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Fractional Supershifts and their associated Cauchy Evolution problems

Natanael Alpay

TL;DR

This work extends the theory of supershifts and superoscillations to fractional Fock spaces associated with Gelfond–Leontiev derivatives, defining fractional supershifts F_{n,φ} and analyzing their evolution under the Schrödinger dynamics. It develops a GL-based fractional differentiation framework, constructs the corresponding fractional Fock space 𝓕_φ with kernel k_φ and weight K_φ via a Mellin-transform relation, and establishes the supershift limit lim_{n→∞} F_{n,φ}(z,a)=K_φ(−i a z). The oscillatory-integral analysis in distributional terms via regularized kernels I_m^ε leads to a rigorous Fourier integral operator framework, enabling the explicit solution of the evolution problem from fractional supershift initial data. The results yield a Hermite-polynomial–based representation of the Schrödinger evolution, illustrating how fractional calculus and RKHS methods interact to describe high-frequency behavior in fractional quantum-like settings and suggesting broader applications in fractional dynamics and sampling theory.

Abstract

In this work, we extend the notion of supershifts and superoscillation sequence to fractional Fock spaces based on Gelfond-Leontiev fractional derivatives. We first introduce the fractional supershifts sequence, and then discuss the associated evolution Cauchy problem with the fractional supershifts as initial condition.

Fractional Supershifts and their associated Cauchy Evolution problems

TL;DR

This work extends the theory of supershifts and superoscillations to fractional Fock spaces associated with Gelfond–Leontiev derivatives, defining fractional supershifts F_{n,φ} and analyzing their evolution under the Schrödinger dynamics. It develops a GL-based fractional differentiation framework, constructs the corresponding fractional Fock space 𝓕_φ with kernel k_φ and weight K_φ via a Mellin-transform relation, and establishes the supershift limit lim_{n→∞} F_{n,φ}(z,a)=K_φ(−i a z). The oscillatory-integral analysis in distributional terms via regularized kernels I_m^ε leads to a rigorous Fourier integral operator framework, enabling the explicit solution of the evolution problem from fractional supershift initial data. The results yield a Hermite-polynomial–based representation of the Schrödinger evolution, illustrating how fractional calculus and RKHS methods interact to describe high-frequency behavior in fractional quantum-like settings and suggesting broader applications in fractional dynamics and sampling theory.

Abstract

In this work, we extend the notion of supershifts and superoscillation sequence to fractional Fock spaces based on Gelfond-Leontiev fractional derivatives. We first introduce the fractional supershifts sequence, and then discuss the associated evolution Cauchy problem with the fractional supershifts as initial condition.
Paper Structure (7 sections, 8 theorems, 111 equations)

This paper contains 7 sections, 8 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized fractional derivatives
  4. Superoscillations
  5. Fractional Supershifts
  6. Oscillatory Integrals and Distributional Limits
  7. Evolution Problems: Schrödinger equation for the free particle

Key Result

Theorem 2.6

The system $\{e_n\}_{n\ge 0}$ with $e_n:=\sqrt{\varphi_n}\,z^n$ is an ONB of $\mathcal{F}_{\varphi}$.

Theorems & Definitions (27)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Theorem 2.6
  • proof
  • Example 2.7
  • Proposition 2.8: AharonovEtAl2017Memoir
  • Corollary 2.9
  • ...and 17 more