Self-adjointness of the semi-relativistic Pauli–Fierz Hamiltonian

Abstract

The spinless semi-relativistic Pauli-Fierz Hamiltonianin quantum electrodynamics is considered. Here p denotes a momentum operator, A a quantized radiation field, M ≥ 0, H f the free hamiltonian of a Boson Fock space and V an external potential. The self-adjointness and essential selfadjointness of H are shown. It is emphasized that it includes the case of M = 0. Furthermore, the self-adjointness and the essential self-adjointness of the semirelativistic Pauli-Fierz model with a fixed total momentum P ∈ R d :is also proven for arbitrary P .

Spectrum of the semi-relativistic Pauli–Fierz model I

“…It is established in [Hir13] that H for M > 0 is essentially self-adjoint on D(|p|) ∩ D(H f ) for external potential V such that D(V ) ⊂ D(|p|) and V f ≤ a |p|f +b f for all f ∈ D(|p|) with 0 ≤ a < 1 and b ≥ 0. We can also show a stronger statement on the self-adjointness of H. This is established in [HH13]. We set…”

Spectrum of the semi-relativistic Pauli–Fierz model II

“…Proof. In the case of V 2 V conf , the lemma was proven by [11]. Since the proof for the case of V 2 V rel is similar, we briefly give an outline of the proof.…”

“…Since the proof for the case of V 2 V rel is similar, we briefly give an outline of the proof. By the definition of V rel , there exist constants 0 < a < 1 and 0 < b such that with some constants C and C 0 (see [11]). Thus, by (7.2), (7.3), and kH 0 ‰k kH m ‰k C kV ‰k;…”

Functional integral approach to semi-relativistic Pauli–Fierz models

“…Then the theorem follows by the Kato-Rellich theorem. ✷ Furthermore in Hidaka and Hiroshima [HH13b] the self-adjointness of H qf for arbitrary m ≥ 0 is established. The key inequality is as follows.…”

“…Then for allp ∈ R d , H(p) is selfadjoint on D(|P f |) ∩ D(H rad ).The proof is similar to that of Theorems 4.5 and 4.7, i.e, it can be show that e −tH(p) leaves D(|P f |) ∩ D(H rad ) invariant fo rm > 0, and that by using the inequality |p − P f |Φ 2 + H rad Φ ≤ C (H(p) + 1l)Φ we can show the self-adjointness of H(p) for m ≥ 0. See[HH13b].…”