2003/10/09 by Vladimir Gerdt, Gerdt, Vladimir, Denis Yanovich +3
Mathematics · Physics and Astronomy · #Commutative Algebra (math.AC) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AC #math.MP
paper · pdf · doi:10.48550/arxiv.math-ph/0310012
18 pages. In: "Computer Algebra in Scientific Computing / CASC 2003", V.G.Ganzha, E.W.Mayr, E.V.Vorozhtsov (Eds.). Institute of Informatics, Technical University of Munich, Garching, 2003, pp.143-162
arxiv created 2003/10/09 · arxiv updated 2009/12/01
General Schrödinger equation is considered with a central polynomial potential depending on 2q arbitrary coupling constants. Its exceptional solutions of the so called Magyari type (i.e., exact bound states proportional to a polynomial of degree N) are sought. In any spatial dimension D ≥ 1, this problem leads to the Magyari's system of coupled polynomial constraints, and only purely numerical solutions seem available at a generic choice of q and N. Routinely, we solved the system by the construction of the Janet bases in a degree-reverse-lexicographical ordering, followed by their conversion into the pure lexicographical Gröbner bases. For very large D we discovered that (a) the determination of the "acceptable" (which means, real) energies becomes extremely facilitated in this language; (b) the resulting univariate "secular" polynomial proved to factorize, utterly unexpectedly, in a fully non-numerical manner. This means that due to the use of the Janet bases we found a new exactly solvable class of models in quantum mechanics.