Quantum complex Hénon–Heiles potentials

Abstract

Quantum-mechanical PT -symmetric theories associated with complex cubic potentials such as V = x 2 + y 2 + igxy 2 and V = x 2 + y 2 + z 2 + igxyz, where g is a real parameter, are investigated. These theories appear to possess real, positive spectra. Low-lying energy levels are calculated to very high order in perturbation theory. The large-order behavior of the perturbation coefficients is determined using multidimensional WKB tunneling techniques.This approach is also applied to the complex Hénon-Heiles potential V = x 2 + y 2 + ig(xy 2 − 1 3 x 3 ).

Effective potential and resummation procedure to multidimensional complex cubic potentials for weak and strong coupling

“…(37) Table 1 and Table 2 show the coefficients up to the 10th order [10,12]; one sees that the results obtained by the Bender-Wu method are in agreement with those obtained by the connected fourth order Feynman diagrams (see (18) and (19)). Table 1 Weak-coupling coefficients for the 2-dimensional potential up to the 10th order, k represents the order of expansion and ǫ k coefficients of the energy corrections.…”

“…(21) In the perturbation theory, the ground-state wave functions are expanded in the form [3][4][5]12] …”

Distributional Borel Summability of Perturbation Theory for the Quantum Hénon-Heiles Model

“…The quantum counterpart of (1.1) (see, e.g., [15], [2], [1], [11], [3], [14]) is represented by the Schrödinger operator in L 2 (R 2 ) formally given by…”

symmetric models in more dimensions and solvable square-well versions of their angular Schr dinger equations

“…The quick progress in the field has led to the sophisticated perturbative analyses of some non-Hermitian PT symmetric models in more dimensions [20]. The "analytic continuation" activities intensify since the Hermitian non-central partial differential Schrödinger equations are attractive by the variability of their possible physical interpretations.…”