Holomorphic anomaly and quantum mechanics. (28th December 2017)

• Record Type: Journal Article
• Authors: Codesido, Santiago; Mariño, Marcos
• Is Part Of: Journal of physics. Volume 51:Number 5(2018)
• Publication Date: 2017-12-28
• Subjects: quantum mechanics -- WKB -- anharmonic oscillators -- resurgence -- topological string theory; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/aa9e77
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: We show that the all-orders WKB periods of one-dimensional quantum mechanical oscillators are governed by the refined holomorphic anomaly equations of topological string theory. We analyze in detail the double-well potential and the cubic and quartic oscillators, and we calculate the WKB expansion of their quantum free energies by using the direct integration of the anomaly equations. We reproduce in this way all known results about the quantum periods of these models, which we express in terms of modular forms on the WKB curve. As an application of our results, we study the large order behavior of the WKB expansion in the case of the double well, which displays the double factorial growth typical of string theory.