Bender Carl M, Brody Dorje C, Müller Markus P
Department of Physics, Washington University, St. Louis, Missouri 63130, USA.
Department of Mathematics, Brunel University London, Uxbridge UB8 3PH, United Kingdom.
Phys Rev Lett. 2017 Mar 31;118(13):130201. doi: 10.1103/PhysRevLett.118.130201. Epub 2017 Mar 30.
A Hamiltonian operator H[over ^] is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. The classical limit of H[over ^] is 2xp, which is consistent with the Berry-Keating conjecture. While H[over ^] is not Hermitian in the conventional sense, iH[over ^] is PT symmetric with a broken PT symmetry, thus allowing for the possibility that all eigenvalues of H[over ^] are real. A heuristic analysis is presented for the construction of the metric operator to define an inner-product space, on which the Hamiltonian is Hermitian. If the analysis presented here can be made rigorous to show that H[over ^] is manifestly self-adjoint, then this implies that the Riemann hypothesis holds true.
构造了一个哈密顿算符(\hat{H})，其性质为：如果本征函数服从适当的边界条件，那么相关的本征值对应于黎曼ζ函数的非平凡零点。(\hat{H})的经典极限是(2xp)，这与贝里 - 基廷猜想一致。虽然(\hat{H})在传统意义上不是厄米算符，但(i\hat{H})是具有破缺PT对称性的PT对称算符，因此(\hat{H})的所有本征值都为实数是有可能的。针对定义内积空间的度规算符的构造给出了一个启发式分析，在该内积空间上哈密顿算符是厄米的。如果这里给出的分析能够严格证明(\hat{H})是明显自伴的，那么这意味着黎曼假设成立。
