多维系统收敛本征能界的产生：一种新的矩表示、代数、量化形式
Generating Converging Eigenenergy Bounds for Multidimensional Systems: A New Moment Representation, Algebraic, Quantization Formalism
摘要：
对于采用矩方程表示的低维系统，直到现在，一个适用于所有离散态的有效的本征能束缚理论的发展仍然是难以捉摸的。而Handy等人（1988 Phys.Rev.Lett. 60 253）证明了基于{ \it矩量问题}的特征值矩量方法（EMM）对于生成多维正玻色基态的任意紧边界的有效性，但其扩展到任意激发态似乎是难以处理的。我们已经发现了一个新的，矩表示为基础，量化形式主义，实现这一点。与EMM不同，不需要凸优化方法。整个公式都是代数的。作为我们的初步调查的结果，我们能够匹配，或超越，优秀的，但复杂的，Kravchenko等人（1996年物理评论A 54 287）的分析，关于二次塞曼效应，为广泛的磁场强度。与他们的分析不同，所提出的方法很简单，不涉及截断，并且量子算符的投影在每个矩子空间内都是精确的。我们的新方法，正交多项式投影量化-定界方法（OPPQ-BM），利用了由Handy和Vrinceanu（2013 J. of Phys. A：Math. Theor. 46 135202）。所出现的是一种全新的分析类型（即约束二次型最小化），它验证了矩方程表示对量化物理系统的重要性。虽然EMM的基本原理保证它比OPPQ-BM更有效，但与追求非线性凸优化方法（可以通过线性规划替代方案放松）相反，实现代数计算的能力推荐OPPQ-BM。我们给出了一个概述的新方法与应用。
英文摘要：
For low dimension systems admitting a moment equation representation (MER), the development of an effective eigenenergy bounding theory applicable to all discrete states had remained elusive, until now. Whereas Handy et al (1988 Phys. Rev. Lett. 60 253) demonstrated the effectiveness of the {\it Moment Problem} based, Eigenvalue Moment Method (EMM), for generating arbitrarily tight bounds to the multidimensional, positive, bosonic ground state, its extension to arbitrary excited states seemed intractable. We have discovered a new, moment representation based, quantization formalism that achieves this. Unlike EMM, no convex optimization methods are required. The entire formulation is algebraic. As a result of our preliminary investigation, we are able to match, 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 a broad range of magnetic field strengths. Unlike their analysis, the proposed method is simple, involves no truncations, and the projection of the quantum operator is exact, within each moment subspace. Our new approach, the Orthogonal Polynomial Projection Quantization-Bounding Method (OPPQ-BM), exploits the implicit bounding capabilities of a previous method developed by Handy and Vrinceanu (2013 J. of Phys. A: Math. Theor. 46 135202). What emerges is a completely new type of analysis (i.e. constrained quadratic form minimization) that validates the importance of moment equation representations for quantizing physical systems. Whereas the underlying principles of EMM guarantee it to be more efficient than OPPQ-BM, the ability to implement algebraic computations, as opposed to pursuing nonlinear convex optimization methods (which can be relaxed through linear programming alternatives) recommends OPPQ-BM. We give an overview of the new method with applications.
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