×

zbMATH — the first resource for mathematics

Spectral theory of Pauli-Fierz operators. (English) Zbl 1034.81016
Pauli-Fierz Hamiltonians are self-adjoint operators that are used in quantum mechanics to describe the interaction of a small system with a Bose field. They are operators of the form \(H=H_{fr}+\lambda V\) on a Hilbert space \(\mathcal H\). This Hilbert space has a distinguished decomposition \({\mathcal H}={\mathcal H}^\nu \oplus {\mathcal H}^{\bar\nu}\). The “free operator” has the form \[ H_{fr} = \begin{pmatrix} H_{fr}^{\nu\nu} &0\\ 0 &H_{fr}^{\bar\nu\bar\nu}\end{pmatrix} \] and the perturbation has the form \[ V = \begin{pmatrix} 0 &V^{\nu\bar\nu}\\ V^{\bar\nu\nu} &V^{\bar\nu\bar\nu}. \end{pmatrix} \] The main results of the paper are the following.
a) Outside of an \(O(\lambda^2)\) neighborhood of \(\sigma(H^{\nu\nu})\) the spectrum of \(H\) is purely absolutely continuous and the Limiting Absorption Principle holds.
b) Let \(k\) be an isolated eigenvalue of \(H^{\nu\nu}\) and suppose the Fermi Golden Rule assumption for \(k\) holds. Then the spectrum of \(H\) is purely absolutely continuous in an \(O(\lambda^2)\) neighborhood of \(k\) and the Limiting Absorption Principle holds.
c) If the Fermi Golden Rule assumption fails, the description of the spectrum of \(H\) in a neighborhood of \(k+\lambda^2 m\) is obtained, where \(m\in\sigma_{disc}(w_k)\cap {\mathbb R}\), \(w_k=p_kw(z+i0)p_k\), \(w(z)=V^{\nu\bar\nu} (zI^{\bar\nu\bar\nu}- H_{fr}^{\bar\nu\bar \nu})^{-1}V^{\bar\nu\nu},\) and \(p_k\) is the projection of \(H_{fr}^{\nu\nu}\) onto \(k\).

MSC:
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47N50 Applications of operator theory in the physical sciences
81U05 \(2\)-body potential quantum scattering theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agmon, S.; Herbst, I.; Skibsted, E., Perturbation of embedded eigenvalues in the generalized N-body problem, Comm. math. phys., 122, 411, (1989) · Zbl 0668.35078
[2] Aguilar, J.; Combes, J.M., A class of analytic perturbations for one-body Schrödinger Hamiltonians, Comm. math. phys., 22, 269, (1971) · Zbl 0219.47011
[3] Amrein, W.; Boutet de Monvel, A.; Georgescu, V., C0-groups, commutator methods and spectral theory of N-body Hamiltonians, Progress in mathematics, (1996), Birkhäuser Basel · Zbl 0962.47500
[4] Arai, A., On a model of a harmonic oscillator coupled to a quantized, massless, scalar field, I, J. math. phys., 22, 2539, (1981) · Zbl 0473.46050
[5] Arai, A.; Hirokawa, M., On the existence and uniqueness of ground states of the spin-boson Hamiltonian, J. funct. anal., 151, 455, (1997) · Zbl 0898.47048
[6] Araki, H.; Woods, E.J., Representations of the canonical commutation relations describing a nonrelativistic infinite free Bose gas, J. math. phys., 4, 637, (1963)
[7] Bach, V.; Fröhlich, J.; Sigal, I., Quantum electrodynamics of confined non-relativistic particles, Adv. math., 137, 299, (1998) · Zbl 0923.47040
[8] Bach, V.; Fröhlich, J.; Sigal, I., Convergent renormalization group analysis for non-selfadjoint operators on Fock space, Adv. math., 137, 205, (1998)
[9] Bach, V.; Fröhlich, J.; Sigal, I., Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field, Comm. math. phys., 207, 249, (1999) · Zbl 0965.81134
[10] V. Bach, J. Fröhlich, and, I. Sigal, Return to equilibrium, preprint, 1999.
[11] Bach, V.; Fröhlich, J.; Sigal, I.; Soffer, A., Positive commutators and the spectrum of pauli – fierz Hamiltonian of atoms and molecules, Comm. math. phys., 207, 557, (1999) · Zbl 0962.81011
[12] Baez, J.C.; Segal, I.E.; Zhou, Z., Introduction to algebraic and constructive quantum field theory, (1991), Princeton Univ. Press Princeton
[13] Balslev, E.; Combes, J.-M., Spectral properties of many-body Schrödinger operators with dilation analytic interactions, Comm. math. phys., 22, 280, (1971) · Zbl 0219.47005
[14] Blanchard, P., Discussion mathématique du modèle de Pauli et Fierz relatif à la catastrophe infrarouge, Comm. math. phys., 15, 156, (1969)
[15] Boutet de Monvel, A.; Georgescu, V., Boundary values of the resolvent of a self-adjoint operator. higher order estimates, (), 9-52 · Zbl 0917.47022
[16] A. Boutet de Monvel, V. Georgescu, and, J. Sahbani, Higher order estimates in the conjugate operator theory, preprint.
[17] Boutet de Monvel, A.; Sahbani, J., On the spectral properties of the spin-boson Hamiltonians, Lett. math. phys., 44, 23, (1998) · Zbl 0911.47007
[18] Brattelli, O.; Robinson, D.W., Operator algebras and quantum statistical mechanics 2, (1981), Springer-Verlag Berlin
[19] Cohen-Tannoudji, C.; Dupont-Roc, J.; Grynberg, G., Photons and atoms—introduction to quantum electrodynamics, (1991), Wiley New York
[20] Cycon, H.; Froese, R.; Kirsch, W.; Simon, B., Schrödinger operators, (1987), Springer-Verlag Berlin
[21] Dereziński, J., Asymptotic completeness in quantum field theory. A class of gallilei-covariant models, Rev. math. phys., 10, 191, (1997) · Zbl 1079.81580
[22] Dereziński, J.; Gérard, C., Asymptotic completeness in quantum field theory. massive pauli – fierz Hamiltonians, Rev. math. phys., 11, 383, (1999) · Zbl 1044.81556
[23] J. Dereziński, V. Jakšić, and, C.-A. Pillet, in preparation.
[24] Dirac, P.A.M., The quantum theory of the emission and absorption of radiation, Proc. roy. soc. Edinburgh sect. A, 114, 243, (1927) · JFM 53.0847.01
[25] Dümcke, R.; Spohn, H., Quantum tunneling with dissipation and the Ising model over R, J. statist. phys., 41, 389, (1985)
[26] Fannes, M.; Nachtergaele, B.; Verbeure, A., The equilibrium states of the spin-boson model, Comm. math. phys., 114, 537, (1988) · Zbl 0653.46064
[27] Friedrichs, K.O., Perturbation of spectra in Hilbert space, (1965), Am. Math. Soc Providence · Zbl 0142.11001
[28] Frigerio, A., Quantum dynamical semigroups and approach to equilibrium, Lett. math. phys., 2, 79, (1997) · Zbl 0381.46037
[29] Froese, R.; Herbst, I., A new proof of the Mourre estimate, Duke math. J., 49, 1075, (1982) · Zbl 0514.35025
[30] Glimm, J.; Jaffe, A., Quantum physics. A functional integral point of view, (1987), Springer-Verlag New York
[31] Gohberg, I.; Goldberg, S.; Kaashoek, M.A., Classes of linear operators, (1993), Birkhäuser Basel
[32] Haag, R., Local quantum physics, (1993), Springer-Verlag New York · Zbl 0843.46052
[33] Heitler, W., The quantum theory of radiation, (1954), Oxford Univ. Press London · Zbl 0055.21603
[34] Howland, J., The livsic matrix in perturbation theory, Math. anal. appl., 50, 415, (1975) · Zbl 0367.47010
[35] Haag, R.; Hugenholtz, N.M.; Winnik, M., On the equilibrium states in quantum statistical mechanics, Comm. math. phys., 5, 215, (1967) · Zbl 0171.47102
[36] Hübner, M.; Spohn, H., Radiative decay: non-perturbative approaches, Rev. math. phys., 7, 363, (1995) · Zbl 0843.35068
[37] Hübner, M.; Spohn, H., Spectral properties of the spin-boson Hamiltonian, Ann. inst. H. poincare probab. statist., 62, 289, (1995) · Zbl 0827.47053
[38] Jakšić, V.; Pillet, C.-A., On a model for quantum friction II: Fermi’s Golden rule and dynamics at positive temperature, Comm. math. phys., 176, 619, (1996) · Zbl 0852.47038
[39] Jakšić, V.; Pillet, C.-A., On a model for quantum friction III: ergodic properties of the spin-boson system, Comm. math. phys., 178, 627, (1996) · Zbl 0864.47049
[40] Jakšić, V.; Pillet, C.-A., Spectral theory of thermal relaxation, J. math. phys., 38, 1757, (1997) · Zbl 0891.47053
[41] Jensen, A.; Mourre, E.; Perry, P., Multiple commutator estimates and resolvent smoothness in quantum scattering theory, Ann. inst. H. Poincaré probab. statist., 41, 207, (1984) · Zbl 0561.47007
[42] Kato, T., Perturbation theory for linear operators, (1976), Springer-Verlag Berlin
[43] Legget, A.J.; Chakravarty, S.; Dorsey, A.T.; Fisher, M.P.A.; Garg, A.; Zwerger, W., Dynamics of the dissipative two-state system, Rev. mod. phys., 59, 1, (1987)
[44] Mennicken, R.; Motovilov, A.K., Operator interpretation of resonances arising in spectral problems for 2×2 operator matrices, Theoret. and math. phys., 116, 867, (1998) · Zbl 0915.47051
[45] Mourre, E., Absence of singular continuous spectrum for certain self-adjoint operators, Comm. math. phys., 78, 391, (1981) · Zbl 0489.47010
[46] Okamoto, T.; Yajima, K., Complex scaling technique in non-relativistic qed, Ann. inst. H. poincare probab. statist., 42, 311, (1985) · Zbl 0594.58057
[47] Pauli, W.; Fierz, M., Nuovo cimento, 15, 167, (1938)
[48] Perry, P.; Sigal, I.; Simon, B., Spectral analysis of N-body Schrödinger operators, Ann. of math. (2), 114, 519, (1981) · Zbl 0477.35069
[49] Robinson, D.W., Return to equilibrium, Comm. math. phys., 31, 171, (1973) · Zbl 0257.46091
[50] Robinson, D.W., C*-algebras in quantum statistical mechanics, () · Zbl 0412.46056
[51] Reed, M.; Simon, B., Methods of modern mathematical physics, I. functional analysis, (1980), Academic Press London
[52] Reed, M.; Simon, B., Methods of modern mathematical physics, II. Fourier analysis, self-adjointness, (1975), Academic Press London · Zbl 0308.47002
[53] Reed, M.; Simon, B., Methods of modern mathematical physics, III. scattering theory, (1978), Academic Press London
[54] Reed, M.; Simon, B., Methods of modern mathematical physics, IV. analysis of operators, (1978), Academic Press London · Zbl 0401.47001
[55] Raggio, G.A.; Zivi, S.H., Semiclassical description of N-level systems interacting with radiation fields, Quantum probability II, Heidelberg 1984, Lecture notes in math., 1136, (1985), Springer-Verlag Berlin/New York · Zbl 0583.60098
[56] Simon, B., Resonances in N-body quantum systems with dilation analytic potential and foundations of time-dependent perturbation theory, Ann. math., 97, 247, (1973) · Zbl 0252.47009
[57] Skibsted, E., Spectral analysis of N-body systems coupled to a bosonic system, Rev. math. phys., 10, 989, (1998) · Zbl 0945.81008
[58] Spohn, H., Ground states of the spin-boson Hamiltonian, Comm. math. phys., 123, 277, (1989) · Zbl 0667.60108
[59] Spohn, H., Ground state of a quantum particle coupled to a scalar Bose field, Lett. math. phys., 44, 9, (1998) · Zbl 0908.60094
[60] Spohn, H., An algebraic condition for the approach to equilibrium of an open N-level system, Lett. math. phys., 2, 33, (1977) · Zbl 0366.47019
[61] Weinberger, H.F., Variational methods for eigenvalue approximations, Cmbs, 15, (1974), Soc. for Industr. and Appl. Math Philadelphia
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.