The interplay between the spectral theory of Schr|dingeroperators and probabilistic considerations forms the maintheme of these notes, written for the non-specialist readerand intended to provide a brief and elementaryintroductionto this field. An attempt is made to show basic ideas instatu nascendi and to follow their evaluation from simplebeginnings through to more advanced results. The term"genetic" in the title refers to this proceedure. The authorconcentrates on 2 topics which, in the history of thesubject, have been of major conceptual importance - on theone hand the Laplacian is a random medium and the left endof its spectrum (leading to large deviation problems forBrownian motion and the link to thenotion of entropy) andon the other, Schr|dinger operators with general ergodicpotentials in one-dimensional space. Ideas and concepts areexplained in the simplest, possible setting and by means ofa few characteristic problems with heuristic argumentspreceding rigorous proofs.

The interplay between the spectral theory of Schr|dingeroperators and probabilistic considerations forms the maintheme of these notes, written for the non-specialist readerand intended to provide a brief and elementaryintroductionto this field. An attempt is made to show basic ideas instatu nascendi and to follow their evaluation from simplebeginnings through to more advanced results. The term"genetic" in the title refers to this proceedure. The authorconcentrates on 2 topics which, in the history of thesubject, have been of major conceptual importance - on theone hand the Laplacian is a random medium and the left endof its spectrum (leading to large deviation problems forBrownian motion and the link to thenotion of entropy) andon the other, Schr|dinger operators with general ergodicpotentials in one-dimensional space. Ideas and concepts areexplained in the simplest, possible setting and by means ofa few characteristic problems with heuristic argumentspreceding rigorous proofs.