelectronics. So it is important to extend as far as possible the class of spectral zone control algorithms. We suggest, in particular, potential transformations leading to given shifts of chosen zone boundaries. We apply our experience (quantum intuition) in the finite interval inverse problem (IP) algorithms [1-4], which allow the construction of the potentials corresponding to a given set of spectral parameters, e.g., bound state energy levels and spectral weights {E λ , c λ }, to periodic case. In the IP formalism [5-8] these parameters are input parameters uniquely determining the potential and so can be considered as spectral " control levers ". Corresponding exact models form the complete sets. The next step is the periodical continuation over the whole axis the potential derived with the inversion procedure on finite interval. So we use the missed possibility to combine inverse problem approach on a finite interval with direct one of constructing solutions for the periodically continued potential (infinitely enriched Kronig-Penney-like procedure). This gives us additional flexibility of the formalism in comparison with the pure inverse finite gap theory of spectral band structures. Let us use, for example, the formulae for energy level shift in infinite rectangular potential well of finite width. Let ψ 0 (x, E n) be corresponding eigenfunction at the energy E n. We assume ¯ ψ 0 (x, E n + t) to be a non-physical auxiliary solution in the initial potential at the shifted energy E n + t with the symmetry being opposite to one of the ψ 0 (x, E n). With these solutions, we can construct the Wronskian