The local Riemann hypothesis states that the zeros of the Mellin transform of a harmonic-oscillator eigenfunction (on a real or p-adic configuration space) have real part 1/2. For the real case, we show that the imaginary parts of these zeros are the eigenvalues of the Berry-Keating hamiltonian H=(xp+px)/2 projected onto the subspace of oscillator eigenfunctions of lower level. This gives a spectral proof of the local Riemann hypothesis for the reals, in the spirit of the Hilbert-Polya conjecture. The p-adic case is also discussed.
We construct and study a version of the Berry-Keating operator with a built-in truncation of the phase space, which we choose to be a two-dimensional torus. The operator is a Weyl quantisation of the classical Hamiltonian for an inverted harmonic oscillator, producing a difference operator on a finite, periodic lattice. We investigate the continuum and the infinite-volume limit of our model in conjunction with the semiclassical limit. Using semiclassical methods, we show that a specific combination of the limits leads to a logarithmic mean spectral density as it was anticipated by Berry and Ke…
The Berry-Keating operator on $L^2(\rz_>,\ud x)$ and on compact quantum graphs with general self-adjoint realizations
Berry-Keating算子$H_{\mathrm{BK}}:= -\ui\hbar(x\frac{\ud\phantom{x}}{\ud x}+{1/2})$ [M. V. Berry and J. P. Keating，SIAM Rev. 41（1999）236]在希尔伯特空间$L^2(\rz_>,\ud x)$和紧量子图上讨论了薛定谔动力学。证明了定义在$L^2(\rz_>,\ud x)$上的$H_{\mathrm{BK}}$的谱是纯连续的，因而这种对$H_{\mathrm{BK}}$的量子化不能产生假设的以Riemann zeta函数的非平凡零点为特征值的Hilbert-Polya算子.给出了作用于紧量子图上的$H_{\mathrm{BK}}$的所有自伴扩张的完全分类，以及相应的以行列式形式表示的久期方程，其零点决定了$H_{\mathrm{BK}}$的离散谱.此外，还导出了特征值计数函数的精确迹公式和Weyl渐近性.此外，我们还引入了“平方”Berry-Keating算子$H_{\mathrm{BK}}^2:= -x^2\frac{\ud^2\phantom{x}}{\ud x^2}-2x\frac{\ud\phantom{x}}{\ud x}-{1/4}$，它是金融期权定价理论中Black-Scholes算子的一个特例。再次，所有的自伴扩张，相应的久期方程，迹公式和W…
We first explain the link between the Berry-Keating Hamiltonian and the spectral realization of zeros of the Riemann zeta function of the first author, and why there is no conflict at the semi-classical level between the "absorption" picture of A. Connes and the semiclassical "emission" computations of M. Berry and J. Keating, while the minus sign manifests itself in the Maslov phases. We then use the quantized calculus to analyse the recent attempt of X.-J. Li at proving Weil's positivity, and understand its limit. We then propose an operator theoretic semi-local framework directly related to…
Hamiltonian with Energy Levels Corresponding to Riemann Zeros
College of Physics and Information Engineering, Zhaotong University（昭通大学物理与信息工程学院）；Institute for Astronomy, School of Physics, Zhejiang University（浙江大学物理学院天体物理研究所）
A Hamiltonian operator $\hat H$ 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 $\hat H$ is $2xp$, which is consistent with the Berry-Keating conjecture. While $\hat H$ is not Hermitian in the conventional sense, ${\rm i}{\hat H}$ is ${\cal PT}$ symmetric with a broken ${\cal PT}$ symmetry, thus allowing for the possibility that all eigenvalues of $\hat H$ are real. A heuristic analysis is presented for the construction of the m…
We analize the Berry-Keating model and the Sierra and Rodríguez-Laguna Hamiltonian within the polymeric quantization formalism. By using the polymer representation, we obtain for both models, the associated polymeric quantum Hamiltonians and the corresponding stationary wave functions. The self-adjointness condition provide a proper domain for the Hamiltonian operator and the energy spectrum, which turned out to be dependent on an introduced scale parameter. By performing a counting of semiclassical states, we prove that the polymer representation reproduces the smooth part of the Riemann-von …
We propose the construction of an ensemble of unitary random matrices (UMM) for the Riemann zeta function. Our approach to this problem is `$p$-iecemeal', in the sense that we consider each factor in the Euler product representation of the zeta function to first construct a UMM for each prime $p$. We are able to use its phase space description to write the partition function as the trace of an operator that acts on a subspace of square-integrable functions on the $p$-adic field. This suggests a Berry-Keating type Hamiltonian. We combine the data from all primes to propose a Hamiltonian and a m…
Quantum gravity is expected to gauge all global symmetries of effective theories, in the ultraviolet. Inspired by this expectation, we explore the consequences of gauging CPT as a quantum boundary condition in phase space. We find that it provides for a natural semiclassical regularisation and discretisation of the continuous spectrum of a quantum Hamiltonian related to the Dilation operator. We observe that the said spectrum is in correspondence with the zeros of the Riemann zeta and Dirichlet beta functions. Following ideas of Berry and Keating, this may help the pursuit of the Riemann hypot…
We construct a formally self-adjoint Hamiltonian whose eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. We consider a two-dimensional Hamiltonian which couples the Berry-Keating Hamiltonian to the number operator on the half-line via a unitary transformation. We demonstrate that the unitary operator, which is composed of squeeze (dilation) operators and an exponential of the number operator, confines the eigenfunction of the Hamiltonian to one dimension as the squeezing parameter tends towards infinity. The Riemann zeta function appears at the boundary of the result…
Odlyzko has computed a data set listing more than $10^9$ successive Riemann zeros, starting at a zero number beyond $10^{23}$. The data set relates to random matrix theory since, according to the Montgomery-Odlyzko law, the statistical properties of the large Riemann zeros agree with the statistical properties of the eigenvalues of large random Hermitian matrices. Moreover, Keating and Snaith, and then Bogomolny and collaborators, have used $N \times N$ random unitary matrices to analyse deviations from this law. We contribute to this line of study in two ways. First, we point out that a natur…
We construct a Hamiltonian H whose discrete spectrum contains, in a certain limit, the Riemann zeros. H is derived from the action of a massless Dirac fermion living in a domain of Rindler spacetime, in 1+1 dimensions, that has a boundary given by the world line of a uniformly accelerated observer. The action contains a sum of delta function potentials that can be viewed as partially reflecting moving mirrors. An appropriate choice of the accelerations of the mirrors, provide primitive periodic orbits associated to the prime numbers $p$, whose periods, measured by the observer's clock, are log…
We investigate the asymptotic behavior of eigenfunctions of the Laplacian on Riemannian manifolds. We show that Benjamini-Schramm convergence provides a unified language for the level and eigenvalue aspects of the theory. As a result, we present a mathematically precise formulation of Berry's conjecture for a compact negatively curved manifold and formulate a Berry-type conjecture for sequences of locally symmetric spaces. We prove some weak versions of these conjectures. Using ergodic theory, we also analyze the connections of these conjectures to Quantum Unique Ergodicity.
我们扩展到前300个黎曼零点，我们在arXiv：math-ph/0606005中报告的分析形式，其中最大的研究涉及前75个零点。再次，我们模型的非光滑波动的Wu-Sprung势的一部分，再现黎曼零点，由交替符号的正弦级数分形Berry和Lewis A（x，g）。将分形维数设置为3/2。我们估计频率参数（g）加上引入的总缩放参数（s）。我们搜索这对参数（g，s），最小化最低的300个特征值的最小二乘拟合-通过求解一维固定（非分形）薛定谔方程与审判潜力（光滑加上非光滑部分）-到前300个黎曼零点。我们在矩形0 < s < 3，0 < g < 25内随机采样值。得到的拟合进行比较，通过简单地使用无任何分形补充的吴-Sprung势的光滑部分。一些有限的改善再次被发现。有两个（主要和次要）相当不同的子域，其中值给予改善拟合集中。
Wu和Sprung（Phys. Rev. E 48，2595（1993））使用一维局部势模型再现了前500个非平凡的Riemann零点。他们得出结论--类似于van Zyl和Hutchinson（Phys. Rev. E 67，066211（2003））--势具有维数d=3/2的分形结构。我们用Berry和Lewis A（x，g）的交替符号正弦级数分形来模拟势的非光滑起伏部分。设置d=3/2，我们估计频率参数（gamma），加上我们引入的总体缩放参数（sigma）。我们搜索这对参数（伽马，西格玛），最小化最小二乘拟合S_{n}（伽马，西格玛）的最低n个特征值-通过求解一维固定（非分形）薛定谔方程与试验潜力（光滑加上非光滑部分）-最低n个黎曼零点n =25。对于我们研究的其他情况，n=50和75，我们简单地设置sigma=1。得到的拟合相比，那些通过使用吴-Sprung势的光滑部分，没有任何分形补充。在我们的（非优化的，计算受限的）搜索程序中发现了一些有限的改善- 5.7261 vs. 6.39207（n=25），11.2672 vs. 11.7002（n=50）和16.3119 vs. 16.6809（n=75）。改进在gamma=3和（其平方）9附近相对较强。此外，我们扩展了吴-Sprung半经典框架，包括高阶修正的黎曼-冯Mangoldt公式（超越领先的，占主导地位的长期）到光…
