On inverse spectral problem for quasi-periodic Schrödinger equation

Monday, December 3, 2012 -
3:15pm to 4:15pm
In this talk I will discuss the main ideas and new technology of our recent joint work with D.Damanik. We study the quasi-periodic Schr\"{o}dinger equation $$-\psi''(x) + V(x) \psi(x) = E \psi(x), \qquad x \in \mathbb{R} $$ in the regime of ``small'' $V$. Let $(E_m',E''_m)$, $m \in \mathbb{Z}^\nu$, be the standard labeled gaps in the spectrum. Our main result says that if $E''_m - E'_m \le \varepsilon \exp(-\kappa_0 |m|)$ for all $m \in \mathbb{Z}^\nu$, with $\varepsilon$ being small enough,depending on $\kappa_0 > 0$ and the frequency vector involved, then the Fourier coefficients of $V$ obey $|c(m)| \le \varepsilon^{1/2} \exp(-\frac{\kappa_0}{2} |m|)$, for all $m \in \mathbb{Z}^\nu$. On the other hand we prove that if $|c(m)| \le \epsilon \exp(-\kappa_0 |m|)$ with $\epsilon$ being small enough, depending on $\kappa_0 > 0$ and the frequency vector involved, then $E''_m - E'_m \le 2 \varepsilon \exp(-\frac{\kappa_0}{2} |m|)$.
Speaker: 
Michael GoldStein
University of Toronto
Event Location: 
Fine Hall 314