Spectral realization of the Riemann zeros by quantizing H = w(x)(p + ℓ2p/p): the Lie-Noether symmetry approach

If tn are the heights of the Riemann zeros 1/2 + itn, an old idea, attributed to Hilbert and Polya [6], stated that the Riemann hypothesis would be proved if the tn 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 + ℓ2p/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 tn. 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 Schrödinger 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].


Introduction
In [2] Michael Berry wrote: Why should a physicist be concerned with the zeros of the Riemann zeta function ζ(s), and in particular the Riemann hypothesis, according to which all complex zeros of ζ(s) have Res = 1 2 ? Not for the reasons that motivate mathematicians, e.g. fluctuations in the distribution of primes. Rather, my interest grew from the study of quantum systems whose classical counterparts possess chaotic trajectories.
One hundred years ago Polya and Hilbert suggested that in order to prove the Riemann hypothesis one has to find a self-adjoint operator whose spectrum contains the imaginary part of the nontrivial Riemann zeros [6]. Michael Berry suggested the existence of a classical Hamiltonian whose quantum version would realize the Polya-Hilbert conjecture. This conjectured Hamiltonian must satisfy the following conditions [25]: (i) be chaotic, with isolated periodic orbits related to the prime numbers, (ii) break time reversal symmetry, to agree with the Gaussian unitary ensemble statistics, (iii) be quasi-one dimensional.
The first classical Hamiltonian proposed by Berry, i.e.
H cl = xp was then shown by Berry and Keating [3] not to fulfill the requirement (i).
Sierra and collaborators [25], [24] and also Berry and Keating [4] have proposed different modifications of the xp Hamiltonian in order to have bounded classical trajectories and a discrete quantum spectrum. The first such Hamiltonian was [25] Others followed, in particular In the present paper we study the Lagrangian equations that derive from those Hamiltonians. In particular, we find their Lie point symmetries, their Jacobi last multipliers, various Lagrangians and the Noether symmetries that they admit. Finally we use those Noether symmetries to straightforwardly construct the Schrödinger equations [18], [19], [20], [7]. The paper is organized in the following way. In the next section we recall the properties of the Jacobi last multiplier, its connection with Lagrangians of second-order equations [12], [27], and the link with Lie symmetries [14], [15], [5]. In section 3 we study the Hamiltonian (1) and in section 4 the Hamiltonian (48). The last section contains some final remarks.

Jacobi last multiplier, Lie symmetries, Lagrangians
The method of the Jacobi Last Multiplier [9]- [12] provides a means to determine all the solutions of the partial differential equation or its equivalent associated Lagrange's system In fact, if one knows the JLM and all but one of the solutions, then the last solution can be obtained by a quadrature. The JLM M is given by where ∂(f, ω 1 , ω 2 , . . . , ω n−1 ) and ω 1 , . . . , ω n−1 are n − 1 solutions of (3) or, equivalently, first integrals of (4) independent of each other. This means that M is a function of the variables (x 1 , . . . , x n ) and depends on the chosen n − 1 solutions, in the sense that it varies as they vary. The essential properties of the JLM are: (a) If one selects a different set of n − 1 independent solutions η 1 , . . . , η n−1 of equation (3), then the corresponding JLM N is linked to M by the relationship: (b) Given a non-singular transformation of variables then the JLM M ′ of A ′ F = 0 is given by: where M obviously comes from the n − 1 solutions of AF = 0 which correspond to those chosen for A ′ F = 0 through the inverse transformation τ −1 . (c) One can prove that each JLM M is a solution of the following linear partial differential equation: or equivalently:  (3), then their ratio is a solution ω of (3), or, equivalently, a first integral of (4). Naturally the ratio may be quite trivial, namely a constant. Viceversa the product of a multiplier M 1 times any solution ω yields another JLM M 2 = M 1 ω.
Since the existence of a solution/first integral is consequent upon the existence of symmetry, an alternate formulation in terms of symmetries was provided by Lie [14], [15]. A clear treatment of the formulation in terms of solutions/first integrals and symmetries is given by Bianchi [5].
If we know n − 1 symmetries of (3)/(4), say Another property of the JLM is its (almost forgotten) relationship with the Lagrangian, L = L(t, q,q), for any second-order equation i.e. [12] (Lecture 10) 1 , where M = M (t, q,q) satisfies the following equation Then equation (11) becomes the Euler-Lagrange equation: The proof is based on taking the derivative of (14) with respect toq and showing that this yields (13). If one knows a JLM, then L can be easily obtained by a double integration, i.e.: where f 1 and f 2 are functions of t and q which have to satisfy a single partial differential equation related to (11) [21]. As it was shown in [22], f 1 , f 2 are related to the gauge function g = g(t, q).
In fact, we may assume where f 3 has to satisfy the mentioned partial differential equation and g is obviously arbitrary. In [21] it was shown that if one knows several (at least two) Lie symmetries of the second-order differential equation (11), i.e.
then many Jacobi Last Multipliers could be derived by means of (10), i.e.
with (n, m = 1, r), and therefore many Lagrangians can be obtained by means of (15). 1 Jacobi's Lectures on Dynamics are finally available in English [13]. 3. The case H = x(p + ℓ 2 p /p) Sierra has proposed the following Hamiltonian [25], [24]: This Hamiltonian yields the Lagrangian equation: This equations admits an eight-dimensional Lie point symmetry algebra sl(3, R) generated by which implies that equation (20) is linearizable by means of a point transformation 2 [15]. In order to find the linearizing transformation we have to look for an abelian intransitive subalgebra of sl(3, R) and, following Lie's classification of two-dimensional algebras in the real plane [15], we have to transform it into the canonical form ∂x,t∂x withx andt the new dependent and independent variables, respectively. We have found that such subalgebra 3 is that generated by Γ 7 and Γ 8 . Then it is easy to derive that and equation (20) is transformed into the free-particle equation, i.e.: Several JLM can be obtained by means (18) and two of the eight symmetries (21). Consequently several Lagrangians can be derived from (15). In particular from Γ 7 and Γ 8 comes the Lagrangian that admits the following five Noether symmetries [17] If we quantize by preserving those Noether symmetries then the following Schrödinger equation is obtained: 2 We prefer to cite the original work by Sophus Lie, although many textbooks/papers on Lie symmetries report his findings. 3 We may find other such subalgebras but they are obviously related by a point transformation since there exist only one abelian intransitive subalgebra of sl(3, R) [26]. For example, two other abelian intransitive subalgebras are those generated by < Γ1, Γ3 >, and < Γ2, Γ5 >, respectively. and its spectrum is obviously a continuum [24]. Indeed equation (27) admits the following five Lie symmetries and also Φ∂ Φ , and φ(t, x)∂ Φ with φ any solution of (27).
From Γ 1 and Γ 3 comes another Lagrangian that admits the following five Noether symmetries [17] If we quantizing by requiring the preservation of the five Noether symmetries (30) then the following parabolic equation is obtained Its Lie symmetries are generated by 4 and can be reduced to its canonical form by the change of independent variable x = ζ exp(2t), i.e. Ω tt − Ω = 0 (33) that can be solved to give with f 1 , f 2 arbitrary functions of ζ. Equation (33) is not surprising. Indeed equation (20) can be transformed intoz by means of the transformation z = x 2 exp(bt) where b is an arbitrary constant. Substituting b = −3 into (35) yields 4 Also Ω∂Ω, ω(t, x)∂Ω with ω any solution of (31). which corresponds exactly to (33). The Lagrangian associated with the Hamiltonian (1) is given by 5 [24]: This Lagrangian admits three Noether symmetries only, i.e.: These symmetries generate the following family of equations: where A is an arbitrary constant. The three symmetries (39) yield three JLM and consequently three Lagrangians. In particular Γ 5 and Γ 8 yields the JLM by means of (18), and the Lagrangian by means of (15). Lagrangian L 58 coincides with L S if A = 0, but in this case M 58 does not exist since in (18) ∆ 5 8 is equal to zero. This means that the JLM that yields the Lagrangian L S does not come from the two symmetries Γ 5 and Γ 8 . If A = 0 then L 58 admits only two Noether symmetries, namely Γ 4 and Γ 5 . If we quantizing by requiring the preservation of the three Noether symmetries (39) then the following parabolic equation is obtained Its Lie symmetries are generated by 6 and can be reduced to its canonical form by the change of independent variable x = ξ exp(t), that can be solved to give with F 1 , F 2 arbitrary functions of ξ. Equation (45) is also not surprising.
which corresponds exactly to (45). 5 The reality of LS implies thatẋ < x. 6 Also Ψ∂Ψ, ψ(t, x)∂Ψ with ψ any solution of (43). Another classical Hamiltonian proposed by Sierra in [24] is This Hamiltonian yields the Lagrangian equation: Since this equation admits an eight-dimensional Lie symmetry algebra then it is linearizable by means of an abelian intransitive subalgebra and we found that the linearizing transformation is that yields the free-particle equation (24). The Lagrangian that derives from the JLM obtained by means of the abelian intransitive subalgebra is and admits five Noether symmetries, while the Lagrangian L B corresponding to the Hamiltonian H B (48), i.e.
admits just three Noether symmetries that actually generate the following family of equations of second orderẍ with A an arbitary constant. Here we do not pursue this case any further since the discussion is analogous to that of the previous section but the formula are quite lenghtier. We observe that the linearizing transformation (50) is singular and resembles somehow the transformation between the free-particle and the linear harmonic oscillator: a similar instance recurs also in the case of a Liénard-type nonlinear oscillator [7]. We will address this issue in future work.

Final Remarks
We have shown that Lie symmetries may explain what goes wrong/right if one takes a classical (even non-physical) problem into the realm of quantum mechanics.
In fact, the application of Lie symmetries yield that: • the Lagrangian equation (20) admits an eight dimensional Lie symmetry algebra, therefore is linearizable and can be transformed to either a free-particle by a t-dependent transformation (23) or a variety of linear equations (35); • there exist many Lagrangians for equation (20) but only one, up to a representation of the two-dimensional abelian intransitive (Type II) subalgebra, admits the maximum number (five) of Noether symmetries; • those five symmetries lead to the derivation of a time-dependent Schrödinger equation (27) with a continuum spectrum; • the Hamiltonian (1) introduced by Sierra corresponds to a t-independent Lagrangian (38) which admits three Noether symmetries Γ 4 , Γ 5 , Γ 8 : those three symmetries generate the complete symmetry group of the family of equations (40); • those three symmetries are not admitted by the Lagrangian (42) of equation (40) unless A = 0: the Lagrangian (42) can be obtained through the JLM coming from the two symmetries Γ 5 and Γ 8 but not if A = 0 • those three symmetries lead to the Schrödinger equation (43) that admits a continuum spectrum.
Indeed we may have transformed one of Berry's quantum obsessions [2] into a Lie symmetry obsession.