Zentralblatt MATH HOME

Your review has been received.

Thank you very much.

(You may want so save a copy of this page for your files.)

Miloslav Znojil
Reviewer number:
Item's zbl-Number:
DE 017 975 937
Takeda, Masayoshi:
Conditional gaugeability and subcriticality of generalized Schroedinger operators
J. Funct. Anal. 191, No. 2, 343-376 (2002).
46G12Measures and integration on abstract linear spaces
Primary Classification:
60H30Applications of stochastic analysis to PDE, etc.
Secondary Classification:
28C20Set functions and measures and integrals in infinite-dimensional spaces Wiener measure, Gaussian measure, etc.
large deviation principle; symmetric Markov processes; Schroedinger operator; Green's function; necessary and sufficient conditions of subcriticality; conditional gaugeability of a measure;

There exists a wealth of connection between the quantum mechanics (or
rather theory of Schroedinger and Schroedinger-type operators) and
statistical physics (in particular, stochastic processes etc). The
paper is devoted to this class of problems. One should not feel
discouraged by an unfortunate presentation of the results: It makes
sense to glimpse in some related references first (and, in
particular, in the last one). Then, still, one should probably skip
the incomprehensible introduction and read the paper in a
non-sequential, ``direct-access" manner. In my case I enjoyed the
reading started by the introductory part of the last section 4 (large
deviation principle for symmetric Markov processes) which set me in
the game. Before waiting till the end of section (Onegin was famous
for that) I was then able to see the point: The paper is about the
existence of Greens functions (disguised under the name of
``subcriticality"). Moving more deeply and technically in the text
one may then pick up any of its subjects: the review of the concept
of gaugeability (section 2) and the study of its conditional version
(in section 3). The reward: One sees the necessary and sufficient
conditions for things named in the title and, finally, indulges
oneself in the subtleties of the key illustrative example 3.10.
Remarks to the editors:

(New formular )