On a multidimensional, algebraic, "shooting method", for generating tight spectral bounds to a large class of strongly coupled, low dimension, quantum systems.

The development of efficient methods for generating tight bounds to the discrete spectrum of quantum operators has been an outstanding problem for over ninety years. Recently, Martinazzo and Pollak (Proc Natl Acad Sci USA 2020 117 (28) 16181-16186), proposed one such approach by devising a method to improve upon the convergence rate of the Temple lower bound method. Despite their achievements, their approach is restricted to hermitian systems. In this work we develop an alternative analysis which is amenable to hermitian and non-hermitian systems. This new approach continues with the general quantization philosophy introduced by Handy and Bessis in the 1980s, exploiting the use of positivity as applied to systems transformable into a moment representation. Their approach required the (pioneering) application of semidefinite programming. We can simplify their analysis through a purely algebraic approach which is tantamount to a multidimensional, "shooting method". Its foundation rests upon a novel use of weighted polynomial expansions. Excellent results have been obtained for the famous quadratic Zeeman effect, surpassing the well known results of Kravchenko et al (1996), and competing with the more recent results of Schimerczek and Wunner (2014).