Equivalent Hermitian Hamiltonian for the non-Hermitian -x4 potential
Equivalent Hermitian Hamiltonian for the non-Hermitian -x4 potential
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DOI：
10.1103/physrevd.73.085002
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发表时间：
2006-04-01
影响因子：
5
通讯作者：
Mateo, J
中科院分区：
文献类型：
--
作者：
Jones, HF;Mateo, J
势函数V（x）=-x（4）在真实的直线上是无界的，当x取在复平面下半部分的一个围线上时，可以得到一个适定的有界态问题.因此它是PT对称的而不是厄米特的。尽管如此，它已被证明有一个真实的频谱数字，并证明现实，涉及常微分方程和可积系统之间的对应关系，随后构建的一般类的潜力-（ix）（N）。对于这样的哈密尔顿算子，自然的PT度量不是正定的，但是可以根据算子Q定义动态定义的正定度量。此外，在该算子的帮助下，可以构造等效的埃尔米特汉密尔顿h。这个程序已经对几个可解模型进行了精确的计算，并且已经找到了势m（2）x（2）+igx（3）的微扰展开式的前几项。然而，到目前为止，-x（4）势已被证明是难以处理的。在本文中，我们给出了明确的，封闭的形式表达式Q和h，这是可能的一个特定的参数化的轮廓在复杂的平面上的问题被定义。这构成了光谱真实性的一个明确证明。由此产生的等效哈密顿量具有正四次项和线性项的势。
The potential V(x)=-x(4), which is unbounded below on the real line, can give rise to a well-posed bound state problem when x is taken on a contour in the lower-half complex plane. It is then PT-symmetric rather than Hermitian. Nonetheless it has been shown numerically to have a real spectrum, and a proof of reality, involving the correspondence between ordinary differential equations and integrable systems, was subsequently constructed for the general class of potentials -(ix)(N). For such Hamiltonians the natural PT metric is not positive definite, but a dynamically-defined positive-definite metric can be defined, depending on an operator Q. Further, with the help of this operator an equivalent Hermitian Hamiltonian h can be constructed. This programme has been carried out exactly for a few soluble models, and the first few terms of a perturbative expansion have been found for the potential m(2)x(2)+igx(3). However, until now, the -x(4) potential has proved intractable. In the present paper we give explicit, closed form expressions for Q and h, which are made possible by a particular parametrization of the contour in the complex plane on which the problem is defined. This constitutes an explicit proof of the reality of the spectrum. The resulting equivalent Hamiltonian has a potential with a positive quartic term together with a linear term.