On Exact Solvability of Anharmonic Oscillators in Large Dimensions

Abstract

General Schr\"{o}dinger 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 Gr\"obner 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.