Spectral Properties of Faddeev Equations in Differential Form

Abstract

Faddeev equations in differential form were introduced by Noyes and Fiedeldey in 1968 [1]

$$ ({H_0} - E){\varphi _\alpha } + {V_\alpha }\sum\limits_{\beta = 1}^3 {{\varphi _\beta } = 0,} $$

(1)

and since that time are used extensively for investigating theoretical aspects of the three-body problem as well as for getting numerical solutions of three-body bound-state and scattering state problems. The simple formula

$$ \sum\nolimits_{\beta = 1}^3 {{\varphi _\beta } = \Psi } $$

allows one to obtain the solution to the three-body Schrödinger equation

$$ ({H_0} + \sum\nolimits_{\beta = 1}^3 {{V_\beta } - E} )\Psi = 0 $$

in the case when

$$ \sum\limits_{\beta = 1}^3 {{\varphi _\beta } \ne 0.} $$

(2)

. Such solutions of (1) can be called physical. The proper asymptotic boundary conditions should be added to Eqs. (1) in order to guarantee (2). These conditions were studied by many authors and are well known [2], so that I will not discuss them here.