Symmetrically anharmonic oscillators

Abstract

We propose a new nonrelativistic Pauli-type equation where some specific small relativistic terms are retained. With the confining potentials ${V}_{\ensuremath{\infty}}(x)$ approximated by the polynomials ${V}_{m}(x)={g}_{0}{x}^{2}+\ensuremath{\cdots}+{g}_{m}{x}^{2m+2}$, ${g}_{m}>0$, the nonzero kinematical corrections ${T}_{m}\ensuremath{-}{T}_{0}$, where ${T}_{m}={h}_{0}{p}^{2}+\ensuremath{\cdots}+{h}_{m}{p}^{2m+2}\ensuremath{\simeq}{T}_{\ensuremath{\infty}} ={({\ensuremath{\mu}}^{2}{c}^{4}+{p}^{2}{c}^{2})}^{\frac{1}{2}}\ensuremath{-}\ensuremath{\mu}{c}^{2}$, are added to the anharmonic-oscillator Schr\"odinger equation, so that the $p\ensuremath{-}x$ symmetry typical for a harmonic-oscillator Hamiltonian is restored. As a consequence of this semirelativistic regularization, the analytic diagonalization of an entirely anharmonic Hamiltonian ${H}_{\mathrm{mm}}={T}_{m}+{V}_{m}$ in terms of the $m\ifmmode\times\else\texttimes\fi{}m$-matrix continued fractions is obtained. Both the auxiliary fractions and the eigenstates converge very quickly. In the cases of the bounded spectrum of ${H}_{\mathrm{mm}}$ ($m=2q$), it is proved exactly for $q=1, 2, \mathrm{and} 3$ and conjectured for $q\ensuremath{\ge}4$.