On Exact Solvability of Anharmonic Oscillators in Large Dimensions

Gerdt, Vladimir; Yanovich, Denis; Znojil, Miloslav

On Exact Solvability of Anharmonic Oscillators in Large Dimensions

Authors

Gerdt, Vladimir

Yanovich, Denis

Znojil, Miloslav

Description

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 $D \geq 1$, 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.
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