[Paper Review] S-Bases in S-Linear Algebra

David Carfì|arXiv (Cornell University)|Apr 17, 2011

Graph theory and applications11 references4 citations

TL;DR

This paper introduces S-bases—continuous, distributional bases indexed by Euclidean space—for the space of tempered distributions, providing a rigorous mathematical framework analogous to Hilbert bases in separable Hilbert spaces. The key contribution is establishing that S-bases are characterized by the superposition operator being a topological isomorphism, enabling a complete, manageable model for physical bases in quantum mechanics.

ABSTRACT

In this paper we define the S-bases for the spaces of tempered distributions. These new bases are the analogous of Hilbert bases of separable Hilbert spaces for the continuous case (they are indexed by m-dimensional Euclidean spaces) and enjoy properties similar to those shown by algebraic bases in the finite dimensional case. The S-bases are one possible rigorous and extremely manageable mathematical model for the "physical" bases used in Quantum Mechanics.

Motivation & Objective

• To formalize a continuous, distribution-based analog of Hilbert and algebraic bases in infinite-dimensional spaces.
• To address the need for a mathematically rigorous model of the 'bases' used in quantum mechanics, particularly in the context of continuous spectra.
• To define and characterize S-linear independence and S-generators as foundational concepts for constructing S-bases.
• To establish topological isomorphism conditions under which an S-family becomes a full S-basis for the space of tempered distributions.

Proposed method

• Define S-linear independence via the condition that a zero S-linear combination (integral over R^m with coefficient distribution a) implies a = 0 in S’_m.
• Introduce the superposition operator ∫(·,v) : S’_m → S’_n, mapping coefficient distributions to their linear combinations over the index space R^m.
• Use the adjoint operator ŵ : S_n → S_m to characterize duality and injectivity conditions for S-bases.
• Apply the Dieudonné-Schwartz theorem to link weak* and strong topologies with surjectivity and closed range of the superposition operator.
• Establish equivalence between S-basis conditions and topological isomorphisms on weak* and strong dual topologies.
• Use examples such as the Dirac family and Fourier families to demonstrate S-linear independence and validate the framework.

Experimental results

Research questions

• RQ1 How can a continuous family of tempered distributions serve as a basis in the absence of finite or countable bases?
• RQ2 What conditions ensure that a family of distributions is S-linearly independent, and how does this differ from standard linear independence?
• RQ3 Under what topological conditions does an S-family generate the entire space of tempered distributions?
• RQ4 How does the superposition operator relate to the structure of S-bases, and when is it a topological isomorphism?
• RQ5 Can physical bases in quantum mechanics, such as momentum or position states, be rigorously modeled as S-bases in tempered distribution spaces?

Key findings

• The Dirac family (δ_x)_{x∈R^n} is S-linearly independent because ∫ a(x)δ_x dx = 0 implies a = 0 in S’_m.
• The (a,b)-Fourier family (a^{-n}e^{-ib(p|·)})_{p∈R^n} is S-linearly independent due to the injectivity of the Fourier-Schwartz transform.
• The family of first derivatives of Dirac distributions (δ’_x)_{x∈R} is linearly independent but not S-linearly independent, as the derivative operator on S’_1 is surjective but not injective.
• An S-basis exists if and only if the superposition operator ∫(·,v) is a topological isomorphism for both weak* and strong topologies on the dual spaces.
• A family v is an S-basis if and only if it is total in both the function space S_n and the coefficient distribution space S’_m.
• The operator ŵ : S_n → S_m is a topological isomorphism if and only if v is an S-basis, establishing a duality between the test function space and the coefficient space.