Convergence proof for optimizedδexpansion: Anharmonic oscillator

Abstract

A recent proof of the convergence of the optimized 6 expansion for one-dimensional non-Gaussian integrals is extended to the finite-temperature partition function of the quantum anharmonic oscillator.The convergence is exponentially fast, with the remainder falling as e ' at order N in the expansion, independently of the size of the coupling or the sign of the mass term. In particular, the approach gives a convergent resummation procedure for the double-well (non-Borel-summable) case.

Convergence of Scaled Delta Expansion: Anharmonic Oscillator

“…(2.72) and (2.73) are positive [4,16], we thus complete the proof that the delta expansion converges to the exact energy eigenvalue for either of these choices for x.…”

“…as well as from the principle of minimal sensitivity for the partition function of anharmonic oscillator [16],…”

“…A particularly intriguing statement made by Duncan and Jones [16] is that the optimized delta expansion (apparently) cures the problem of large order divergences even in non-Borel summable cases. If this were true in field theoretic models it could shed some important light on the instanton physics and vacuum structure of Quantum Chromodynamics.…”

“…An important step forward has been made recently [16]: It was demonstrated that the optimized delta expansion converges to the correct answer in the quantum mechanical anharmonic oscillator as well as in the double well, at finite temperatures. At zero temperature (hence for energy eigenvalues, Green functions, etc.…”

“…Recently, there has been a considerable interest in the so-called optimized linear delta expansion [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. In simple quantum mechanical models, the method involves a rearrangement of the Hamiltonian as H = p 2 2 + V (q) (1.1)…”