On the Inverse Spectral Problem for the Quasi-Periodic Schr\"odinger Equation

TL;DR

This paper establishes a quantitative relationship between the decay of spectral gaps and the Fourier coefficients of the potential in the quasi-periodic Schrödinger equation, providing a two-way spectral and potential reconstruction link.

Contribution

It proves that small spectral gaps imply exponentially decaying Fourier coefficients of the potential, and vice versa, advancing inverse spectral theory for quasi-periodic Schrödinger operators.

Findings

Spectral gap size bounds Fourier coefficients of the potential.

Fourier decay bounds spectral gap size.

Results depend on smallness parameter and frequency vector.

Abstract

We study the quasi-periodic Schr\"odinger equation $$ -\psi"(x) + V(x) \psi(x) = E \psi(x), \qquad x \in \IR $$ in the regime of "small" $V$. Let $(E_m',E"_m)$, $m \in \zv$, be the standard labeled gaps in the spectrum. Our main result says that if $E"_m - E'_m \le \ve \exp(-\kappa_0 |m|)$ for all $m \in \zv$, with $\ve$ being small enough, depending on $\kappa_0 > 0$ and the frequency vector involved, then the Fourier coefficients of $V$ obey $|c(m)| \le \ve^{1/2} \exp(-\frac{\kappa_0}{2} |m|)$ for all $m \in \zv$. On the other hand we prove that if $|c(m)| \le \ve \exp(-\kappa_0 |m|)$ with $\ve$ being small enough, depending on $\kappa_0 > 0$ and the frequency vector involved, then $E"_m - E'_m \le 2 \ve \exp(-\frac{\kappa_0}{2} |m|)$.