Ultrafunctions and applications

Benci, Vieri; Baglini, Lorenzo Luperi

doi:10.3934/dcdss.2014.7.593

Cited by 13 publications

(28 citation statements)

References 14 publications

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“…The proof of the existence of such a field is an easy consequence of the Krull -Zorn theorem. It can be found, e.g., in [24,13,14,26]. In this paper, we also use the complexification of E, denoted 5 by…”

Section: The Field Of Euclidean Numbersmentioning

confidence: 99%

“…An explicit technical construction of (several) ultrafunctions spaces has been provided in Ref. [13,16,17,14,15,19,18] by various reformulations of nonstandard analysis. However, we prefer to pursue an axiomatic approach to underscore the key properties of ultrafunctions needed for our aims since such technicalities are not important in the applications of the ultrafunctions spaces.…”

Section: Ultrafunctionsmentioning

confidence: 99%

“…In this paper, we build a non-Archimedean approach to quantum mechanics in a simpler way through a new space, which can be used as a basic construction in the description of a physical system, by analogy with the Hilbert space in the standard approach. This space is called the space of ultrafunctions, a particular class of non-Archimedean generalized functions [13,14,15,16,17,18,19]. The ultrafunctions are defined on the hyperreal field R * , which extends the reals R by including infinitesimal and infinite elements into it.…”

Section: Introductionmentioning

confidence: 99%

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Infinitesimal and infinite numbers as an approach to quantum mechanics

Benci

Baglini

Simonov

2019

Quantum

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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].

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Section: The Field Of Euclidean Numbersmentioning

confidence: 99%

Section: Ultrafunctionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Infinitesimal and infinite numbers as an approach to quantum mechanics

Benci

Baglini

Simonov

2019

Quantum

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“…In this paper we deal with ultrafunctions, which are a kind of generalized functions that have been introduced recently in [1] and developed in [2,[4][5][6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Generalized solutions of variational problems and applications

Benci¹,

Baglini

Squassina

2018

Advances in Nonlinear Analysis

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Ultrafunctions are a particular class of generalized functions defined on a hyperreal field R * ⊃ R that allow to solve variational problems with no classical solutions. We recall the construction of ultrafunctions and we study the relationships between these generalized solutions and classical minimizing sequences. Finally, we study some examples to highlight the potential of this approach.2010 Mathematics Subject Classification. 03H05, 26E35, 28E05, 46S20.

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“…Colombeau [13] and M. Sato [18,19]. This paper deals with a new kind of generalized functions, called "ultrafunctions", which have been introduced recently in [2] and developed in [5,6,7,8,9]. They provide generalized solutions to certain equations which do not have any solution, not even among the distributions.…”

Section: Introductionmentioning

confidence: 99%

The Caccioppoli ultrafunctions

Benci

Berselli

Grisanti

2017

Advances in Nonlinear Analysis

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Ultrafunctions are a particular class of functions defined on a hyperreal field R * ⊃ R. They have been introduced and studied in some previous works ([2],[5],[6]). In this paper we introduce a particular space of ultrafunctions which has special properties, especially in term of localization of functions together with their derivatives. An appropriate notion of integral is then introduced which allows to extend in a consistent way the integration by parts formula, the Gauss theorem and the notion of perimeter. This new space we introduce, seems suitable for applications to PDE's and Calculus of Variations. This fact will be illustrated by a simple, but meaningful example.

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Ultrafunctions and applications

Cited by 13 publications

References 14 publications

Infinitesimal and infinite numbers as an approach to quantum mechanics

Infinitesimal and infinite numbers as an approach to quantum mechanics

Generalized solutions of variational problems and applications

The Caccioppoli ultrafunctions

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