The translation invariant massive Nelson model. III: asymptotic completeness below the two-boson threshold
Based on results in Part II of this series [the second author and \textit{M. G. Rasmussen}, ibid. 14, No. 4, 793--852 (2013; Zbl 1272.81077)] and Mourre estimate [\textit{E. Mourre}, Commun. Math. Phys. 78, 391--408 (1981; Zbl 0489.47010)], asymptotic completeness of two body scattering for a class of translation invariant models describing a single quantum particle (the electron) linearly coupled to a massive scalar field (bosons) is shown (Theorem 2.3. Corollary 2.4). The models cover wide classes such as massive \textit{E. Nelson} model [``Interaction of nonrelativistic particles with a quantum scalar filed'', J. Math. Phys. 5, 1190--1197 (1964)] and the \textit{H. Frölich} polaron model [``Electron in lattice filed'', Adv. Phys. 3. 325--362 (1954)]. Precisely, the Hilbert space of the system is \(\mathcal{H}=\mathcal{K}\otimes\mathcal{F}\). Here \(\mathcal{K}=L^2(\mathbb{R}_y^\nu)\), is the Hilbert space of a quantum mechanical particle moving in \(\mathbb{R}^\nu\), whose position is denoted by \(y\) and whose moment is \(D_y=-\mathrm{i}\nabla_y\) and \(\mathcal{F}\) is the Fock space \(\Gamma(\mathfrak{h})=\bigoplus^\infty_{n=0}\Gamma^{(n)}(\mathfrak{h})\), \(\Gamma^{(n)}(\mathfrak{h})=\mathfrak{h}^{\otimes_s n}\), where \(\mathfrak{h}=L^2(\mathbb{R}^\nu_k)\) is the Hilbert space of a single boson, whose dispersion relation is denoted by \(\omega(k)\). The total energy and momentum operators of bosons are given by \(H_{\mathrm{ph}}=d\Gamma(\omega)\) and \(P_{\mathrm{ph}}=d\Gamma(k)\). The Hamiltonian, which governs the dynamics of the system, is \[ H=\Omega(D_y)\otimes 1+1\otimes H_{\mathrm{ph}}+\phi(G_y) \] (\S2. (2.4).). In Part II, it is shown under minimal conditions (described as Condition 1 in \S2. The authors remark that this conditions are a little stronger than those of in Part II), \(H\) is essentially self-adjoint on \(C_0^\infty(\mathbb{R}^\nu)\otimes\mathcal{C}\) (detailed definition of \(\mathcal{C}=\Gamma_{\mathrm{fin}}(C_0^\infty(\mathbb{R}^\nu))\) is given in Appendix A). Extended Hilbert spaces \(\mathcal{F}^{\mathrm{ex}}=\mathcal{F}\otimes\mathcal{F}\) and \(\mathcal{H}^{\mathrm{ex}}=\mathcal{H}\otimes\mathcal{F}\), and extended Hamiltonian and the momentum operator \(H^{\mathrm{ex}}=H\otimes 1+1\otimes d\Gamma(\omega))\) and \(P^{\mathrm{ex}}=P\otimes 1+1\otimes d\Gamma(k)\), which was introduced in the context of spectral and scattering theory [\textit{J. Dezerezinski} and \textit{C. Gerard}, ``Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonian'', Rev. Math. Phys. 213, 39--125 (1999)], are also introduced. Under minimal conditions, extended operators are essentially selfadjoint on \(C_0^\infty(\mathbb{R}^\nu)\times (\mathcal{C}\otimes\mathcal{C})\). The Domains of fiber Hamiltonians \(H(\xi)\), etc. do not depend on \(\xi\). Since the Hamiltonian and momentum operators commute, they have a common spectrum. Spectral properties of these operators are studied in Part II assuming spectral theory conditions (described as Condition 2 in \S2.3 of this paper). Especially, it is shown that the fiber Hamiltonian has no singular continuous spectrum below the two-boson threshold. Including this, results in Part II are summarized as Theorem 2.2. These results allows to apply Mourrre estimate (Theorem 2.2. (d)) in the later discussions of this paper. Let \(E\) be the spectral resolution, \(\Sigma_{\mathrm{pp}}\) the set of pure-point spectrum and let \(\mathcal{R}\) be the energy momentum regime below the two-boson threshold. Let the space of bounded state be \(\mathcal{H}_{\mathrm{bnd}}=E(\Sigma_{\mathrm{pp}})\mathcal{H}\) and let \[ \mathcal{H}_\pm=\mathcal{H}_{\mathrm{bnd}}\otimes\mathcal{F}. \] Then this paper proves under assumptions Conditions 1 and 2, the wave operator \[ \Omega_\mathcal{R}^\pm=s-\lim_{t\to \pm\infty}e^{itH}\check{\Gamma}(1,1)^\ast e^{-itH^{\mathrm{ex}}}, \] exists and unitary as maps from \(E^{\mathrm{ex}}(\mathcal{R})\mathcal{H}_\pm\) to \(E(\mathcal{R})\mathcal{H}\). The scattering operator \(S_\mathcal{R}=(\Omega_\mathcal{R}^-)^\ast\Omega_\mathcal{R}^+\) is also unitary (Theorem 2.3). Here \(\check{\Gamma}(1,1)^\ast\) is the map given by \[ \check{\Gamma}(1,1)^\ast(\psi\otimes a^\ast(h_1)\cdots a^\ast(h_n)|0\rangle)=a^\ast(h_1)\cdots a^\ast(h_n)\Gamma(1)\psi, \] where \(a^\ast(k)\) is the annihilation operator (for the detail, see Appendix A). From Theorem 2.3, the asymptotic completeness of the two body scattering follows (Corollary 2.4. It asserts the span of scattering states and elements of \(E(\mathcal{R})\mathcal{H}_{\mathrm{bnd}}\) is dense in \(E(\mathcal{R})\mathcal{H}\).). To prove Theorem 2.3, first expressions for Heisenberg derivative \(\mathbf{D}\Phi(t)=\partial_t\Phi(t)+\mathrm{i}[H(\xi),\Phi(t)]\), etc., are computed in \S3, with technical details in Appendixes C,D, G and H. By these computations and Mourre estimate, minimal velocity estimates are obtained in \S4 with some technical details in Appendix F (Proposition 4.1 and 4.2; \S1. (1.9)). Using these estimates, the existence of suitable asymptotic observables is proved in \S5 with some technical details in Appendix K (Theorem 5.1 and 5.2). After these preparations, the localized wave operator is defined (Theorem 6.6) and its properties are studied in \S6. Then, Theorem 2.3 is proved in \S7 (Theorem 7.2), the last section. Some auxiliary results related to the proof of Theorem 2.3 are given in the appendix. The last Appendixes L and M, give some results on the isolated spectrum and the structure of the spectrum of the extended Hamiltonian.
translation invariant model, massive Nelson model
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