Manuscript Hawk02-D@1 · Hawk02-D.tex · section · e6e4b72f

Many-particle Hamiltonian and its domain

The autonomous fermionic source–reservoir model › Many-particle Hamiltonian and its domain · lines 454–499

Hawk02-D@1#eq:many-particle-H

On fermionic Fock space

F_{f} (h) = k = 0 ⨁^{\infty} ⋀ k h,

define

H = d Γ (h) .

The $k$ -particle restriction is the sum of $h$ over the $k$ antisymmetrized slots, with norm at most $k ∥ h ∥$ . The direct sum defines a positive self-adjoint operator on

D (H) = {ψ = (ψ_{k}) : k \sum ∥ H_{k} ψ_{k} ∥^{2} < \infty} .

Its unitary group is

e^{- i t H} = Γ (e^{- i t h}) .

This constructs the infinite-reservoir dynamics without treating a formal infinite Hamiltonian sum as a proof of existence.

In creation and annihilation notation, the same operator is formally

H = ω_{0} a = 1 \sum^{n} j \geq 0 \sum f_{j, a}^{†} f_{j, a} + c a = 1 \sum^{n} j \geq 0 \sum (f_{j, a}^{†} f_{j + 1, a} + f_{j + 1, a}^{†} f_{j, a}),

with the rigorous meaning given by (eq:many-particle-H). It preserves total particle number.

Displayed formulas in this passage

Hawk02-D@1#eq:many-particle-H · line 463
$H = d Γ (h) .$
unlabeled · line 457
$F_{f} (h) = k = 0 ⨁^{\infty} ⋀ k h,$
unlabeled · line 470
$D (H) = {ψ = (ψ_{k}) : k \sum ∥ H_{k} ψ_{k} ∥^{2} < \infty} .$
unlabeled · line 478
$e^{- i t H} = Γ (e^{- i t h}) .$
unlabeled · line 486
$H = ω_{0} a = 1 \sum^{n} j \geq 0 \sum f_{j, a}^{†} f_{j, a} + c a = 1 \sum^{n} j \geq 0 \sum (f_{j, a}^{†} f_{j + 1, a} + f_{j + 1, a}^{†} f_{j, a}),$