On the location of the essential spectrum of Schrödinger operators (Q2781349)

For a quantum particle moving in an \(N\)-dimensional potential a few lower estimates are derived in this paper for the bottom of the essential energy spectrum. The method is based on a Poincaré-type inequality and the authors generalize their recent work which paid attention to polynomial potentials. Historically, the motivation of similar studies dates back to the B. Simon's answers to some problems which originated in quantum field theory and where the classically escaping particles proved bound due to the ``too narrow'' character of the escape tubes, i.e., in effect, via uncertainty principle. Mathematically, the idea of such a ``quantum escape protection'' is probably due to F. Rellich who offered several results of this type, e.g., on p. 339 in ``Studies and Essays'' ed. by K. Friedrichs et al in 1948 in Interscience, New York. NEWLINENEWLINEIn the text of the present paper the idea of ``escape tubes'' is efficiently generalized in the measure-theoretic manner. The possible relation of this approach to the more usual capacity methods is discussed and its merits and a relatively more elementary character are emphasized. Marginally I would like to note that some of the old Simon's methods remain applicable also to the asymptotically strongly repulsive potentials which could model an explosion via change of couplings (an elementary explicit example may even be found in my own paper ``Quantum exotic: a repulsive and bottomless confining potential'' in J. Phys. A, Math. Gen. 31, No. 14, 3349--3355 (1998; Zbl 0928.34064) so that, personally, I feel a regret that the authors assume that all their potentials remain bounded from below.