A large, and physically important, class of linear hermitian/non-hermitian, elliptic partial differential equations are transformable into a representation corresponding to a finite difference equation for the power moments of the physical, exponentially bounded, solutions. We refer to this as the Moment Equation Representation (MER). We show that for such MER systems, the use of a novel, non-orthonormal but complete, weighted polynomial expansion procedure yields tight bounds to the discrete state eigenvalues. Our formalism is referred to as the Orthonormal Polynomial Projection Quantization Bounding Method (OPPQ-BM) [C. R. Handy, 2021 Phys. Scr. 96 075201]. It exploits certain positivity (nonnegativity) structures inherent to the formalism, combined with the exactness of the OPPQ projection analysis. Bounding methods are important for delicate systems where traditional methods can yield varying results. These are usually characterized by strong coupling, short range, interactions, requiring the application of singular perturbation analysis (SPA). The relevance of power moments and SPA has long been appreciated (C. R. Handy, 1981 Phys. Rev. D 24, 378); as well as their relevance for function-wavelet reconstruction (C. R. Handy and R. Murenzi 1996 Phys. Rev. A 54, 3754). We discuss both OPPQ-BM for one dimensional and two dimensional examples. This result coincides with a similar breakthrough, using more traditional approaches (and limited to Hermitian systems), by Martinazzo and Pollak [2020 Proc. of the Natl. Acad. Sc., 117 16181]. The OPPQ-BM approach presented, is based, in part, on an eigenvalue approximation ansatz by Handy and Vrinceanu [2013 J.Phys. A: Math. Theo. 46 135202; 2013 J. Phys. B: At. Mol. Opt. Phys, 46, 115002). This work is the culmination of a formalism introduced in 1985 that was the first to exploit the Moment Problem theorems as applied to certain Sturm-Liouville systems, yielding tight bounds, for some of the discrete state eigenvalues. This method is referred to as the Eigenvalue Moment Method.
(EMM) and was one of the first to introduce Semidefinite Programming (Convex Optimization) related analysis to the quantization problem for physical systems [Handy and Bessis (et al), 1985 Phys Rev. Lett. 55 931;
1988 Phys. Rev. Lett. 60 253]. We emphasize that power moments define an affine map representation, in which scaling, translating, rotating become important transformations, and enables the results achieved through EMM (an affine map invariant variational procedure) and OPPQ. Affine map transforms are also important for affine map generated fractals (i.e. iterated function system, a la M. Barnsley), and may offer a better representation in which to quantize gravity. All of this augur the importance of the methods introduced here.
Physicist Carlos Handy was born in Havana, Cuba, to a Cuban mother and an American father. English was his second language. His grandfather, W.C. Handy, is known as “Father of the Blues.” Growing up in New York City, Handy attended George Washington High School where he was a top math student (and captain of the Academic team that appeared on NBC’s TV-Show: “It’s Academic”). In 1972, Handy earned his B.A. degree in physics from Columbia College in New York. As a freshman, he programmed the mathematical equations of Martin Gutzwiller (IBM-Watson Labs, who would become the “Father of Quantum Chaos”). He then continued his studies at Columbia University, earning the Ph.D. degree in theoretical particle physics in 1978. One of his mentors was John R. Klauder, noted mathematical physicist, famous for Coherent States.
From 1978 to 1981, Handy worked as a postdoctoral research associate at Los Alamos National Laboratory focusing on the use of extensive (non-local) representations for quantizing strong coupling (short range) systems. This led to exploiting the use of moment representations with which to better understand (i.e. regulate) multiscale dynamics. This approach, when implemented through the scaling transform, readily yields the continuous wavelet representation; however, Handy preferred the analytical implications made possible through the underlying power moment dependencies. In 1983, Handy was hired by Clark Atlanta University as an associate professor of physics. During his time there, he received grant money from the National Science Foundation (NSF), which led to his discovery of the Eigenvalue Moment Method (EMM) technique. This work was the first to use Moment Problem (positivity) theorems in quantizing physical systems, and introduced convex optimization analysis formalisms (both linear programming and semidefinite programming analysis) in the algorithmic implementation of the underlying theory. This resulted in some of the most accurate calculations for the energy levels of hydrogenic atoms in super-strong magnetic fields (i.e. neutron stars). In these endeavors, his association with Daniel Bessis proved invaluable.
EMM corresponds to an affine map invariant variational procedure with remarkable properties based on exploiting positivity, combined with the exact representation afforded by working with power moment representations. Affine maps are relevant in many other important areas including (iterated function based) fractal generation and inversion, and its relevance for quantizing gravity.
With a second grant from the NSF, Handy (together with A. Msezane) established the Center for Theoretical Studies of Physical Systems at Clark Atlanta University (CAU), a CREST research and student mentoring center. While at CAU he raised over $16,000,000 in grants (NSF, DARPA, NASA, etc.). In addition to producing various Ph.D. students, Handy also oversaw the mentoring of various young professionals including Romain Murenzi, presently Executive Director of the World Academy of Sciences, Triest, Italy. In 2005, Handy left Clark Atlanta University and became the head of the physics department at Texas Southern University, making it one of the more research active departments, through the hiring of young, dynamic, investigators.
Throughout his career, Handy published numerous research articles. The most recent of these was an extension of EMM to determining the symmetry breaking regime of an important pseudo-hermitian system, and application to Regge pole scattering analysis in atomic and molecular physics. The present
work is the culmination of this decades long search for extending the EMM philosophy to arbitrary states of low dimension systems, either bosonic or fermionic, regardless of their hermitian or non-hermitian character. His professional concerns include the need for modern facilities in physics education as well as students’ early mastery of calculus.