Spectral realization of the Riemann zeros by quantizing H = w(x)(p + ?_p~2/p): the Lie-Noether symmetry approach

摘要

If t_n are the heights of the Riemann zeros 1/2 + it_n, an old idea, attributed to Hilbert and Polya [6], stated that the Riemann hypothesis would be proved if the t_n could be shown to be eigenvalues of a self-adjoint operator. In 1986 Berry [1] conjectured that tn could instead be the eigenvalues of a deterministic quantum system with a chaotic classical counterpart and in 1999 Berry and Keating [3] proposed the Hamiltonian H = xp, with x and p the position and momentum of a one-dimensional particle, respectively. This was proven not to be the correct Hamiltonian since it yields a continuum spectrum [23] and therefore a more general Hamiltonian H = w(x)(p + ?_p~2/p) was proposed [25], [4], [24] and different expressions of the function w(x) were considered [25], [24], [16] although none of them yielding exactly t_n. We show that the quantization by means of Lie and Noether symmetries [18], [19], [20], [7] of the Lagrangian equation corresponding to the Hamiltonian H yields straightforwardly the Schr?dinger equation and clearly explains why either the continuum or the discrete spectrum is obtained. Therefore we infer that suitable Lie and Noether symmetries of the classical Lagrangian corresponding to H should be searched in order to alleviate one of Berry's quantum obsessions [2].