Hamiltonian for the Zeros of the Riemann Zeta Function.
Hamiltonian for the Zeros of the Riemann Zeta Function.
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DOI：
10.1103/physrevlett.118.130201
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发表时间：
2016-08
影响因子：
8.6
通讯作者：
C. Bender;D. Brody;Markus P. Müller
中科院分区：
文献类型：
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作者：
C. Bender;D. Brody;Markus P. Müller
构造了一个哈密顿算子H[over ^]，其性质是：如果本征函数服从适当的边界条件，则相关的本征值对应于Riemann zeta函数的非平凡零点。H[over ^]的经典极限是2xp，这与Berry-Keating猜想是一致的。虽然H[over ^]在传统意义上不是埃尔米特的，但iH[over ^]是PT对称的，具有破缺的PT对称性，因此允许H[over ^]的所有本征值是真实的的可能性。给出了构造度量算子以定义一个内积空间的启发式分析，在该内积空间上的哈密顿量是厄米的。如果这里给出的分析可以被严格地证明H[over ^]明显是自伴的，那么这意味着黎曼假设成立。
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.