论文标题
Riemann假设，修饰的Morse电位和超对称量子力学
Riemann Hypothesis, Modified Morse Potential and Supersymmetric Quantum Mechanics
论文作者
论文摘要
在本文中，我们讨论了与Riemann Zeta函数和Riemann XI函数有关的各种潜力。这些电势是修改的摩尔斯电势版本，也可能与径向谐波振荡器的修改形式和改良的库仑电位有关。我们使用超对称量子力学来构建其基态波函数和基态的傅立叶变换以展示Riemann零。这使我们能够根据基态波函数在动量空间中的位置来制定Riemann假设。我们还讨论了这些势与一个和两个矩阵积分的关系，并构建了一些与矩阵模型相关的正交多项式。我们将动量空间中的schrodinger方程与动量空间中的有限差方程与无限数量的术语相关联。我们计算了与这些电势和基础状态以及香农信息熵相关的不确定性关系，并与未修改的摩尔斯和谐波振荡器电位进行了比较。最后，我们讨论了这些方法的扩展到由Dirichlet系列（例如Ramanujan Zeta函数）定义的其他函数。
In this paper we discuss various potentials related to the Riemann zeta function and the Riemann Xi function. These potentials are modified versions of Morse potentials and can also be related to modified forms of the radial harmonic oscillator and modified Coulomb potential. We use supersymmetric quantum mechanics to construct their ground state wave functions and the Fourier transform of the ground state to exhibit the Riemann zeros. This allows us to formulate the Riemann hypothesis in terms of the location of the nodes of the ground state wave function in momentum space. We also discuss the relation these potentials to one and two matrix integrals and construct a few orthogonal polynomials associated with the matrix models. We relate the Schrodinger equation in momentum space to and finite difference equation in momentum space with an infinite number of terms. We computed the uncertainty relations associated with these potentials and ground states as well as the Shannon Information entropy and compare with the unmodified Morse and harmonic oscillator potentials. Finally we discuss the extension of these methods to other functions defined by a Dirichlet series such as the the Ramanujan zeta function.
