Generalized functions beyond distributions

Baglini

Simonov

2019

Quantum

Self Cite

Non-Archimedean mathematics is an approach based on fields which contain infinitesimal and infinite elements. Within this approach, we construct a space of a particular class of generalized functions, ultrafunctions. The space of ultrafunctions can be used as a richer framework for a description of a physical system in quantum mechanics. In this paper, we provide a discussion of the space of ultrafunctions and its advantages in the applications of quantum mechanics, particularly for the Schrödinger equation for a Hamiltonian with the delta function potential.Notice that, trivially, every superreal field is non-Archimedean. Infinitesimals allow introducing the following equivalence relation, which is fundamental in all non-Archimedean settings.Definition 2. We say that two numbers ξ, ζ ∈ K are infinitely close if ξ −ζ is infinitesimal. In this case we write ξ ∼ ζ.In the superreal case, ∼ can be used to introduce the fundamental notion of "standard part" 2 . Theorem 3. If K is a superreal field, every finite number ξ ∈ K is infinitely close to a unique real number r ∼ ξ, called the the standard part of ξ.Following the literature, we will always denote by st(ξ) the standard part of any finite number ξ. Moreover, with a small abuse of notation, we also put st(ξ) = +∞ (resp. st(ξ) = −∞) if ξ ∈ K is a positive (resp. negative) infinite number. Definition 4. Let K be a superreal field, and ξ ∈ K a number. The monad of ξ is the set of all numbers that are infinitely close to it,1 Without loss of generality, we assume that Q ⊆ K.2 For a proof of the following simple theorem, the interested reader can check, e.g., [21].

Section: Ultrafunctionsmentioning

confidence: 99%

Section: Potential Barrier (τ > 0)mentioning

confidence: 99%

“…(42) reveals the self-adjoint representation of the Hamiltonian in Eq. (19) in the ultrafunctions setting.…”

Section: The Ultrafunctions Approachmentioning

confidence: 99%

Section: Splitting Of the Ultrafunction Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Infinitesimal and infinite numbers as an approach to quantum mechanics

Baglini

Simonov

2019

Quantum

Self Cite

A Topological Approach to Non-Archimedean Mathematics

Springer Proceedings in Mathematics &Amp; Statistics

Baglini

2016

Self Cite

Non-Archimedean mathematics (in particular, nonstandard analysis) allows to construct some useful models to study certain phenomena arising in PDE's; for example, it allows to construct generalized solutions of differential equations and variational problems that have no classical solution. In this paper we introduce certain notions of non-Archimedean mathematics (in particular, of nonstandard analysis) by means of an elementary topological approach; in particular, we construct non-Archimedean extensions of the reals as appropriate topological completions of R. Our approach is based on the notion of Λ-limit for real functions, and it is called Λ-theory. It can be seen as a topological generalization of the α-theory presented in [6], and as an alternative topological presentation of the ultrapower construction of nonstandard extensions (in the sense of [22]). To motivate the use of Λ-theory for applications we show how to use it to solve a minimization problem of calculus of variations (that does not have classical solutions) by means of a particular family of generalized functions, called ultrafunctions.

An Improved Setting for Generalized Functions: Fine Ultrafunctions

2022

Milan J. Math.

Self Cite

Ultrafunctions are a particular class of functions defined on a Non Archimedean field $${\mathbb {E}}\supset {\mathbb {R}}$$ E ⊃ R . They have been introduced and studied in some previous works (Benci, in Adv Nonlinear Stud 13:461–486, 2013; Benci and Luperi Baglini, in Electron J Differ Equ Conf 21:11–21, 2014; Benci et al., in Adv Nonlinear Anal 10. https://doi.org/10.1515/anona-2017-0225.2; Benci et al., in Adv. Nonlinear Anal 9, 2018). In this paper we develop the notion of fine ultrafunctions which improves the older definitions in many crucial points. Some applications are given to show how ultrafunctions can be applied in studing Partial Differential Equations. In particular, it is possible to prove the existence of ultrafunction solutions to ill posed evolution poblems.