What question did this study set out to answer?

This research aims to establish the boundedness of wave operators for fourth order Schrödinger operators under specific conditions regarding real potentials.

April 26, 2026

The L p -boundedness of wave operators for 4-th order schrodinger operators on ℝ 2 , I. regular case

Key Points

This research aims to establish the boundedness of wave operators for fourth order Schrödinger operators under specific conditions regarding real potentials.
Examined wave operators for fourth order Schrödinger operators with real potentials.
Considered conditions for regularity and the absence of positive eigenvalues.
Analyzed the mapping properties with respect to Fourier multipliers.
Wave operators are shown to be bounded in Lp for all p if regular conditions are met.
Absence of non-trivial solutions confirms regular behavior at zero.
Positive eigenvalues do not arise, affirming the stability of the operator's properties.

Abstract

We prove that wave operators of scattering theory for fourth order Schrödinger operators Formula: see text on Formula: see text with real potentials Formula: see text such that Formula: see text and Formula: see text for an Formula: see text, Formula: see text, are bounded in Formula: see text for all Formula: see text if Formula: see text is regular at zero in the sense that there are no non-trivial solutions to Formula: see text such that Formula: see text and if positive eigenvalues are absent from Formula: see text. This reduces Formula: see text-mapping properties of functions Formula: see text of Formula: see text to those of Fourier multipliers Formula: see text.

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