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Radial anharmonic oscillator: perturbation theory, new semiclassical expansion, approximating eigenfunctions. I: Generalities, cubic anharmonicity case. (English) Zbl 1471.81019

Summary: For the general \(D\)-dimensional radial anharmonic oscillator with potential \(V(r) = \frac{1}{g^2} \hat{V}(g r)\) 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 \((g r)\)-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 \((g r)\)-space, respectively, exploring the logarithmic derivative of wave function \(y\). 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 Approximant), which should uniformly approximate the exact eigenfunction at \(r \in [0, \infty)\) for any coupling constant \(g \geq 0\) and dimension \(D > 0\). As a concrete application, the low-lying states of the cubic anharmonic oscillator \(V = r^2 + g r^3\) are considered. 3 free parameters of the Approximant are fixed by taking it as a trial function in variational calculus. It is shown that the relative deviation of the Approximant from the exact ground state eigenfunction is \(\lesssim 1 0^{- 4}\) 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 \(D = 1, 2, \ldots\).

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q15 Perturbation theories for operators and differential equations in quantum theory
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35A15 Variational methods applied to PDEs
81S40 Path integrals in quantum mechanics
70J35 Forced motions in linear vibration theory
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
35P15 Estimates of eigenvalues in context of PDEs
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