A unified approach to Schrödinger evolution of superoscillations and supershifts

Aharonov, Yakir; Behrndt, Jussi; Colombo, Fabrizio; Schlosser, Peter

doi:10.1007/s00028-022-00770-1

Cited by 12 publications

(5 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let Ψ : x −→ Ψ(x) be the spectrum of a compactly supported integrable signal. For any (τ, α, ω) ∈ R 3 and (t, x) ∈ R, for any ξ ∈ Supp Ψ, α ∈ R and (t, x) ∈ R, formula (5.12) follows from the continuity properties of Fresnel type operators on Exp (C) = A 1 (C), see [16, §5.1] or also [3].…”

Section: Examples Related To Harmonic Analysismentioning

confidence: 99%

“…Recently, there has been widespread research on the evolution of superoscillations as initial conditions for Schrödinger equations [3,4,7,12,14,35,33,18] leading to the emergence of new inquiries and questions.…”

Section: Introductionmentioning

confidence: 99%

“…C) = {F ∈ H(C) : |F (z)| = O(exp(B|z| 2 ) for some B > 0}, thus lie beyond A 1 (C) = {F ∈ H(C) : |F (z)| = O(exp(B|z|) for some B > 0},which is the continuity domain for the Fresnel type operator given by convolution with G 0 considered for example in [16, §5.1] or[3].…”

mentioning

confidence: 99%

See 2 more Smart Citations

Superoscillating Sequences and Supershifts for Families of Generalized Functions

Colombo

Sabadini

Struppa

et al. 2022

Complex Anal. Oper. Theory

View full text Add to dashboard Cite

We construct a large class of superoscillating sequences, more generally of $${\mathscr {F}}$$ F -supershifts, where $${\mathscr {F}}$$ F is a family of smooth functions in (t, x) (resp. distributions in (t, x), or hyperfunctions in x depending on the parameter t) indexed by $$\lambda \in {\mathbb {R}}$$ λ ∈ R . The frame in which we introduce such families is that of the evolution through Schrödinger equation $$(i\partial /\partial t - {\mathscr {H}}(x))(\psi )=0$$ ( i ∂ / ∂ t - H ( x ) ) ( ψ ) = 0 ($${\mathscr {H}}(x) = -(\partial ^2/\partial x^2)/2 + V(x)$$ H ( x ) = - ( ∂ 2 / ∂ x 2 ) / 2 + V ( x ) ), V being a suitable potential). If $${\mathscr {F}}= \{(t,x) \mapsto \varphi _\lambda (t,x)\,;\, \lambda \in {\mathbb {R}}\}$$ F = { ( t , x ) ↦ φ λ ( t , x ) ; λ ∈ R } , where $$\varphi _\lambda $$ φ λ is evolved from the initial datum $$x\mapsto e^{i\lambda x}$$ x ↦ e i λ x , $${\mathscr {F}}$$ F -supershifts will be of the form $$\{\sum _{j=0}^N C_j(N,a) \varphi _{1-2j/N}\}_{N\ge 1}$$ { ∑ j = 0 N C j ( N , a ) φ 1 - 2 j / N } N ≥ 1 for $$a\in {\mathbb {R}}{\setminus }[-1,1]$$ a ∈ R \ [ - 1 , 1 ] , taking $$C_j(N,a) =\left( {\begin{array}{c}N\\ j\end{array}}\right) (1+a)^{N-j}(1-a)^j/2^N$$ C j ( N , a ) = N j ( 1 + a ) N - j ( 1 - a ) j / 2 N . Our results rely on the fact that integral operators of the Fresnel type govern, as in optical diffraction, the evolution through the Schrödinger equation, such operators acting continuously on the weighted algebra of entire functions $$\mathrm{Exp}({\mathbb {C}})$$ Exp ( C ) . Analyzing in particular the quantum harmonic oscillator case forces us, in order to take into account singularities of the evolved datum that occur when the stationary phasis in the Fresnel operator vanishes, to enlarge the notion of $${\mathscr {F}}$$ F -supershift, $${\mathscr {F}}$$ F being a family of $$C^\infty $$ C ∞ functions or distributions in (t, x), to that where $${\mathscr {F}}$$ F is a family of hyperfunctions in x, depending on t as a parameter.

show abstract

Section: Examples Related To Harmonic Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Superoscillating Sequences and Supershifts for Families of Generalized Functions

Colombo

Sabadini

Struppa

et al. 2022

Complex Anal. Oper. Theory

View full text Add to dashboard Cite

show abstract

“…The properties of superoscillations have been the object of extensive study, both from a mathematical [52][53][54][55][56][57][58][59][60][61][62][63] and a physical [64][65][66][67][68][69][70][71][72][73][74][75][76][77] perspective. However, even with all the progress that has been made so far, a clear-cut, rigorous definition of what it means for a function to display superoscillations is still arguably missing.…”

Section: Introductionmentioning

confidence: 99%

A proposal to characterize and quantify superoscillations

Li,

Polo-Gómez,

Martín-Martínez

2024

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…Superoscillation [1][2][3][4][5][6][7][8][9][10][11] is a wave phenomenon which occurs in both classical and quantum contexts wherein the local oscillation frequency or wave vector of a wavefunction exceeds the global band limit of that function (e.g. a wave superposition composed entirely of frequencies of smaller magnitude than f 0 may, in some small region, oscillate faster than f 0 ).…”

Section: Introductionmentioning

confidence: 99%

Madelung mechanics and superoscillations

Waegell

2024

New J. Phys.

View full text Add to dashboard Cite

In single-particle Madelung mechanics, the single-particle quantum state $\Psi(\vec{x},t) = R(\vec{x},t) e^{iS(\vec{x},t)/\hbar}$ is interpreted as comprising an entire conserved fluid of classical point particles, with local density $R(\vec{x},t)^2$ and local momentum $\vec{\nabla}S(\vec{x},t)$ (where $R$ and $S$ are real). The Schr"{o}dinger equation gives rise to the continuity equation for the fluid, and the Hamilton-Jacobi equation for particles of the fluid, which includes an additional density-dependent quantum potential energy term $Q(\vec{x},t) = -\frac{\hbar^2}{2m}\frac{\vec{\nabla}R(\vec{x},t)}{R(\vec{x},t)}$, which is all that makes the fluid behavior nonclassical. In particular, the quantum potential can become negative and create a nonclassical boost in the kinetic energy. This boost is related to superoscillations in the wavefunction, where the local frequency of $\Psi$ exceeds its global band limit. Berry showed that for states of definite energy $E$, the regions of superoscillation are exactly the regions where $Q(\vec{x},t)<0$. For energy superposition states with band-limit $E_+$, the situation is slightly more complicated, and the bound is no longer $Q(\vec{x},t)<0$. However, the fluid model provides a definite local energy for each fluid particle which allows us to define a local band limit for superoscillation, and with this definition, all regions of superoscillation are again regions where $Q(\vec{x},t)<0$ for general superpositions. An alternative interpretation of these quantities involving a \textit{reduced quantum potential} is reviewed and advanced, and a parallel discussion of superoscillation in this picture is given. Detailed examples are given which illustrate the role of the quantum potential and superoscillations in a range of scenarios.

show abstract

A unified approach to Schrödinger evolution of superoscillations and supershifts

Cited by 12 publications

References 40 publications

Superoscillating Sequences and Supershifts for Families of Generalized Functions

Superoscillating Sequences and Supershifts for Families of Generalized Functions

A proposal to characterize and quantify superoscillations

Madelung mechanics and superoscillations

Contact Info

Product

Resources

About