×

Noether invariants and complete Lie-point symmetries for equations of the Hill type. (English) Zbl 1098.58503

Summary: We carry out a systematic analysis of second-order differential equations of Hill type in the framework of the Lie group theory of point transformations. Both the homogeneous and the inhomogeneous cases are treated. We find the complete Lie-point symmetry group to be associated with these equations. This group contains, as subgroups, \(\text{SO}(2,1)\) and \(E_ 2\), which are important in evaluating the energy spectrum, as well as the degeneracy of levels of quantum-mechanical systems related to Hill equations. A set of Noether invariants which come from a symmetry subgroup endowed with five linearly independent generators is also determined.

MSC:

58J70 Invariance and symmetry properties for PDEs on manifolds
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
PDFBibTeX XMLCite
Full Text: DOI