Multi-instantons and exact results II: Specific cases, higher-order effects, and numerical calculations
By J. Zinn-Justin, U. Jentschura. 2004
133 citations recorded by the source corpus.
Abstract In this second part of the treatment of instantons in quantum mechanics, the focus is on specific calculations related to a number of quantum mechanical potentials with degenerate minima. We calculate the leading multi-instanton contributions to the partition function, using the formalism introduced in the first part of the treatise [Ann. Phys. (N. Y.) (previous issue) (2004)]. The following potentials are considered: (i) asymmetric potentials with degenerate minima, (ii) the periodic cosine potential, (iii) anharmonic oscillators with radial symmetry, and (iv) a specific potential which bears an analogy with the Fokker–Planck equation. The latter potential has the peculiar property that the perturbation series for the ground-state energy vanishes to all orders and is thus formally convergent (the ground-state energy, however, is non-zero and positive). For the potentials (ii), (iii), and (iv), we calculate the perturbative B-function as well as the instanton A-function to fourth order in g. We also consider the double-well potential in detail, and present some higher-order analytic as well as numerical calculations to verify explicitly the related conjectures up to the order of three instantons. Strategies analogous to those outlined here could result in new conjectures for problems where our present understanding is more limited.
Subfields: Applied Mathematics, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/359b245220089c19de30237537e1fbe098b4b9ff
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
In Atomic and Molecular Physics, and Optics
In Statistical and Nonlinear Physics
Citations in the corpus
- Uniform WKB, Multi-instantons, and Resurgent Trans-Series (2014, methodology): “For the DW problem, ZJJ express their exact quantization condition as [20] 1 √ 2π Γ ( 1 2 −B(E, g(2)) )( 2 g2 )B(E,g2) e−A(E,g 2)/2 = ±i (83)”
- Nonperturbative effects and nonperturbative definitions in matrix models and topological strings (2008, background): “This has been also pointed out for the case of the double-well potential in quantum mechanics, where the multi-instanton corrections play a crucial role in reconstructing the exact answer for the energies [82, 83].”