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

One-particle Hamiltonian

The autonomous fermionic source–reservoir model › One-particle Hamiltonian · lines 422–453

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

Fix

n \geq 1, c > 0, ω_{0} > 2 c .

Let

h = ℓ^{2} (N_{0}) \otimes C^{n} .

The index $j = 0$ is the source site; $j \geq 1$ are reservoir sites. The second index labels $n$ independent chains.

Let $A$ be the half-line adjacency operator:

A e_{0} = e_{1}, A e_{j} = e_{j - 1} + e_{j + 1} (j \geq 1) .

Define

h = (ω_{0} I + c A) \otimes I_{n} .

Since $∥ A ∥ \leq 2$ ,

(ω_{0} - 2 c) I \leq h \leq (ω_{0} + 2 c) I .

Thus $h$ is bounded, self-adjoint, and strictly positive.

Displayed formulas in this passage

Hawk02-D@1#eq:one-particle-H · line 444
$h = (ω_{0} I + c A) \otimes I_{n} .$
unlabeled · line 425
$n \geq 1, c > 0, ω_{0} > 2 c .$
unlabeled · line 429
$h = ℓ^{2} (N_{0}) \otimes C^{n} .$
unlabeled · line 438
$A e_{0} = e_{1}, A e_{j} = e_{j - 1} + e_{j + 1} (j \geq 1) .$
unlabeled · line 449
$(ω_{0} - 2 c) I \leq h \leq (ω_{0} + 2 c) I .$