### Abstract

A quantum anharmonic oscillator with a polynomial self-interaction is defined in coordinate space by a Hamiltonian of the form H = -d ^{2}/dx^{2} + 1/4x^{2} + g[(1/2x^{2}) ^{N} + a(1/2x^{2})^{N-1} + b(1/2x^{2}) ^{N-2}.ast; + ⋯]. Using WKB techniques we derive a secular equation which determines the eigenvalues of H for small |g|. We find that the qualitative analytic structure of these eigenvalues as functions of complex g remains unchanged for all fixed values of a, b, . . . , including a = b = ⋯ = 0. The secular equation also implies an elegant theorem which predicts how the a, b, ⋯ terms in H affect the large-order growth of perturbation theory. We use this theorem to compare the perturbative behavior of non-Wick-ordered and Wick-ordered field theories in one-dimensional space-time. In particular, we show that the perturbation series Σ A_{n}g ^{n}; and Σ B_{n}g^{n} for the energy levels of the (gψ^{2n})_{1} and (:gψ^{2N}:), field theories differ in large order by an over-all multiplicative constant lim _{n→∞} A_{n}/B_{n} = exp[N(2N - 1)/(2N - 2)].

Original language | English (US) |
---|---|

Pages (from-to) | 1320-1324 |

Number of pages | 5 |

Journal | Journal of Mathematical Physics |

Volume | 13 |

Issue number | 9 |

DOIs | |

State | Published - Jan 1 1972 |

Externally published | Yes |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*13*(9), 1320-1324. https://doi.org/10.1063/1.1666140

}

*Journal of Mathematical Physics*, vol. 13, no. 9, pp. 1320-1324. https://doi.org/10.1063/1.1666140

**Anharmonic oscillator with polynomial self-interaction.** / Banks, Thomas; Bender, Carl M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Anharmonic oscillator with polynomial self-interaction

AU - Banks, Thomas

AU - Bender, Carl M.

PY - 1972/1/1

Y1 - 1972/1/1

N2 - A quantum anharmonic oscillator with a polynomial self-interaction is defined in coordinate space by a Hamiltonian of the form H = -d 2/dx2 + 1/4x2 + g[(1/2x2) N + a(1/2x2)N-1 + b(1/2x2) N-2.ast; + ⋯]. Using WKB techniques we derive a secular equation which determines the eigenvalues of H for small |g|. We find that the qualitative analytic structure of these eigenvalues as functions of complex g remains unchanged for all fixed values of a, b, . . . , including a = b = ⋯ = 0. The secular equation also implies an elegant theorem which predicts how the a, b, ⋯ terms in H affect the large-order growth of perturbation theory. We use this theorem to compare the perturbative behavior of non-Wick-ordered and Wick-ordered field theories in one-dimensional space-time. In particular, we show that the perturbation series Σ Ang n; and Σ Bngn for the energy levels of the (gψ2n)1 and (:gψ2N:), field theories differ in large order by an over-all multiplicative constant lim n→∞ An/Bn = exp[N(2N - 1)/(2N - 2)].

AB - A quantum anharmonic oscillator with a polynomial self-interaction is defined in coordinate space by a Hamiltonian of the form H = -d 2/dx2 + 1/4x2 + g[(1/2x2) N + a(1/2x2)N-1 + b(1/2x2) N-2.ast; + ⋯]. Using WKB techniques we derive a secular equation which determines the eigenvalues of H for small |g|. We find that the qualitative analytic structure of these eigenvalues as functions of complex g remains unchanged for all fixed values of a, b, . . . , including a = b = ⋯ = 0. The secular equation also implies an elegant theorem which predicts how the a, b, ⋯ terms in H affect the large-order growth of perturbation theory. We use this theorem to compare the perturbative behavior of non-Wick-ordered and Wick-ordered field theories in one-dimensional space-time. In particular, we show that the perturbation series Σ Ang n; and Σ Bngn for the energy levels of the (gψ2n)1 and (:gψ2N:), field theories differ in large order by an over-all multiplicative constant lim n→∞ An/Bn = exp[N(2N - 1)/(2N - 2)].

UR - http://www.scopus.com/inward/record.url?scp=0001305437&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001305437&partnerID=8YFLogxK

U2 - 10.1063/1.1666140

DO - 10.1063/1.1666140

M3 - Article

AN - SCOPUS:0001305437

VL - 13

SP - 1320

EP - 1324

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 9

ER -