Gamow states as continuous linear functionals over analytical test functions

Abstract

The space of analytical test functions ξ, rapidly decreasing on the real axis (i.e., Schwartz test functions of the type 𝒮 on the real axis), is used to construct the rigged Hilbert space (RHS) (ξ,ℋ,ξ′). Gamow states (GS) 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.