Gamow states as continuous linear functionals
over analytical test functions
C. G. Bollini,
and A. L. De Paoli
Department of Physics, University of La Plata, C.C. 67 (1900) La Plata, Argentina
M. C. Rocca
Department of Physics, University of La Plata and Department of Mathematics,
University of Centro de la Pcia de Bs. As., Pintos 390, C.P. 7000, Tandil, Argentina
͑Received 20 February 1996; accepted for publication 20 May 1996͒
The space of analytical test functions
, rapidly decreasing on the real axis ͑i.e.,
Schwartz test functions of the type S on the real axis͒, is used to construct the
rigged Hilbert space ͑RHS͒͑
͒. Gamow states ͑GS͒ can be deﬁned in RHS
starting from Dirac’s formula. It is shown that the expectation value of a self-
adjoint operator acting on a GS is real. We have computed exactly the probability
of ﬁnding a system in a GS and found that it is ﬁnite. The validity of recently
proposed approximations to calculate the expectation value of self-adjoint operators
in a GS is discussed. © 1996 American Institute of Physics.
The proper treatment of the continuum and the inclusion of decaying states belonging to it in
the deﬁnition of Green’s functions of physical interest is a long-standing problem in various ﬁelds
of physics. The mathematical consequences of the inclusion of the continuum in the scattering of
particles by a central potential have been explored by Gamow long ago.
A modern review of the
scattering theory can be found in Ref. 2 where the basic elements of the involved radial differen-
tial equations are presented in great detail. The use of these states, as it has been shown by
is of central importance in building the physical interpretation of the
-decay mode of
heavy atomic nuclei.
The so-called Gamow resonant states ͓for simplicity Gamow states ͑GS͔͒
fulﬁll purely outgoing boundary conditions with an exponential behavior at inﬁnity.
methods have been proposed since the publication of Gamow’s work,
particularly in connection
with the normalizability of Green’s functions in the presence of GS and in the treatment of
The mathematical equivalence between some of these methods has been
discussed recently and the correspondence between Bergreen’s and Mittag-Leﬂer’s representations
has been explored at length.
Presently a rich literature is available regarding the application of
these concepts to nuclear reactions and to nuclear structure problems.
The amount of information about mathematical properties of representations which include
GS is also very rich. The use of decaying states of complex energy in the framework of the
Hamiltonian formalism, and the use of deformed contours to compute survival amplitudes, has
been reported in Ref. 8 by Sudarshan and co-workers. The formulation of quantum mechanics in
the rigged Hilbert space ͑RHS͒ has been also studied by Bohm.
In Ref. 10 it is shown that
idealized resonances are described, within the RHS, by generalized eigenvectors of a self-adjoint
Hamiltonian with complex eigenvalues and a Breit–Wigner energy distribution. Similar argu-
ments have been advanced by Gadella.
Among the difﬁculties posed by the use of GS one
can mention the appearance of the exponential catastrophe and the need to include nonphysical
Fellow of the CIC, Pcia, Bs. As., Argentina.
Fellow of the CONICET, Argentina.
Fellow of the J. S. Guggenheim Foundation.
4235J. Math. Phys. 37 (9), September 1996 © 1996 American Institute of Physics