Direct Fisher inference of the quartic oscillator's eigenvalues

It is well known that a suggestive connection links Schr\"odinger's equation (SE) and the information-optimizing principle based on Fisher's information measure (FIM). It has been shown that this entails the existence of a Legendre transform structure underlying the SE. Such a structure leads to a first order partial differential equation (PDE) for the SE's eigenvalues from which a complete solution for them can be obtained. As an application we deal with the quantum theory of anharmonic oscillators, a long-standing problem that has received intense attention motivated by problems in quantum field theory and molecular physics. By appeal to the Cramer Rao bound we are able to Fisher-infer the particular PDE-solution that yields the eigenvalues without explicitly solving Schr\"odinger's equation. Remarkably enough, and in contrast with standard variational approaches, our present procedure does not involve free fitting parameters.


INTRODUCTION
It is well-known that a strong link exists between Fisher' information measure (FIM) I [1] and Schrödinger wave equation (SE) [2][3][4][5][6][7][8]. In a nutshell, this connection is based upon the fact that a constrained Fisher-minimization leads to a SE-like equation [1][2][3][4][5][6][7][8]. In turn, this implies the existence of intriguing relationships between various SE-facets, and Jaynes's maximum entropy principle. In particular, basic SE-consequences such as the Hellmann-Feynman and the Virial theorems can be re-interpreted in terms of a special kind of reciprocity relations between relevant physical quantities, similar to the ones exhibited by the thermodynamics' formalism via its Legendre-invariance property [5,6]. This fact demonstrates that a Legendre-transform structure underlies the non-relativistic Schrödinger equation. As a consequence, the possible energy-eigenvalues are now seen to be constrained by such structure in a rather unsuspected way [5][6][7][8], a fact that allows one to obtain a first-order differential equation, unrelated to Schroedinger's equation [7,8], that energy eigenvalues must necessarily satisfy. The predictive power of this new equation will be explored here.
We will apply our formalism here to the quantum anharmonic oscillator, which is the paradigmatic testing-ground for new approaches to Schroedinger eigenvalue equation. Besides their intrinsic conceptual and mathematical interest, anharmonic oscillators have received considerable attention over the years due to their practical relevance in connection with several areas of physics, such as quantum field theory and molecular physics, among others. In this kind of systems, the most intense focus has been traditionally concentrated upon the quartic oscillator. General accounts containing illuminating references on this problem may be found, for instance, in [9,10]. Note that a perturbation series solution to this problem in powers of the anharmonicity-parameter λ is divergent-asymptotic for all λ > 0 [11]. Specifically, we will apply our procedure to treat the quartic anharmonic oscillator. By appeal to the Cramer Rao bound we obtain the particular solution that leads to the system's eigenvalues without need of explicitly solving Schrödinger's equation. More importantly, we do not need to appeal to any arbitrary or empirical parameter, as is common practice in other treatments [12].

BASIC IDEAS
Let x be a stochastic variable and f (x, θ) the probability density function (PDF) for this variable, which depends on the parameter θ. If an observer were to make a measurement of x and had to best infer θ from such measurement, calling the resulting estimateθ =θ(x), one might ask how well could θ be determined. Estimation theory [1] tells us that the best possible estimatorθ(x), after a very large number of x-samples is examined, suffers a meansquare error ∆x from θ obeying the rule I (∆x) 2 = 1, where I is an information quantifier called the Fisher information measure (FIM), a non linear functional of the PDF that reads Any other estimator must have a larger mean-square error (all estimators must be unbiased, i.e., satisfy θ (x) = θ). Thus, FIM has a lower bound. No matter what the parameter θ might be, I has to obey the celebrated Cramer-Rao bound [1].
In the case of physical Fisher applications the particular instance of translational families merits special consideration. These are mono-parametric distribution families of the form , known up to the shift parameter θ. All family members exhibit identical shape. For such families we get Focus attention now a system that is specified by a set of M physical parameters µ k . We . The set of µ k -values is to be regarded as our prior knowledge. It represents our available empirical information. Let the pertinent probability distribution function (PDF) be f (x). Then, In this context it can be shown (see for example [2,3]) that the physically relevant PDF f (x) minimizes FIM subject to the prior conditions and the normalization condition. Normalization entails dxf (x) = 1, and, consequently, our Fisher-based extremization problem adopts the appearance where we have introduced the (M + 1) Lagrange multipliers λ k (λ 0 = α). In Ref. [2] on can find the details of how to go from (5) to a Schrödinger's equation (SE) that yields the desired PDF in terms of the amplitude and can be formally interpreted as the (real) Schrödinger equation (SE) for a particle of unit mass (h = 1) moving in the effective, "information-related pseudo-potential" U(x) [2] in which the normalization-Lagrange multiplier (α/8) plays the role of an energy eigenvalue.
The λ k are fixed, of course, by recourse to the available prior information. In the case of one-dimensional scenarios, ψ(x) is real [13] and so that using the SE (6) we obtain Legendre structure The connection between the variational solutions f and thermodynamics was established in Refs. [2] and [4] in the guise of reciprocity relations that express the Legendre-transform structure of thermodynamics. They constitute its essential formal ingredient [14] and were re-derivedà la Fisher in [2] by recasting (8) in a fashion that emphasizes the role of the relevant independent variables, Obviously, the Legendre transform main goal is that of changing the identity of our relevant variables. As for I we have so that we encounter the three reciprocity relations (proved in [2]) the last one being a generalized Fisher-Euler theorem.

FISHER MEASURE AND QUANTUM MECHANICAL CONNECTION
The potential function U(x) belongs to L 2 and thus admits of a series expansion in the basis x, x 2 x 3 , etc. [15]. The A k (x) themselves belong to L 2 as well and can also be series-expanded in similar fashion. This enables us to base our future considerations on the assumption that the a priori knowledge refers to moments x k of the independent variable, i.e., A k = x k , and that one possesses information about M of these moments x k . Our "information" potential U then reads We will assume that the first M terms of the above series yield a satisfactory representation of U(x). Consequently, the Lagrange multipliers are identified with U(x)'s series-expansion's coefficients.
In this Schrödinger-scenario the virial theorem states that [5] and thus, from (7) and (13) Also, substituting the above I-expression into (8) and solving for α, we obtain α (I) is explicit function of the M Lagrange multipliers -U(x)'s series-expansion coefficients λ k (associated to the physical parameters x k ). Eqs. (14) and (15) encode the information provided by the virial theorem [5,6].

Fisher-Schröedinger Legendre structure
Interestingly enough, the reciprocity relations (RR) (11) can be re-derived on a strictly pure quantum mechanical basis [5], starting from 1. the quantum Virial theorem [which leads to Eqs. (14) and (15)] plus 2. information provided by the quantum Hellmann-Feynman theorem.
This fact indicates that a Legendre structure underlays the one-dimensional Schröedinger equation [5]. Thus, with A k = x k , our "new" reciprocity relations are given by FIM expresses a relation between the independent variables or control variables (the prior information) and I. Such information is encoded into the functional form I = I( x 1 , ..., x M ).
For later convenience, we will also denote such a relation or encoding as {I, x k }. We see that the Legendre transform FIM-structure involves eigenvalues of the "information-Hamiltonian" and Lagrange multipliers. Information is encoded in I via these Lagrange multipliers, i.e., α = α(λ 1 , ...λ M ), together with a bijection {I, x k } ←→ {α, λ k }.

Two scenarios
In a I, x k -scenario, the λ k are functions dependent on the x k -values. As shown in [6], substituting the RR given by (16) in (14) one is led to a linear, partial differential equations (PDE) for I, and a complete solution is given by where C k are positive real numbers (integration constants). The I -domain is D I = ( x 1 , ..., x M )/ x k ∈ ℜ o . Eq. (18) states that for x k > 0, I is a monotonically decreasing function of x k , and as one expects from a "good" information measure [1], I is a convex function. We may obtain λ k from the reciprocity relations (16). For x k > 0 one gets, and then, using (8), we obtain the α -normalization Lagrange multiplier.
The general solution for the I -PDE does exist and its uniqueness has been demonstrated via an analysis of the associated Cauchy problem [6]. Thus, Eq. (18) implies what seems to be a kind of "universal" prescription, a linear PDE that any variationally (with constraints) obtained FIM must necessarily comply with.
In the {α, λ k } scenario, the x k are functions that depend on the λ k -values. As we showed in [7], an analog α-PDE exists. Substituting the RR given by (16) in (15) we are led to and a complete solution is given by where the D k s are positive real numbers (integration constants). The α-domain is D α = {(λ 1 , · · · , λ M )/λ k ∈ ℜ} = ℜ M . Also, Eq.(21) states that for λ k < 0, α is a monotonically decreasing function of the λ k , and as one expect from the Legendre transform of I, we end up with a concave function. We may obtain the x k 's from the reciprocity relations (16).
For λ k < 0 one gets and then, using (8) one us able to build up I.
The general solution for α -PDE exists. Uniqueness is, again, proved from an analysis of the associated Cauchy problem [7]. Thus, Eq. (18) implies once more a kind of "universal" prescription, a linear PDE that all SE-eigenvalues must necessarily comply with.
The mathematical structure of the Legendre transform leads to a relation between the integration constants C k and D k pertaining to the I and α expressions, respectively, given by (18) and (21). In [7] we studied with some detail this relation. In our two scenarios, I, x k and {α, λ k }, we have [7] Consequently, expressions (18) and (21) take the form, The reciprocity relations (19) and (22) can thus be economically summarized in the fashion

PRESENT RESULTS
The reference quantities F k The essential FIM feature is undoubtedly its being an estimation measure known to obey the Cramer Rao (CR) bound of Eq. (2) [1]. Accordingly, since our partial differential equation has multiple solutions, it is natural to follow Jaynes's MaxEnt ideas and select amongst them the one that optimizes the CR bound, that constitutes the informational operative constraint in Fisher's instance. Of course, Jaynes needs to maximize the entropy instead. We will also, without loss of generality, renormalize the reference quantities F k . This procedure is convenient because it allows us to regard these quantities as statistical weights that optimize the CR-bound. In other words, our procedure entails that we extremize with the constraint We are going to apply now the preceding considerations so as to obtain the eigenvalues of the quartic anharmonic oscillator.
Accordingly, we have, in the {α, λ k } -scenario [Cf. (24)], The functions f and φ defined by (26) and (27) After these preparatory moves we can recast our methodology in a convenient specialized fashion, suitable for the task at hand. We just face the simple two-equations system: from which we obtain F 2 and F 4 . Substituting them into (29) we determine α and, of course, the eigenvalue E = α/8. Consider now our SE (28), taking k = 1 and a given value of λ (0.0001 ≤ λ ≤ 10000). The function [Cf. Eq. 26] f (F 2 , F 4 ) = I x 2 exhibits, as a function of its arguments, a unique "critical" point that satisfies (31). Using f = f critical , that optimizes the CR-bound, we find a ground-state eigenvalue that is in good agreement the literature. In this way, after properly dealing with (24), with the F k regarded as "FIM statistical weights" that optimize the Cramer Rao inequalities, we determine α as a function of the λ k without passing first through a Schrödinger equation, which is a notable aspect of the present approach.
Interestingly enough, the Cramer-Rao inequality us equivalent to the quantum uncertainty principle (see the Appendix for details and references). Thus, our methodology actually employs Heisenberg's celebrated principle to pick up just one solution among the several ones that our partial differential equation possesses.

CONCLUSIONS
On the basis of a variational principle based on Fisher's information measure, free of adjustable parameters, we have obtained the Schrödinger energy-eigenvalues for the fundamental state of the quartic anharmonic oscillator (for several anharmoniticy-values). Our theoretical results, obtained without passing first through a Schrödinger equation, are in a good agreement with those of the literature. This constitutes an illustration of the power of information-related tools in analyzing physical problems.
Thus, we have in this communication introduced a new general technique for eigenvalueproblems of linear operators, whose use seems to constitute a promising venue, given the results here displayed.
Acknowledgments-This work was partially supported by the Projects FQM-2445 and FQM-207 of the Junta de Andalucia (Spain, EU).