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Journal of Mathematical Physics

Exact Quantization Conditions. II

Joseph B. Krieger

Published Aug 1, 1969• 26 citations
Publisher

Abstract

By employing coordinate transformations, a generalized WKB quantization condition is derived which includes a modified WKB quantization rule and the related higher-order integrals. It is observed that a necessary condition for the first-order integral to give an exact quantization rule is the vanishing of the higher-order integrals. When this condition is satisfied, the resulting quantization condition is of the form of all previously known exact-quantization rules. The higher-order integrals are shown to vanish for certain cases of interest. An error in Paper I [C. Rosenzweig and J. B. Krieger, J. Math. Phys. 9, 849 (1968)] is noted and an examination of the higher-order integrals shows that a proposed quantization rule given there is not exact.

DOI
10.1063/1.1664990

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hep-thCross-listedarXiv:2511.20778•Nov 27, 2025
By employing coordinate transformations, a generalized WKB quantization condition is derived which includes a modified WKB quantization rule and the related higher-order integrals. It is observed that a necessary condition for the first-order integral to give an exact quantization rule is the vanishing of the higher-order integrals. When this condition is satisfied, the resulting quantization condition is of the form of all previously known exact-quantization rules. The higher-order integrals are shown to vanish for certain cases of interest. An error in Paper I [C. Rosenzweig and J. B. Krieger, J. Math. Phys. 9, 849 (1968)] is noted and an examination of the higher-order integrals shows that a proposed quantization rule given there is not exact.
The method of Froman and Froman for proving exact quantization conditions is reviewed. This formalism, unlike the usual WKB approximation to which it bears a close resemblance, requires consideration of the behavior of the potential everywhere it is defined. This approach leads to proofs that certain quantization conditions are exact without having to compare the results to solutions of the Schrödinger equation obtained by other means. Using the formalism, we prove the correctness of all previously known exact quantization rules for the one-dimensional and radial cases. Furthermore, it is shown that exact quantization rules can be proved for two other potentials. For one of these, no analytic solutions to the Schrödinger equation are known. For the latter case, the proof is checked by numerical integration of the Schrödinger equation for a special case.
First, two conditions are specified for the lowest order Wentzel–Kramers–Brillouin quantization rule to yield exact results. These rules are related to the periodic orbit decomposition of the quantum density of states. This approach is then applied to supersymmetric quantum mechanics. It leads to a new derivation of the result that shape-invariant potentials give exact results when the classical action is calculated with the square of the super potential, but without the Maslov index or the Langer correction.PACS Nos.: 03.65.Sq, 12.60.Jv
The WKB quantization condition can be used to estimate the quantized bound state energies, En, in a potential V(x). If a small perturbation of the form λV(x) is added to this potential, we show that the resulting shift in the quantized energy eigenvalues is given by a form which is quite similar to standard first-order perturbation theory. Several examples are presented in which the large n form of the lowest-order energy shift is more easily evaluated using WKB-perturbation techniques than by using standard first-order perturbation methods.
We discuss the exact quantization of general one-dimensional potentials in view of the exact-WKB formalism. Building on our previous work, we perform analytic continuations across different sectors via the complexification to the spectral (energy) parameter uuu and identify continuous and discontinuous transitions of the exact spectrum for generic potentials. When the transition is discontinuous, it is characterized by the Stokes phenomena, inducing different exact (median) quantization conditions, thereby distinct trans-series structures valid in different sectors. We analyze two illustrative examples, namely asymmetric triple-well (ATW) and tilted double-well (TDW), and verify the general qualitative analysis by deriving exact (median) quantization conditions in each sector. Moreover, by obtaining the trans-series solutions for each system, we identify bion/bounce configurations and show that the trans-series of ATW is organized in accordance with the cluster expansion of the bion gas and there should exist a previously neglected complex saddle in the TDW system. These identifications further strengthen the link between path integral and exact-WKB formalisms, while also demonstrating the predictive power of the latter. In parallel, for the P-NP relations of genus-1 systems, we derive transformation rules between any perturbative and non-perturbative pair of WKB-cycles. Our results show that the entire resurgence data of a genus-1 system transforms only by the change of classical parameters, i.e. frequencies and bion/bounce actions, and the perturbative energy series. This also reveals the underlying reasons of the previously found SSS-duality transformations.