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

If t_n are the heights of the Riemann zeros 1/2 + it_n, an old idea, attributed toH ilbert 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 t_n 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 + ℓ_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¨odinger 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].