One dimensional Anderson model with non-homogeneous disorder

We consider the random discrete Schr枚dinger operator $H=L+V$ on the one dimensional integer lattice $Z$. The operator $L$ is the discrete laplacian, $(LF)(x)=f(x-1)+f(x+1)$, and $V$ is a potantial, $Vf(x)=v(x)f(x)$, where $v(x)$ is a family of independent random variables. We will discuss a new method to establish localization, i.e. that generically the eigenfunctions of $H$ decay exponentially. The method is robust enough to allow $v(x)$ to have different probability distributions for different lattice points $x$. Moreover, the method allows to obtain lower bounds for the rate of decay of the eigenfunctions. The talk will be given in the language of finite dimensional matrices and basic probability theory.