ON BIFURCATIONS IN THE SCHRÖDINGER PROBLEM ADMITTING OF AN EXISTENCE OF NONLINEAR SHIFT OPERATORS

Abstract

The problem of reconstruction of one-dimensional potentials with a given spectrum that is strictly equidistant except for one gap of a specified size and location is solved. This problem leads to a necessity of a nonlinear generalization of spectral shift operators. The solvability condition for an equation for these operators generates a dynamical system for possible potentials. Two classes of symmetric and regular in ℛ1 potentials obeying this equation are found and investigated in detail. The first class is a countable set of one-parameter families of anharmonic oscillators with a spectrum as described above. The second class is a countable set of two-parameter families of complicated potentials with asymptotics that oscillate with linearly increasing frequency and amplitude. The spectrum of that class is a combination of three strictly equidistant sequences of levels with the ground states shifted arbitrarily relative to each other. Bifurcations of those potentials under changes of structure parameters as well as bifurcations of corresponding spectra have been studied.