On the self-adjointness of H+A*+A

Let $H:D(H)\subseteq{\mathscr F}\to{\mathscr F}$ be self-adjoint and let $A:D(H)\to{\mathscr F}$ (playing the role of the annihilator 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 Krein-type formula, we build self-adjoint realizations $\widehat H$ of the formal Hamiltonian $H+A^{*}+A$ with $D(H)\cap D(\widehat H)=\{0\}$. We give the explicit characterization of $D(\widehat H)$ and provide a formula for the resolvent difference $(-\widehat H+z)^{-1}-(-H+z)^{-1}$. Moreover, we consider the problem of the description of $\widehat H$ as a (norm resolvent) limit of sequences of the kind $H+A^{*}_{n}+A_{n}+E_{n}$, where the $A_{n}\!$'s are bounded operators approximating $A$ and the $E_{n}$'s are suitable renormalizing bounded operators. These results show the connection between the construction of singular perturbations of self-adjoint operators by Krein's resolvent formula and the nonperturbative theory of renormalizable models in Quantum Field Theory.


Introduction
In the last few years several works appeared where questions about the characterization of the self-adjointness domains of some renormalizable quantum fields Hamiltonians and their spectral properties were addressed (see [7], [8], [6], [13], [12], [10], [11], [22], [23]). In such papers (see also [16], [26], [27] for some antecedent works considering simpler models) the operator theoretic framework much resembles the one involved in the construction of singular perturbations of self-adjoint operators (a.k.a. self-adjoint extensions of symmetric restrictions) by Kreȋn's type resolvent formulae (see [18] and references therein). The correspondence is exact as regards the Fermi polaron model considered in [6] (see the remark following [6,Corollary 4.3] and our Remark 2.21); instead, as regards the Nelson model studied in [12] (this paper was our main source of inspiration), the self-adjointness domain of the Nelson Hamiltonian H Nelson there provided does not correspond, even if it has a similar structure, to the domain of a singular perturbation of the non-interacting Hamiltonian H free . Indeed, if that where so, by [18,Remark 2.10], dom(H Nelson ) = {ψ ∈ F : ψ 0 := ψ + (AH −1 free ) * φ ∈ dom(H free ) , Aψ 0 = Θφ, φ ∈ dom(Θ)}, for some self-adjoint operator Θ (here A denotes the annihilation operator) while, by [12], dom(H Nelson ) = {ψ ∈ F : ψ + (AH −1 free ) * ψ ∈ dom(H free )}. If the two domain representation coincided, then Θ = A − A(AH −1 free ) * , which, beside containing the ill-defined operator A(AH −1 free ) * , is not even formally symmetric. The lack of a direct correspondence between the two approaches apparently prevents the writing of a formula for the resolvents difference (−H Nelson + z) −1 − (−H free + z) −1 . Such a kind of resolvent formula can help the study, beside of the spectrum, of the scattering theory for the couple (H free , H Nelson ) (see [15] and reference therein, also see Remark 3.6).
Our main aim here is to show that H Nelson can be still obtained using the theory of singular perturbations (thus providing a resolvent formula) by applying Kreȋn's formula twice: at first one singularly perturbs H free obtaining a polaron-type Hamiltonian and then one singularly perturbs the latter obtaining the Nelson Hamiltonian (such a strategy is suggested by the use of an abstract Green-type formula, see Lemma 3.1); since for both the two operators Kreȋn's resolvent formula holds, by inserting the resolvent of the first operator in the resolvent formula for the second one, re-arranging and using operator block matrices, at the end one obtains a final formula for the resolvent difference (−H Nelson + z) −1 − (−H free + z) −1 only containing the resolvent of H free and the extension parameter (which is a suitable operator in Fock space).
We consider also the problem of the description of H Nelson as a (norm resolvent) limit of sequences of the kind H n := H free +A * n +A n +E n , where the A n 's are the bounded annihilation operators corresponding with an ultraviolet cutoff at frequencies less than n and the E n 's are suitable renormalizing constants. We approach this problem by employing the resolvent formula for H Nelson here obtained and an analogous one for the approximating H n ; this shows the role of the ever-present term of the kind A n H −1 free A * n : it is due to the difference between the so-called Weyl functions (see (2.3)) in the resolvents of the H n 's and the limit one. The Weyl function of H Nelson contains A((−AH −1 free ) * − (A(−H free + z * ) −1 ) * ) and (−AH −1 free ) * plays the role of a regularizing term: indeed the operator difference (−AH −1 free ) * − (A(−H free + z * ) −1 ) * has range in the domain of A while the ranges of the single terms never are. Contrarily the Weyl function of H n only contains −A n (−H free + z) −1 A * n without the need of adding the balancing term −A n H −1 free A * n . This explain why one has to take into account such an addendum (and also a renormalizing counterterm E n since A n H −1 free A * n does not converge when the ultraviolet cutoff is removed) in order to approximate H Nelson in norm resolvent sense (see Theorem 3.9 and Subsection 3.1).
In the present paper we embed the previous discussion in an abstract framework; thus we consider a general self-adjoint operators H (playing the role of the free Hamiltonian H free ) in an abstract Hilbert space F (playing the role of the Fock space) and an abstract annihilation operators A. In Section 2 we provide a self-contained presentation (with some simplifications and generalizations) of (parts of) our previous results contained in the papers [18], [19], [20], [21] that we will need later and give a results of the approximation (in norn resolvent sense) by regular perturbations of the singular perturbations here provided. In particular, in Subsection 2.1, we consider the problem of the construction, by providing their resolvents, of the self-adjoint extensions of the symmetric restriction S := H| ker(Σ), where Σ : dom(H) → X is bounded with respect to the graph norm in dom(H) and X is an auxiliary Hilbert space. Successively, in Section 3, we apply the previous results to the case where X = F and Σ = A. This provides a family H T of self-adjoint extension of S, where the parametrizing operator T is self-adjoint in F . This, in the case H = H free , provides a polaron-like Hamiltonian (see Remark 2.21). Then, we apply again the results in Subsection 2.1 now to the case where H = H T and Σ = 1 − A * , A * a suitable left inverse of (A(−H + z * ) −1 ) * . The final self-adjoint operator H T is the one we were looking for: it can be represented as H T = H + A * + A T , where H is a (no more F -valued) suitable closure of H such that H + A * is F -valued when restricted to dom(S * ) and A T is an extension of the abstract annihilation operator A. By inserting the resolvent Kreȋn formula for H T into the one for H T one gets a Kreȋn resolvent formula for the difference (− H T + z) −1 − (− H + z) −1 which contains only the resolvent of H and the operator T (see Theorem 3.4 and Remark 3.5). Since A T has the additive representation A T = A 0 + T , where A 0 corresponds to the case T = 0, T enters in an additive way in the definition of H T , i.e., H T = H 0 + T and so one can relax the self-adjointness hypothesis on T , and suppose that T is symmetric and H 0 -bounded with relative bound a < 1 (see Theorem 3.8). In Theorem 3.9 we address the problem of the approximation of H T by a sequence of bounded perturbations on H. Finally, in Subsection 3.1, we show how, by the suitable choice T = T Nelson provided in [12], one obtains H T Nelson = H Nelson , where the self-adjoint Hamiltonian H Nelson is the one constructed in the seminal paper [17]; the same kind of analysis can be applied to other renormalizable quantum field models. Acknowledgements. The author thanks Jonas Lampart for some useful explanations and bibliographic remarks.

Singular perturbations.
For convenience of the reader, in this subsection we provide a compact (almost) self-contained presentation (with some simplifications and generalizations) of parts of the results from papers [18], [19], [20], [21] that we will need in the next section; we also refer to papers [20] and [21] for the comparison with other formulations (mainly with boundary triple theory, see, e.g., [5,Section 7.3], [2, Chapter 2]) which produce some similar outcomes. Let be a self-adjoint operator in the Hilbert space F with scalar product ·, · ; just in order to simplify the exposition, we suppose that ̺(H) ∩ R = ∅ (without this hypothesis some formulae become a bit longer). We introduce the following definition: H 1 denotes the Hilbert space given by dom(H) endowed with the scalar product ·, · 1 , ψ 1 , ψ 2 1 := (H 2 + 1) 1/2 ψ 1 , (H 2 + 1) 1/2 ψ 2 ; H 1 coincides, as a Banach space, with dom(H) equipped with the graph norm. Given a bounded linear map Σ : H 1 → X , X an auxiliary Hilbert space with scalar product (·, ·), for any z ∈ ̺(H) we define the linear bounded operator By first resolvent identity one has Hence ran(G w − G z ) ⊆ H 1 , and the linear operator (playing the role of what is called a Weyl operator-valued function in boundary triple theory, see [20], [5,Section 7 is well defined and bounded; by (2.2) it can be re-written as Then Proof. Let z ∈ Z Σ,Θ . Since Θ * = Θ and M z is bounded, by the first equality in (2.5), Theorem 2.2. Let Σ : H 1 → X be bounded and let Θ : dom(Θ) ⊆ X → X be self-adjoint. Suppose that Then and one has the λ-independent characterization Proof. At first let us notice that, by ran(G − G z ) ⊆ H 1 , (2.7) implies that the same relations hold for G z for any z ∈ ̺(H). By (2.5), the operator family on the righthand side of (2.8) (here denoted byȒ z ) is a pseudo-resolvent (i.e., it satisfies the first resolvent identity) and By (2.7), this gives φ z = φ w , i.e., the definition of φ z is z-independent. Thus, posing ψ 0 := ψ z + (G z − G)φ, one has ψ = ψ 0 + Gφ, with ψ 0 ∈ H 1 and Remark 2.4. Obviously λ ∈ Z Σ,Θ whenever 0 ∈ ̺(Θ). In this case, whenever (2.7) holds, Regarding hypotheses (2.6) and (2.7), one has the following sufficient conditions: Since Σ is surjective, G * z = ΣR z * has a closed range and so G z has closed range as well by the closed range theorem. Therefore, since, by point 1), ker . Thus, by (2.9), Θ + M z has a bounded inverse and, by [9, Thm. 5.2, Chap. IV], has a closed range. Therefore, by (2.9) again, Remark 2.7. Remark 3.5 below shows that one can still have a self-adjoint operator with a resolvent given by a formula like (2.8) (see (3.11)) even if hypothesis (2.7) does not hold true.
In the following by symmetric operator we mean a (not necessarily densely defined) linear operator S : dom(S) ⊆ F → F such that Sψ 1 , ψ 2 = ψ 1 , Sψ 2 for any ψ 1 and ψ 2 belonging to dom(S); whenever S is densey defined, S * denotes its adjoint.
Lemma 2.8. Let S be the symmetric operator S := H| ker(Σ). Suppose that ran(G) ∩ H 1 = {0} and define the (λ-independent) linear operator and (2.7) hold then S ⊆ H Θ ⊆ S × and so H Θ is a self-adjoint extension of S.
We start by applying the results in the previous section to the case Hence, supposing that hypotheses (2.6) and (2.7) hold, one gets a self-adjoint extension H T of the symmetric operator S = H| ker(A). Using here the notations one has (see (2.11) and (2.12)), whenever ψ = ψ 0 + Gφ,
The operator in (3.2) seems to be different from what we are looking for, i.e., an operator of the kind H + A * + A. However, the difference is not so big: by the definition of A T and by Green's formula (2.10), for any ψ, ϕ ∈ dom(A T ) ⊆ dom(S × ) one has (here T symmetric would suffice) This gives the following T extends a restriction of a self-adjoint operator: Therefore we can try to apply the formalism recalled in Subsection 2.1 to the case H = H T and Σ = Σ in order to build self-adjoint extensions of S. If for some of such self-adjoint extensions H one has H ⊆ S × T , then, since S × T is symmetric by Lemma 3.1, H = S × T and so S × T itself is self-adjoint. To apply such a strategy, we need to check the validity of hypotheses in Theorem 2.2.
Since ker(Q) = H 1 = ran(R z ) and A * is a left inverse of G z (see Remark 2.11), for any z ∈ Z Σ,T , one has Thus Σ : dom(H T ) → F is bounded w.r.t. the graph norm in dom(H T ) and, for any z ∈ ̺(H T ) one can define the bounded operator By (3.5), for any z ∈ Z Σ,T , one has Regarding the validity of hypothesis (2.7), one has the following: Proof. At first notice that, since is not empty (this hypothesis is not necessary, it is used in order to simplify the exposition), pick λ there and set Then Lemma 3.3. One has dom( S × ) ⊆ dom(S × ) and Proof. At first notice that, for any ψ ∈ dom( S × ) decomposed as ψ = ψ 0 + Gφ, where ψ 0 ∈ dom(H T ) and φ ∈ F , one has, since dom(H T ) = ker(A T ) and A T G = 1 (see the proof of Lemma 3.2), Since, by (3.6),   Here P ac and P ac are the orthogonal projectors onto F ac and F ac , the absolutely continuous subspaces relative to H and H T respectively.
In order to apply Theorem 3.4 one needs to show that there exists at least one z • ∈ ̺(H) such that Σ G z• has a bounded inverse. A simple criterion is provided in the next Lemma. We premise a definition: let H s , s ≥ 0, be the scale of Hilbert spaces defined by H s := dom(H s ) endowed with the scalar product ψ 1 , ψ 2 s := (H 2 + 1) s/2 ψ 1 , (H 2 + 1) s/2 ψ 2 .
Proof. Since (we take |y| ≥ 1 in the second inequality) one gets, by interpolation, This shows that both 1 − G z ± and 1 − G * z ∓ have bounded inverses whenever |y| is sufficiently large. Since Z A,−T = ∅, by [4,Theorem 2.19 and Remark 2.20], A T G z has a bounded inverse for any z ∈ ̺(H) ∩ ̺(H T ) ⊆ C\R and so whenever |y| is sufficiently large. Hence, whenever |y| is sufficiently large, Σ G z ± has a bounded inverse given by Since the operator T enters as an additive perturbation in the definition of H T , one can eventually avoid the self-adjointness hypothesis on it and work with H 0 alone: Theorem 3.8. Let A ∈ B(H s , F ) for some 0 < s < 1 and such that both ker(A|H 1 ) and ran(A|H 1 ) are dense in F . Then and resolvent given, for any z ∈ C such that µ + z ∈ ̺(H) ∩ ̺( H 0 ), µ ∈ R\{0}, by If is self-adjoint with resolvent given by (3.14) for some µ ∈ R. Let {A n } ∞ 1 be a sequence of bounded operators in F such that
We take where Γ b (L 2 (R 3 )) denotes the boson Fock space over L 2 (R 3 ), and Here with selfadjointness domain the Sobolev space H 2 (R d ) and dΓ b (L) denotes the boson second quantization of L (see, e.g., [1,Chapter 5]). Since 0 ∈ ̺(H free ), we can take λ = 0 in the definition of G (see (2.1)), so that G = −(AH −1 free ) * . In order to define the appropriate annihilator operator A we use the identification L 2 (R 3M ) ⊗ Γ b (L 2 (R 3 )) ≃ L 2 (R 3M ; Γ b (L 2 (R 3 ))) which maps ψ ⊗ Φ to x → Ψ(x) := ψ(x)Φ. Given v := (−∆ (3) + m 2 ) −1/4 δ 0 , δ 0 ∈ S ′ (R 3 ) denoting the Dirac mass at the origin, we define where v x (y) := v(x − y) and denotes the bosonic annihilator operator with test vector v x k (see, e.g. [1,Chapter 5] , is bounded for any power s > 1/2 and ker(A|dom(H free )) is dense in L 2 (R 3n ) ⊗ Γ b (L 2 (R 3 )). Since ran(A|dom(H free )) is dense in L 2 (R 3M ) ⊗ Γ b (L 2 (R 3 )) (it suffices to consider states with a finite number of bosons), Theorem 3.8 applies and defines a self-adjoint operator H T for any symmetric operator T which is H 0 -bounded with relative bound a < 1. By Remark 3.10, T should be a suitable regularization of the ill-defined operator −AH −1 free A * ; for A given in (3.23), the right choice, consisting in a regularization of the diagonal (with respect to the direct sum structure of F in (3.22)) part of −AH −1 free A * , is provided in [12, equations (29)-(31)]. Here we denote such an operator by T = T Nelson ; it is infinitesimally H 0 -bounded by [12, Lemma 3.10] (let us notice that, by Remark 3.11, our H 0 coincides with the operator there written as (1 − G * )H free (1 − G)).
Given the sequence v n ∈ L 2 (R 3 ), such that v n = χ n v, where denotes the Fourier transform and χ n denotes the characteristic function of a ball of radius R = n (this provides an ultraviolett cutoff on the boson frequencies), let us denote by A n the sequence of bounded operators in L 2 (R 3M ) ⊗ Γ b (L 2 (R 3 )) defined as A in (3.23) with v replaced by v n . Since (3.16) is equivalent to H −1 free A * n − (AH −1 free ) * F,F → 0, (3.16) holds by [12,Proposition 3.2]. Let E n be the sequence of bounded symmetric operators in L 2 (R 3n ) ⊗ Γ b (L 2 (R 3 )) corresponding to the multiplication by the real constant given by (minus) the leading order term in the expansion in the coupling constant g of the the ground state energy at zero total momentum of the regularized Hamiltonian H free + A * n + A n (see, e.g., [24,Section 19.2]):

Defining then
T n := E n − A n H −1 free A * n , by [22, Proposition 3.1] (see also the proof of Theorem 1.4 in [12]), one has T n → T Nelson in norm as operators in B(dom(T Nelson ), L 2 (R 3M ) ⊗ Γ b (L 2 (R 3 )); thus hypothesis (3.19) holds. Hypothesis (3.18) holds since the estimates in [12] with v replaced by v n are bounded by the integrals with v (see in particular the arguments given in the proof of [12,Theorem 1.4]). Therefore, by Theorem 3.9,