REGIONS OF STABILITY OF THE NONLINEAR SCHRODINGER-EQUATION WITH A POTENTIAL HILL

The problem of finding eigenvalues of the static nonlinear Schrodinger equation with a potential is numerically investigated. The change of eigenfunctions resulting from the transformation of a potential well into a potential hill is studied. Unlike the linear Schrodinger equation, continuous and square-integrable solutions exist, not only for potential wells, but also for potential hills. For potential hills there may exist a few different eigenfunctions with the same number of nodes, whereas eigenfunctions for potential wells are single valued. Moreover, regions of stability are discovered where a continuum of eigenfunctions exist. In these regions, eigenfunctions may continuously transform one into another in certain energy intervals. The possible practical use of these solutions is discussed.