Zeta-regularization for exact-WKB resolution of a general 1D Schrödinger equation
Zeta-regularization for exact-WKB resolution of a general 1D Schrödinger equation
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一般一维薛定谔方程的精确 WKB 分辨率的 Zeta 正则化
DOI：
10.1088/1751-8113/45/37/374007
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
A. Voros
中科院分区：
文献类型：
--
作者：
A. Voros
本文综述了一般一维量子非谐振子的精确解析求解方法：多项式势的定态薛定谔方程。它是WKB处理的一种精确形式，涉及“谱”（通常）与“经典”（较新）并行的zeta正则化。中心的结果是一组玻尔-索末菲，但精确的量子化条件，直接从朗斯基恒等式，并出现扩展的可积系统的贝特安纳公式。这样的精确量子化条件不仅选择本征值，还计算了谱参数在一般位置的谱行列式，而其他的计算了波函数。这篇文章是一个特殊问题的一部分杂志物理A：数学和理论在纪念斯图尔特Dowker的75岁生日致力于“应用zeta函数和其他光谱函数在数学和物理”。
We review an exact analytical resolution method for general one-dimensional quantal anharmonic oscillators: stationary Schrödinger equations with polynomial potentials. It is an exact form of WKB treatment involving ‘spectral’ (usual) versus ‘classical’ (newer) zeta-regularizations in parallel. The central results are a set of Bohr–Sommerfeld-like but exact quantization conditions, directly drawn from Wronskian identities, and appearing to extend the Bethe-ansatz formulae of integrable systems. Such exact quantization conditions do not just select the eigenvalues; some evaluate the spectral determinants, and others the wavefunctions, for the spectral parameter in general position. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker’s 75th birthday devoted to ‘Applications of zeta functions and other spectral functions in mathematics and physics’.