Singular integral equations in the bound-state problem

A new class of Volterra-type integral equations from relativistic quantum physics

“…1. For all three equations (13)- (15), the domain of integration is now effectively finite, as it is finite in the time directions and as the Green's functions G ret 1 , G ret 2 are only supported along (and possibly inside) the backward light cones PLC(…”

“…3. From the integral equations (13)- (15) we can read off that ψ satisfies the initial value problem ψ(0, x 1 , 0, x 2 ) = ψ free (0, x 1 , 0, x 2 ). In other words, ψ is subject to a Cauchy problem "at the Big Bang."…”

“…It turns out that we need to utilize the more specific structure of the kernels of Eqs. (13)- (15). This will be done in Sec.…”

“…The integral equations (13), (14) in d = 1 and d = 2 correspond to this structure for N = 2 and M = 2 or M = 4 with special (but not square-integrable) integral kernels. The integral equation (15) in d = 3 is different because of the time shifts occurring in the integral.…”

“…1 Note that several works in the physics literature on the BS equation study (special) solutions of that equation (see e.g. [11][12][13][14][15][16] and references therein). However, these works seem to be of limited significance for the mathematical theory of (2), for the following reasons.…”