关于tridiagonal matrices的一个特殊性质对偶拟精确可解六次势在量子力学中的应用
On a special property of tridiagonal matrices. Application to dual quasi-exactly solvable sextic potentials in quantum mechanics
摘要：
我们提出并证明了一个简单的定理，即三对角矩阵的特征值改变其符号（作为一个集合），一旦矩阵的对角元素的符号被改变。我们还提供了这个定理在量子物理中应用的一个例子。具体来说，我们介绍了一个六次多项式拟精确可解潜力的对偶性和自对偶性的概念，并证明了代数部分的频谱的对偶潜力有相反的符号彼此（作为集）。我们的定理给出了这一陈述的简洁的一行证明。此外，我们还通过纯粹的量子力学手段证明了这一点-这是一种远不那么简单的方法，需要一些努力。
英文摘要：
We put forward and prove a simple theorem stating that the eigenvalues of a tridiagonal matrix change their sign (as a set), once the signs of the diagonal elements of the matrix are changed. We also provide an example of application of this theorem in quantum physics. Specifically, we introduce the notion of duality and self-duality for a sextic-polynomial quasi-exactly-solvable potential, and demonstrate that the algebraic parts of the spectrum of the dual potentials have signs opposite to one another (as sets). Our Theorem furnishes an elegant one-line proof of this statement. In addition, we also prove it by purely quantum-mechanical means - a far less straightforward method that requires some effort.
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