Analysis of Superoscillatory Wave Functions

Abstract

Surprisingly, differentiable functions are able to oscillate arbitrarily faster than their highest Fourier component would suggest. The phenomenon is called superoscillation. Recently, a practical method for calculating superoscillatory functions was presented and it was shown that superoscillatory quantum mechanical wave functions should exhibit a number of counter-intuitive physical effects. Following up on this work, we here present more general methods which allow the calculation of superoscillatory wave functions with custom-designed physical properties. We give concrete examples and we prove results about the limits to superoscillatory behavior. We also give a simple and intuitive new explanation for the exponential computational cost of superoscillations.

20 pages, several figures

References in corpus (4)

• Black Holes, Bandwidths and Beethoven
• A Covariant Information-Density Cutoff in Curved Space-Time
• Quantum limitations on superluminal propagation
• Trans-Planckian Tail in a Theory with a Cutoff

Cited by in corpus (9)

• Classical Field Approach to Quantum Weak Measurements
• "Superluminal paradox" in wavepacket propagation and its quantum mechanical resolution
• Accurate "superluminal" transmission via entanglement, superoscillations and quasi-Dirac distributions
• Scaling Properties of Superoscillations and the Extension to Periodic Signals
• Abrupt Rabi oscillations in a superoscillating electric field
• Non-resonant excitation of a two-level atom driven by a superoscillating field
• Structures far below sub-Planck scale in quantum phase-space through superoscillations
• Superoscillating response of a nonlinear system on a harmonic signal
• A proposal to characterize and quantify superoscillations