One-dimensional Schrödinger operators with singular potentials: A Schwartz distributional formulation

Dias, Nuno Costa; Jorge, Cristina; Prata, João Nuno

doi:10.1016/j.jde.2016.01.005

Cited by 13 publications

(18 citation statements)

References 44 publications

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“…In fact, as shown below, the point interactions on these three sets are related to those studied byŠeba in [18] (see Theorem 3 therein). First, we note that in the limit as ε → 0, the divergence of the elementsλ 21 given by (24) and (26) can be excluded at the line L S and in the region Q 2 if the following (resonance) conditions K 2 (a 1 , a 2 ; c)| c=0 =: L 2 (a 1 , a 2 ) = a 1 + a 2 = 0 for N = 2, K 2 (a 1 , a 2 , a 3 ; c)| c=0 =: L 3 (a 1 , a 2 , a 3 ) = a 1 + a 2 + a 3 = 0 for N = 3 (36) hold true, being just a 'linearized' version of equations (32). In what follows we refer the line a 1 + a 2 = 0 (green in figure 2) and the plane a 1 + a 2 + a 3 = 0 (figure 3) to as L 2 -and L 3 -sets, respectively.…”

Section: šEba's Transition At the Line L Smentioning

confidence: 99%

Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials

Zolotaryuk¹

2017

J. Phys. A: Math. Theor.

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Abstract. Several families of one-point interactions are derived from the system consisting of two and three δ-potentials which are regularized by piecewise constant functions. In physical terms such an approximating system represents two or three extremely thin layers separated by some distance. The two-scale squeezing of this heterostructure to one point as both the width of δ-approximating functions and the distance between these functions simultaneously tend to zero is studied using the power parameterization through a squeezing parameter ε → 0, so that the intensity of each δ-potential is c j = a j ε 1−µ , a j ∈ R, j = 1, 2, 3, the width of each layer l = ε and the distance between the layers r = cε τ , c > 0. It is shown that at some values of intensities a 1 , a 2 and a 3 , the transmission across the limit point interactions is non-zero, whereas outside these (resonance) values the one-point interactions are opaque splitting the system at the point of singularity into two independent subsystems. Within the interval 1 < µ < 2, the resonance sets consist of two curves on the (a 1 , a 2 )-plane and three disconnected surfaces in the (a 1 , a 2 , a 3 )-space. While approaching the parameter µ to the critical value µ = 2, three types of splitting these sets into countable families of resonance curves and surfaces are observed.

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Section: šEba's Transition At the Line L Smentioning

confidence: 99%

Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials

Zolotaryuk¹

2017

J. Phys. A: Math. Theor.

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“…The two identities (2.11) and (2.12) were already proved in [12,Theorem 3.3] for the case F = = i c i δ (i) (x). The extension to the case where has support on more than one point (but on a finite set) was also discussed in [12] and is strainghtforward. We then focus on the remaining case F = f ∈ C ∞ p , and consider the simplest example where sing supp f = {0}.…”

Section: Proofmentioning

confidence: 78%

“…In this section we review some basic notions about Schwartz distributions and present the main properties of the multiplicative product * . For details and proofs the reader should refer to [10,12]. We also discuss a smooth regularization of the product * , and prove a new result (Theorem 2.9) that will be used in the next section to prove the equivalence of Eqs.…”

Section: A Multiplicative Product Of Schwartz Distributionsmentioning

confidence: 94%

“…ODEs formally of this form appear naturally in models of non-smooth systems and of systems with singularities (e.g. systems with point interactions in quantum mechanics [1,2,12], beams with structural cracks [7,19,31] in the classical theory of solids, etc). The main problem in these cases is how to define the formal equation (1.1) precisely.…”

Section: Introductionmentioning

confidence: 99%

“…Most significative are the generalized functions formulation in the sense of Colombeau [8,9,17,25], the formulations in terms of distributions acting on discontinuous test functions [2,23,24], and the intrinsic approaches using suitable multiplicative products of Schwartz distributions, e.g. [7,12,19,25,27,31]. 1 In the latter case the entire formulation is strictly defined within the standard space of Schwartz distributions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Ordinary Differential Equations with Singular Coefficients: An Intrinsic Formulation with Applications to the Euler–Bernoulli Beam Equation

Dias¹,

Jorge²,

Prata³

2020

J Dyn Diff Equat

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We study a class of linear ordinary differential equations (ODEs) with distributional coefficients. These equations are defined using an intrinsic multiplicative product of Schwartz distributions which is an extension of the Hörmander product of distributions with nonintersecting singular supports (Hörmander in The analysis of linear partial differential operators I, Springer, Berlin, 1983). We provide a regularization procedure for these ODEs and prove an existence and uniqueness theorem for their solutions. We also determine the conditions for which the solutions are regular and distributional. These results are used to study the Euler-Bernoulli beam equation with discontinuous and singular coefficients. This problem was addressed in the past using intrinsic products (under some restrictive conditions) and the Colombeau formalism (in the general case). Here we present a new intrinsic formulation that is simpler and more general. As an application, the case of a non-uniform static beam displaying structural cracks is discussed in some detail.

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An existence and uniqueness result about algebras of Schwartz distributions

Dias,

Jorge,

Prata

2023

Monatsh Math

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We prove that there exists essentially one minimal differential algebra of distributions $$\mathcal A$$ A , satisfying all the properties stated in the Schwartz impossibility result [L. Schwartz, Sur l’impossibilité de la multiplication des distributions, 1954], and such that $$\mathcal C_p^{\infty } \subseteq \mathcal A\subseteq \mathcal D' $$ C p ∞ ⊆ A ⊆ D ′ (where $$\mathcal C_p^{\infty }$$ C p ∞ is the set of piecewise smooth functions and $$\mathcal D'$$ D ′ is the set of Schwartz distributions over $$\mathbb R$$ R ). This algebra is endowed with a multiplicative product of distributions, which is a generalization of the product defined in [N.C.Dias, J.N.Prata, A multiplicative product of distributions and a class of ordinary differential equations with distributional coefficients, 2009]. If the algebra is not minimal, but satisfies the previous conditions, is closed under anti-differentiation and the dual product by smooth functions, and the distributional product is continuous at zero then it is necessarily an extension of $$\mathcal A$$ A .

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One-dimensional Schrödinger operators with singular potentials: A Schwartz distributional formulation

Cited by 13 publications

References 44 publications

Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials

Families of one-point interactions resulting from the squeezing limit of the sum of two- and three-delta-like potentials

Ordinary Differential Equations with Singular Coefficients: An Intrinsic Formulation with Applications to the Euler–Bernoulli Beam Equation

An existence and uniqueness result about algebras of Schwartz distributions

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