On the inverse spectral problem for the quasi-periodic Schrödinger equation

Damanik, David¹  ; Goldstein, Michael²

¹ Department of Mathematics, Rice University 6100 S. Main St. 77005-1892 Houston TX USA
² Department of Mathematics, University of Toronto Bahen Centre, 40 St. George St. M5S 2E4 Toronto Ontario Canada

Publications Mathématiques de l'IHÉS, Tome 119 (2014), pp. 217-401

Résumé

We study the quasi-periodic Schrödinger equation

$$-{\psi }^{\text{'}\text{'}}\left(x\right)+V\left(x\right)\psi \left(x\right)=E\psi \left(x\right),x\in 𝐑$$

in the regime of “small” $V$. Let $(E_m^{\text{'}},E_m^{\text{'}\text{'}})$, $m\in {𝐙}^{\nu }$, be the standard labeled gaps in the spectrum. Our main result says that if ∈ $E_m^{\text{'}\text{'}}-E_m^{\text{'}}\le \epsilon exp(-{\kappa }_0\left|m\right|)$ for all $m\in {𝐙}^{\nu }$, with $\epsilon$ being small enough, depending on κ₀>0 and the frequency vector involved, then the Fourier coefficients of $V$ obey $\left|c\left(m\right)\right|\le {\epsilon }^{1/2}exp(-\frac{{\kappa }_0}{2}\left|m\right|)$ for all $m\in {𝐙}^{\nu }$. On the other hand we prove that if |c(m)|≤εexp(−κ₀|m|) with $\epsilon$ being small enough, depending on κ₀>0 and the frequency vector involved, then $E_m^{\text{'}\text{'}}-E_m^{\text{'}}\le 2\epsilon exp(-\frac{{\kappa }_0}{2}\left|m\right|)$.

Détail

DOI : 10.1007/s10240-013-0058-x

Keywords: Implicit Function Theorem, Inductive Assumption, Simple Eigenvalue, Principal Point, Inverse Spectral Problem