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Miloslav Znojil
Reviewer number:
Item's zbl-Number:
DE 017 210 934
Metafune, Giorgio; Pallara, Diego:
On the location of the essential spectrum of Schroedinger operators.
Proc. Am. Math. Society 130, No. 6, 1779 - 1786 (2002).
35J10Schroedinger operator
Primary Classification:
35P15Estimation of eigenvalues, upper and lower bounds
Secondary Classification:
47N50Applications in quantum physics
Schroedinger operators in N dimensions; essential spectrum; lower estimates; Lebesgue-measure method; Poincare inequality

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 Poincare-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 (cf ref. [10]
for more details) 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. In 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 (1998) 3349) so that, personally, I
feel a regret that the authors assume that all their potentials
remain bounded from below.
Remarks to the editors:

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