This Hamiltonain can be solved for an arbitrary number of particles spin up
and spins down . The corresponding integral equations
are (
)[Yan67]:
The limit correspond to
. In this limit one
can simplify further the system of integral equations by introducing a Fourier
transformation (see also discrete lattice model [Col74]):
By multiplying first equation from (A.8) by and integrating over one obtains expression for the 10.1
Setting one immediately sees that number of spin-down particles is half of the total number of particles .
Inserting (A.10) into second equation from (A.8),
taking into account formula (A.11) and carrying out two
integrations one obtains the integral equation involving only
(10.13) |
Ones this equation is solved the density and energy are given by
(10.14) | |||
(10.15) |
In the strongly interacting limit
and the kernel can be simplified
The energy per particle in units of
is given by
(10.17) |
It is possible to express the kernel in terms of -function (see
Gradstein-Ryzhik). Taking into account the series representation
one obtain following result from the
(exact) sum (A.16)
(10.18) |
The -function is defined using the digamma function . The digamma function is defined as logarithmic derivative of the Gamma function .
The kernel can be expanded at small and large values of the argument:
(10.19) | |||
(10.20) |