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For low dimension systems admitting a moment equation representation (MER),the development of an effective eigenenergy bounding theory applicable to alldiscrete states had remained elusive, until now. Whereas Handy et al (1988Phys. Rev. Lett. 60 253) demonstrated the effectiveness of the {\it MomentProblem} based, Eigenvalue Moment Method (EMM), for generating arbitrarilytight bounds to the multidimensional, positive, bosonic ground state, itsextension to arbitrary excited states seemed intractable. We have discovered anew, moment representation based, quantization formalism that achieves this.Unlike EMM, no convex optimization methods are required. The entire formulationis algebraic. As a result of our preliminary investigation, we are able tomatch, or surpass, the excellent, but intricate, analysis of Kravchenko et al(1996 Phys. Rev. A 54 287) with respect to the quadratic Zeeman effect, for abroad range of magnetic field strengths. Unlike their analysis, the proposedmethod is simple, involves no truncations, and the projection of the quantumoperator is exact, within each moment subspace. Our new approach, theOrthogonal Polynomial Projection Quantization-Bounding Method (OPPQ-BM),exploits the implicit bounding capabilities of a previous method developed byHandy and Vrinceanu (2013 J. of Phys. A: Math. Theor. 46 135202). What emergesis a completely new type of analysis (i.e. constrained quadratic formminimization) that validates the importance of moment equation representationsfor quantizing physical systems. Whereas the underlying principles of EMMguarantee it to be more efficient than OPPQ-BM, the ability to implementalgebraic computations, as opposed to pursuing nonlinear convex optimizationmethods (which can be relaxed through linear programming alternatives)recommends OPPQ-BM. We give an overview of the new method with applications.

Generating Converging Eigenenergy Bounds for Multidimensional Systems: A New Moment Representation, Algebraic, Quantization Formalism