AbstractLet $H:\text {dom}(H)\subseteq \mathfrak {F}\to \mathfrak {F}$
H
:
dom
(
H
)
⊆
F
→
F
be self-adjoint and let $A:\text {dom}(H)\to \mathfrak {F}$
A
:
dom
(
H
)
→
F
(playing the role of the annihilation operator) be H-bounded. Assuming some additional hypotheses on A (so that the creation operator A∗ is a singular perturbation of H), by a twofold application of a resolvent Kreı̆n-type formula, we build self-adjoint realizations $\widehat H$
H
̂
of the formal Hamiltonian H + A∗ + A with $\text {dom}(H)\cap \text {dom}(\widehat H)=\{0\}$
dom
(
H
)
∩
dom
(
H
̂
)
=
{
0
}
. We give an explicit characterization of $\text {dom}(\widehat H)$
dom
(
H
̂
)
and provide a formula for the resolvent difference $(-\widehat H+z)^{-1}-(-H+z)^{-1}$
(
−
H
̂
+
z
)
−
1
−
(
−
H
+
z
)
−
1
. Moreover, we consider the problem of the description of $\widehat H$
H
̂
as a (norm resolvent) limit of sequences of the kind $H+A^{*}_{n}+A_{n}+E_{n}$
H
+
A
n
∗
+
A
n
+
E
n
, where the An’s are regularized operators approximating A and the En’s are suitable renormalizing bounded operators. These results show the connection between the construction of singular perturbations of self-adjoint operators by Kreı̆n’s resolvent formula and nonperturbative theory of renormalizable models in Quantum Field Theory; in particular, as an explicit example, we consider the Nelson model.
From Quantum Entanglement to Spatiotemporal Distance