Spectral analysis of the spin-boson Hamiltonian with two photons for arbitrary coupling

We study the spectrum of the spin-boson model with two photons in $\mathbb{R}^d$ for arbitrary coupling $\alpha>0$. It is shown that the discrete spectrum is finite and the essential spectrum consists of a half-line the bottom of which is a unique zero of a simple Nevanlinna function. Besides the simplicity and more abstract nature of our approach, the main novelty is the achievement of these results under"minimal"regularity conditions on the photon dispersion and the coupling function.


Introduction
The spin-boson model is a well-known quantum-mechanical model which describes the interaction between a two-level atom and a photon field. We refer to [26] and [20] for excellent reviews from physical and mathematical perspectives, respectively.
Despite the formal simplicity of the spin-boson model (from the physics viewpoint), its dynamics is rather complicated and rigorous spectral and scattering results are usually very difficult to obtain, especially in the case when the number of photons is unbounded. For weak coupling, starting with the pioneering work of Hübner and Spohn [21] spectral and scattering properties of the full spin-boson model as well as of its finite photon approximations have been investigated extensively; see, for example, [31,33,21,27,16,4,8,30,3,18,2,1,17,11,5,9] and the references therein.
In this paper we are concerned with the case of two photons which resembles the standard three-body situation. The Hilbert space of our system is C 2 ⊗ F 2 s , where F 2 s is the truncated Fock space with L 2 s (R d × R d ) standing for the subspace of L 2 (R d × R d ) consisting of symmetric functions, and equipped with the inner product (

1.2)
Date: November 9, 2018. The author is grateful to Professor Alexander V. Sobolev for fruitful discussions, reading the preliminary version of this manuscript and making useful suggestions and thanks the Department of Mathematics at University College London for the kind hospitality. The financial support of the Swiss National Science Foundation, SNF, through the Early Postdoc.Mobility grant No. 168723 is gratefully acknowledged.
where σ = ± is the discrete variable, the Hamiltonian of our system is given by the formal expression Here ±ε (ε > 0) are the energy levels of the atom corresponding to its excited and ground states, respectively, ω(k) = |k| is the photon dispersion relation, α > 0 is the coupling constant and λ is the coupling function given by the product of ω(k) with a cut-off function for large k. The spatial dimension, d ≥ 1, plays no particular role in our analysis and is left arbitrary.
In general, the dispersion relation ω ≥ 0 and the coupling function λ are fixed by the physics of the problem. Motivated by different applications of (1.3) (e.g. in solid state physics, see [31]), one likes to consider them as free parameter functions and impose only some general conditions such as whenever inf ω = 0. To the best of our knowledge, every study on the spectrum of the spin-boson model in the up-to-date literature assumes at least (1.4) or its strengthened version where the second condition in (1.4) is replaced by which is known as the infrared regularity condition (see, for example, [18,10]). There are many papers in the current literature containing rigorous results on the spectrum of H α for small coupling. It is known that, under appropriate conditions in addition to (1.4)-(1.5), there exists a sufficiently small coupling constant α 0 > 0 such that for all α ∈ (0, α 0 ) the spectrum of H α consists of a unique simple discrete eigenvalue separated by a gap from the purely absolutely continuous part of the spectrum which is a half-line. This result was obtained in [27] by means of the scattering theory and also in [21] by means of the conjugate operator method (here we would like to note that [21] as well as [19,20] treat also the case of arbitrarily finite number of photons).
On the other hand, not much is known for general coupling, without any assumptions about its smallness. There are rather few papers devoted to the spectral theory of the spin-boson model (with or without particle number cut-off) as well as more general abstract models sometimes called Pauli-Fierz models or generalized spin-boson models; see, for instance, [20,19,4,12,15,17]. In these studies the location of the essential spectrum is given for any value of the coupling constant, and results including finiteness of the point spectrum or absence of the singular continuous spectrum are proven under various assumptions in addition to (1.4). We would also like to mention [25] where the authors study the dynamics of the spin-boson model at arbitrary coupling strength. There is a recent related work [28] on the location of the bottom of the essential spectrum and the finiteness of the number of eigenvalues on the left of it. Although [28] does not need assumptions such as (1.5), it is required that the underlying domain is a compact subset of the real line and that the parameter functions λ and ω are real-analytic.
In the present paper we aim to establish the finiteness of the discrete spectrum along with an explicit description of the essential spectrum under fairly weak regularity conditions on the parameter functions ω and λ by developing a new, simple and instructive approach. It turns out that the continuity of ω and the condition λ ∈ L 2 (R d ) are sufficient to achieve our goal. In particular, neither the infrared regularity (1.5) nor the second condition in (1.4) is needed. We would also like to emphasize that our main result -Theorem 1 -holds even if the level set of ω corresponding to the photon mass is of positive Lebesgue measure, whereas it was always assumed to be a finite or a Lebesgue null set in the up-to-date literature. The methods we employ to achieve these results are direct and of abstract nature, allowing for simpler proofs. In particular, we benefit from block operator matrix techniques involving Schur complements and the corresponding Frobenius-Schur factorizations combined with the standard perturbation theory.
The paper is organized as follows. In Section 2 we give precise formulations of our main results (Theorems 1-2). In Section 3.1 we explain the reduction of the problem to the spectral analysis of a 2 × 2 operator matrix and describe the Schur complement of the latter. The detailed proofs of the main results are given in Sections 3.2-3.3.
Throughout the paper we adopt the following notations. For a self-adjoint operator T acting in a Hilbert space and a constant µ ∈ R such that µ ≤ min σ ess (T ), we denote by N (µ; T ) the number of the eigenvalues of T less than µ (counted with multiplicities). Note that N (µ; T ) coincides with the dimension of the spectral subspace of T corresponding to the interval (−∞, µ), see [7,Section IX]. The integrals with no indication of the limits imply the integration over the whole space R d or R d × R d . The Euclidean and operator norms are denoted by | · | and · , respectively.

Summary of the main results
Throughout the paper we assume the following hypotheses.
Assumption (A). The parameter ε > 0 is fixed, the photon dispersion relation ω : R d → R is an unbounded continuous function with the coupling function λ : R d → C is not identically zero and satisfies Remark 1. If λ ≡ 0 on R d , then the photons do not couple to the atom and the description of the spectrum is straightforward. As the relativistic photon dispersion relation ω(k) = √ k 2 + m 2 suggests, the constant m defined in (2.1) may play a role of the "mass" of the photon.
The natural domain of the unperturbed operator H 0 is given by with H 1 and H 2 standing for the weighted L 2 -Hilbert spaces and The condition (2.2) implies the boundedness of the perturbation and thus the expression for H α given in (1.3) generates a self-adjoint operator in the Hilbert space C 2 ⊗ F 2 s on the natural domain of H 0 (see [24,Theorem V.4.3]). For notational convenience, we denote the corresponding self-adjoint operator again by H α .
Consider the continuous functions with the discrete variable σ = ± and the constant ε > 0 corresponding to the excited state energy of the atom. It is easy to see that Φ (σ) α are strictly decreasing and Moreover, the monotone convergence theorem implies the existence of the (possibly improper) limits We distinguish the two cases: In this case the limits in (2.8) are negative infinity for all α > 0. Hence, (2.7) and the monotonicity imply that each of the continuous functions Φ In this case the limits in (2.8) are finite for all α > 0 and given by We distinguish the two subcases: a) Either m ≥ 2ε and α > 0 is arbitrary, or m < 2ε and α > α cr , where . (2.13) In this case Φ (σ) α (m + σε) < 0 for each σ = ±. Hence, (2.7) and the monotonicity again imply that each of the continuous functions Φ (σ) α has a unique zero (again denoted by) E σε (α) in the interval (−∞, m + σε). Once again, we define E(α) as in (2.10).
b) The case m < 2ε and 0 < α ≤ α cr . In this case Φ Consequently, (2.7) and the monotonicity of the continuous functions Φ 14) The following theorems summarize the main results of the present paper.
Let the coupling constant α > 0 be arbitrary and let E(α) be defined as in one of (2.10) and (2.14). Then i) the essential spectrum of H α is given by The next result provides more explicit description of the bottom of the essential spectrum for weak coupling.
Then, for all sufficiently small α > 0, we have Moreover, the asymptotic expansion holds Remark 2. i) Whenever (2.11) is satisfied, one can make it more explicit how small should the coupling constant be so that E(α) = E ε (α) holds. In fact, it follows from the proof of Theorem 2 that E(α) = E ε (α) for all α > 0 satisfying . (2.18) In the massless case (2.18) is equivalent to α ≤ α cr and E(α) = E ε (α) holds trivially (see (2.14)). ii) There is no difficulty in verifying that E(α) corresponds to the ground state energy of the one-boson Hamiltonian (see Section 4.2). In the massless case m = 0 Theorem 1i) is thus an analogue of the Hunziker-van Winter-Zhislin (HVZ) theorem of the standard three-body Schrödinger operators, see [29, Section XIII] and [20].

Proofs of the main results
Unless specified, we assume throughout this section that the discrete variable σ = ± is fixed.

Preliminaries. By means of the unitary transformation
where is the tridiagonal operator matrix acting in the truncated Fock space F 2 s on the domain with the weighted Hilbert spaces H 1 , H 2 given in (2.4)-(2.5). The operator entries of H where The second operator on the right-hand-side of (3.6) maps F 2 s onto the two-dimensional subspac C ⊕ Span{λ} ⊕ {0} ⊂ F 2 s . Since the essential spectrum as well as the finiteness of the discrete spectrum of self-adjoint operators are invariant with respect to finite-rank perturbations (see, for example, [7, Chapter 9]), in view of (3.2), it follows that for all z ≤ min σ ess (H α ) provided that the right-hand-side of (3.9) is finite. That is why, from now on we completely focus on the 2 × 2 operator matrix H (σ) α defined in (3.7) and acting in the Hilbert space It is easy to see that H (σ) 22 : As it was mentioned in Introduction, our approach is based on the so-called Schur complement and the corresponding Frobenius-Schur factorization. This has proven to be a powerful and natural tool when dealing with spectral properties of 2 × 2 operator matrices such as H (σ) α , see e.g. [32]. We would like to mention that the Schur complement method employed in this paper was used previously in the spectral analysis of Pauli-Fierz Hamiltonians under the name Feshbach method or Feshbach-Schur method in [6] and [13], for instance.
For the rest of this subsection we assume z ∈ (−∞, 2m + σε) = ρ(H  and is the integral operator with the kernel , the following Frobenius-Schur factorization holds where the last equality is an immediate consequence of (3.21).

Proof of Theorem 1. We recall that the function Φ
Conceptually, the proof of the first part of Theorem 1 is similar to that of [22, Theorem 3.1], but the technical details are quite different.

3.2.1.
Proof of Theorem 1i). The proof will be done in three steps.
Step 1: In this step we show that Let z 0 ∈ (2m+σε, ∞) be arbitrary. Then z 0 = 2ω(k 0 )+σε for some k 0 ∈ R d . Further, let ϕ ∈ C ∞ (R d ) be an arbitrary function supported in the annulus {k ∈ R d : 0.5 ≤ k < 1} and such that ϕ L 2 (R d ) = 1. Then the sequence {ϕ n } n∈N defined by is an orthonormal system in L 2 (R d ). It follows that the sequence {ψ n } n∈N defined by is an orthonormal system in L 2 s (R d × R d ) and the Lebesgue's dominated convergence theorem implies that On the other hand, in view of (2.2), the weak convergence (to zero) of the orthonormal system {ϕ n } n∈N in L 2 (R d ) yields It is obvious that the sequence {Ψ n } n∈N := {(0, ψ n ) t } n∈N is an orthonormal system in [7,Theorem IX.1.2]. Because z 0 ∈ (2m + σε, ∞) was arbitrary, the claim follows from the closedness of the essential spectrum.
Step 3: In this step we combine previous two steps to derive (2.15). To this end, first we recall that m ≥ 0 and ε > 0. Hence, the two steps yield It is easy to see that m − ε > E ε (α) for all α > 0, which is a simple consequence of the relation If (2.11) holds together with m < 2ε and 0 < α ≤ α cr , then the claim follows by (3.9) if we show only the relation N m + E ε (α); H (+) α < ∞. To see this one should observe that m + E ε (α) < 2m − ε for all α > 0, see (3.36), and recall that the eigenvalues of a self-adjoint operator cannot accumulate below the bottom of its essential spectrum. Otherwise, the claim follows by (3.9) if we show both of the relations (3.37) In fact, there is no loss of generality in focusing on the latter case as the proof we give below does not require any restrictions in the case σ = +. To this end, we fix the discrete variable σ and denote m + E σε (α) by z 0 for notational convenience. Further, let z ≤ z 0 be fixed for a moment. Since (−∞, Let us denote by K (σ) 1 (z) and K (σ) 2 (z) the integral operators in L 2 (R d ) whose kernels are respectively the functions (k 1 , k 2 ) → α 2 λ(k 1 )λ(k 2 )Ψ (σ) 1 (k 1 , k 2 ; z) and (k 1 , k 2 ) → α 2 λ(k 1 )λ(k 2 )Ψ (σ) 2 (k 1 , k 2 ; z). Then, we have the corresponding decomposition Since each term of the right-hand-side of (3.39) is uniformly bounded by the constant 1 2m+σε−z , we infer from (2.2) that K (σ) 1 (z) is a well-defined rank-two operator in L 2 (R d ). In fact, its range coincides with the subspace of L 2 (R d ) spanned by the functions λ and λ ω+m+σε−z . Therefore, Lemma 3 combined with [7, Theorem IX. (3.42) In particular, (3.37) will follow immediately if we show that α (E σε (α)) and doing some elementary calculations, we obtain .
and ∆ (σ) α ( · ; z 0 ) ≡ 0 on the level set of ω corresponding to m. In particular, the restriction of ∆ (σ) α (z 0 ) to L 2 (R d \ Ω) is the zero operator. Moreover, it is easy to see from (3.39)-(3.40) that the restriction of K To show the latter, first we recall the monotonicity of the function Φ (σ) α in (−∞, m+ σε). For all z < z 0 , this together with (3.24) Hence, the multiplication operator ∆ (σ) α,Ω (z) is positive for all z < z 0 . Since the integral operator K (σ) 2,Ω (z) : L 2 (Ω) → L 2 (Ω) is well-defined and Hilbert-Schmidt, it follows that the operator is also well-defined and Hilbert-Schmidt for all z < z 0 . Next, we define T (σ) α (z 0 ) to be the integral operator in L 2 (Ω) with the kernel We have the following elementary, yet quite important inequality which holds for all real numbers a ≥ 0, b ≥ 0 and c > 0. Its proof is left to the "Appendix". Applying this inequality with a = ω(k 1 ) − m, b = ω(k 2 ) − m and c = 2m + σε − z 0 , we obtain the estimate for all k 1 , k 2 ∈ Ω. Using (3.45) and (3.51), we can estimate the kernel Θ (σ) α in (3.49) as follows and thus we infer from (2.2) that Θ     Remark 4. The finiteness of the discrete spectrum can be shown by even simpler way whenever the condition (2.11) is satisfied. We can proceed in the same way as in the above proof, the only exception being that there is no need for the "special" decomposition (3.41). Instead of (3.48), we can consider the operator function for z < z 0 and continuously extend it up to z 0 by defining T (σ) α (z 0 ) to be the integral operator in L 2 (R d ) with the kernel .