Construction of the Resolvent for a Few-Body System

The rate at which a system undergoes a transition to (or within) the continuum can be expressed in terms of the resolvent without knowing the detailed asymptotic form of the final-state wavefunction; only a crude approximation to this wavefunction, one which satisfies a simple constraint, is needed. For this and other reasons it is of considerable interest to have a method for efficiently calculating the resolvent, defined as G(E)≡1/(E1−H) for a system whose Hamiltonian and energy are H and E, respectively. We develop an approach based on exploiting the underlying time scale, t0 say, which is exposed by writing G(E) as the limit T→∞ of G(T)(E)≡−i∫0Tt0dteiEte−iHt. In reality, only times Tt0 up to some finite value, beyond which the probability of a transition is negligible, need be considered. Introducing a complex unit of time tϕ≡t0eiϕ, and expanding the time-translation operator e−iHt in powers of (t+itϕ)/(t−itϕ), we show that G(T)(E) can be reexpressed as (1)G(T)(E)=−itϕe−12tϕHI0(2Etϕ,T−ϕ)−2tϕH×∑n=1∞1nIn(2Etϕ,T−ϕ)Ln−1(1)(2tϕH)e−12tϕH, where T−ϕ=Te−iϕ, and where the coefficients In(2Etϕ,T−ϕ) satisfy a simple three-term recurrence formula, as do the Laguerre polynomials Ln−1(1)(2tϕH). The pre- and post-factors e−12tϕH serve to cut off the high-energy components of the spectrum if |ϕ|<π/2, so H can be treated as a bounded operator, and therefore can be represented by a finite matrix. The times t0 and Tt0 are associated with two branch points of (matrix elements of) e−itH in the complex-time-plane; Tt0 is associated with a branch point at infinity which sets the “arrow of time,” and t0 is associated with a branch point at ∼it0 which sets the time scale of evolution in the direction of this arrow. The branch structure of the resolvent is embedded in the coefficients In(2Etϕ,T−ϕ). The series can be terminated after roughly (Et0)T2 terms, and convergence to the limit T→∞ can be accelerated by averaging over the oscillations at large values of T. It is useful to break H into two parts, the Hamiltonian Hc for free-body motion distorted by a weak Coulomb tail, and a short-range interaction H−Hc. It is only necessary to evaluate Gc(E)≡1/(E1−Hc) since G(E)=1−Gc(E)(H−Hc)−1Gc(E). Using the series representation, and representing Hc on a basis of harmonic oscillator eigenfunctions with a dynamic length scale that is proportional to T (to conform with unitarity), Gc(E) can be evaluated efficiently. However, Gc(E) should be re-represented on a set of basis functions that decrease exponentially in r (rather than in r2) as r increases since the large-distance behavior of the response functions 〈r→|Gc(E)|ψ〉 and 〈r→|G(E)|ψ〉 is exponential in position space. Moreover, such a basis is more suitable for representing the core interaction H−Hc, which in general has a Coulomb singularity at the origin. This singularity is not of great physicalimportance but it is numerically significant since it magnifies errors. However, if the basis functions are chosen to be the eigensolutions of a Sturm–Liouville eigenvalue problem tailored to the core interaction, the Coulomb singularity can be effectively removed. The singularity can also be removed if the analytically known Coulomb–Sturmian functions are employed, but additional matrix elements must be evaluated in this case. We illustrate the method using examples of a one-body system, for which exact results are known. In particular, we demonstrate that the rate for photoionization of hydrogen can be estimated to high accuracy using a final-state wavefunction whose asymptotic form is incorrect in all directions but one.