Abstract
AbstractLet Q and P be the position and momentum operators of a particle in one dimension. It is shown that all compact operators can be approximated in norm by linear combinations of the basic resolvents $$(aQ + bP - i r)^{-1}$$
(
a
Q
+
b
P
-
i
r
)
-
1
for real constants $$a,b,r \ne 0$$
a
,
b
,
r
≠
0
. This implies that the basic resolvents form a total set (norm dense span) in the C*-algebra $$\mathfrak {R}$$
R
generated by the resolvents, termed resolvent algebra. So the basic resolvents share this property with the unitary Weyl operators, which span the Weyl algebra. These results are obtained for finite systems of particles in any number of dimensions. The resolvent algebra of infinite systems (quantum fields), being the inductive limit of its finitely generated subalgebras, is likewise spanned by its basic resolvents.
Funder
Australian Research Council
Georg-August-Universität Göttingen
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
Reference8 articles.
1. Buchholz, D.: The resolvent algebra of non-relativistic Bose fields: observables, dynamics and states. Commun. Math. Phys. 362, 949–981 (2018)
2. Buchholz, D., Grundling, H.: The resolvent algebra: a new approach to canonical quantum systems. J. Funct. Anal. 254, 2725–2779 (2008)
3. Buchholz, D., Grundling, H.: Quantum systems and resolvent algebras. In: Blanchard, P., Fröhlich, J. (eds.) The Message of Quantum Science - Attempts Towards a Synthesis. Lect. Notes Phys. 899, pp. 33-45. Springer, Berlin (2015)
4. Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. Studies in Mathematics and its Applications. Vol. 1. North-Holland, Amsterdam (1976)
5. Reed, M., Simon, B.: Methods of Mathematical Physics III: Scattering Theory. Academic Press, New York (1978)