General Hamiltonians and Model Hamiltonians of the Theory of Superconductivity and Superfluidity in the Hilbert Space of Translation-Invariant Functions

Abstract

In many exactly solvable models of quantum statistical mechanics, an interaction Hamiltonian is equal to an integral of a product of operators of annihilation and creation and a potential divided by a volume of a system raised to a certain power. For example, the interaction Hamiltonian of the BCS model of superconductivity has the form

$$ {H_{I,M}} = \frac{g}{{2V}}\iint {\iint {\Phi ({x_1} - {x_2},{{x'}_1} - {{x'}_2}){\psi ^ + }({x_2})\psi ({{x'}_1})\psi ({{x'}_2}})d{x_1}d{x_2}d{{x'}_1}d{{x'}_2},} $$

((1))

where integration is carried out over the whole three-dimensional Euclidean space ℝ³, V is the volume of ℝ³, ψ⁺ and ψ^- are operators of creation and annihilation of Fermi particles, and Φ is a potential that satisfies the conditions

$$ \Phi ({x_1} - {x_2},{x'_1} - {x'_2}) = \Phi (\overline {{x_1} - {x_2},{{x'}_1} - {{x'}_2}} ) = \Phi ({x'_2} - {x'_1},{x_2} - {x_2}),\Phi ({x_2} - {x_1},{x'_1} - {x'_2}) = \Phi ({x_2} - {x_1},{x'_2} - {x'_1}) $$

((2))

and is a test function with respect to x ₁-x ₂ and ${x'_1} - {x'_2}$. The dependence on spin is omitted here.