Radial Anharmonic Oscillator: Perturbation Theory, New Semiclassical Expansion, Approximating Eigenfunctions. I. Generalities, Cubic Anharmonicity Case

For the general $D$-dimensional radial anharmonic oscillator with potential $V(r)= \frac{1}{g^2}\,\hat{V}(gr)$ the Perturbation Theory (PT) in powers of coupling constant $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) is developed constructively in $r$-space and in $(gr)$ space, respectively. The Riccati-Bloch (RB) equation and Generalized Bloch (GB) equation are introduced as ones which govern dynamics in coordinate $r$-space and in $(gr)$-space, respectively, exploring the logarithmic derivative of wavefunction $y$. It is shown that PT in powers of $g$ developed in RB equation leads to Taylor expansion of $y$ at small $r$ while being developed in GB equation leads to a new form of semiclassical expansion at large $(g r)$: it coincides with loop expansion in path integral formalism. In complementary way PT for large $g$ developed in RB equation leads to an expansion of $y$ at large $r$ and developed in GB equation leads to an expansion at small $(g r)$. Interpolating all four expansions for $y$ leads to a compact function (called the {\it Approximant}), which should uniformly approximate the exact eigenfunction at $r \in [0, \infty)$ for any coupling constant $g \geq 0$ and dimension $D > 0$. Free parameters of the Approximant are fixed by taking it as a trial function in variational calculus. As a concrete application the low-lying states of the cubic anharmonic oscillator $V=r^2+gr^3$ are considered. It is shown that the relative deviation of the Approximant from the exact ground state eigenfunction is $\lesssim 10^{-4}$ for $r \in [0, \infty)$ for coupling constant $g \geq 0$ and dimension $D=1,2,\ldots$. In turn, the variational energies of the low-lying states are obtained with unprecedented accuracy 7-8 s.d. for $g \geq 0$ and $D=1,2,\ldots$.