On the smooth Feshbach–Schur map

Griesemer, Marcel; Hasler, David

doi:10.1016/j.jfa.2008.01.015

Cited by 39 publications

(64 citation statements)

References 12 publications

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“…There are further small differences between our presentation here and the one of [3], which are explained in [12].…”

Section: The Smooth Feshbach Mapcontrasting

confidence: 58%

“…Section 4 describes the smoothed Feshbach transform F χ (H) of an operator H and the isomorphism Q(H) between the kernels of F χ (H) and H (isospectrality of the Feshbach transform). The transform H → F χ (H) was discovered in [3], and generalized to the form needed here in [12]. In Section 5 we perform a first Feshbach transformation on H(s) − z to obtain an effective Hamiltonian In Section 7 it is shown that the analyticity of a family of Hamiltonians is preserved under the renormalization transformation.…”

Section: Important Properties Of H(s) Are That H(s) = H(s)mentioning

confidence: 99%

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Analytic Perturbation Theory and Renormalization Analysis of Matter Coupled to Quantized Radiation

Griesemer

Hasler

2009

Ann. Henri Poincaré

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Abstract.For a large class of quantum mechanical models of matter and radiation we develop an analytic perturbation theory for non-degenerate ground states. This theory is applicable, for example, to models of matter with static nuclei and non-relativistic electrons that are coupled to the UV-cutoff quantized radiation field in the dipole approximation. If the lowest point of the energy spectrum is a non-degenerate eigenvalue of the Hamiltonian, we show that this eigenvalue is an analytic function of the nuclear coordinates and of α 3/2 , α being the fine structure constant. A suitably chosen ground state vector depends analytically on α 3/2 and it is twice continuously differentiable with respect to the nuclear coordinates.

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“…There are further small differences between our presentation here and the one of [3], which are explained in [12].…”

Section: The Smooth Feshbach Mapcontrasting

confidence: 58%

Section: Important Properties Of H(s) Are That H(s) = H(s)mentioning

confidence: 99%

Analytic Perturbation Theory and Renormalization Analysis of Matter Coupled to Quantized Radiation

Griesemer

Hasler

2009

Ann. Henri Poincaré

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“…For more details see [4,33]. Next, we introduce the scaling transformation S ρ : B[H] → B[H], which acts on the particle component of H := H p ⊗ H f by identity and on the field one, by…”

Section: Elimination Of Particle and High-photon Energy Degrees Of mentioning

confidence: 99%

Ground State and Resonances in the Standard Model of the Non-Relativistic QED

Sigal

2009

J Stat Phys

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We prove existence of a ground state and resonances in the standard model of the non-relativistic quantum electro-dynamics (QED). To this end we introduce a new canonical transformation of QED Hamiltonians and use the spectral renormalization group technique with a new choice of Banach spaces. I IntroductionProblem and outline of the results. Non-relativistic quantum electro-dynamics (QED) describes the processes of emission and absorption of radiation by systems of matter, such as atoms and molecules, as well as other processes arising from interaction of the quantized electro-magnetic field with non-relativistic matter. The mathematical framework of this theory is well established. It is given in terms of the time-dependent Schrödinger equation,

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“…One can prove similarly that F P (θ) (H g (θ) − z, H 0 (θ) − z) is well-defined and satisfies all the assumptions of [11]. The isospectrality then follows by [11].…”

Section: The Smooth Feshbach Map Applied To H G (θ)mentioning

confidence: 72%

“…Hence, the proof of the theorem is complete, except for the fact that and that gρ 0 −1/2 is sufficiently small. Moreover, by [11], H (0) [ · ] is isospectral to the initial Hamiltonian H g (θ), in the sense that…”

Section: A Banach Space Of Hamiltoniansmentioning

confidence: 99%

Resonances of the Confined Hydrogen Atom and the Lamb–Dicke Effect in Non-Relativistic QED

Faupin

2008

Ann. Henri Poincaré

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Abstract. We study a model describing a system of one dynamical nucleus and one electron confined by their center of mass and interacting with the quantized electromagnetic field. We impose an ultraviolet cutoff and assume that the fine-structure constant is sufficiently small. Using a renormalization group method (based on [3, 4]), we prove that the unperturbed eigenvalues turn into resonances when the nucleus and the electron are coupled to the radiation field. This analysis is related to the Lamb-Dicke effect.

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On the smooth Feshbach–Schur map

Cited by 39 publications

References 12 publications

Analytic Perturbation Theory and Renormalization Analysis of Matter Coupled to Quantized Radiation

Analytic Perturbation Theory and Renormalization Analysis of Matter Coupled to Quantized Radiation

Ground State and Resonances in the Standard Model of the Non-Relativistic QED

Resonances of the Confined Hydrogen Atom and the Lamb–Dicke Effect in Non-Relativistic QED

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