Spectral theory of the operator (p 2+m 2)1/2−Ze 2/r

Herbst, Ira

doi:10.1007/bf01609852

“…It now follows as in Ref. [52] that the essential spectra of H;(o) and Hi(e) coincide. (There is a misprint in the proof of Lemma 2.6, which contains the relevant ideas to show this:…”

Section: Spectral Properties Of H"mentioning

confidence: 54%

“…It then follows as in Ref. [52] that 55) and that the essential spectrum of H" is [m, co), so that the spectrum of H" in (0, m) consists of finitely degenerate eigenvalues without accumulation point in [0, m). (Of course, H"(h) > m if X is negative.)…”

Section: Spectral Properties Of H"mentioning

confidence: 82%

“…Qualitatively the spectrum of Hs may be largely determined in a rigorous fashion by using recent results of Herbst [52] concerning the operator (-.4 + m2)V2 -X j x 1-l on L2(dx), which is the Fourier transform of our HO(A). Indeed, since the largest eigenvalue of the matrix k;;2"(@[k"] + @[&I) is bounded above by 22s for s > 4, the relation [/ E,1'2voE,1'211 = fn (5.52) (which is Eq.…”

Section: Spectral Properties Of H"mentioning

confidence: 99%

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A positive energy dynamics and scattering theory for directly interacting relativistic particles

1980

Annals of Physics

0

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Starting from the tensor product of N irreducible positive energy representations of the PoincarC group describing N free relativistic particles with arbitrary spins and positive masses, we construct an interacting positive energy representation by modifying the total 4-momentum operator. We first make a transformation to a Hilbert space on which the free total 4-momentum operator equals the product of a dimensionless center-of-mass 4-vector ((I k I2 + 1)1/2, k) and a free "reduced Hamiltonian"Hro, which is a positive operator acting only on internal variables, and then replace HrO by an interacting reduced Hamiltonian H, = H> + V, where V commutes with the Lorentz group and is such that H, is a positive operator. The resulting product form is shown to imply that the wave operators intertwine the free and interacting representations so that the s-operator is Lorentz invariant. From a physical point of view the scheme is related to the framework first introduced by Bakamjian and Thomas, in which the Hamiltonian and boost generators are modified, but the above procedure makes a mathematically rigorous discussion much simpler. In the spin-zero case we introduce a natural generalization of the pair potentials of nonrelativistic N-particle Schrodinger theory to the present relativistic setting, study its scattering theory, and point out some problems that do not have analogs at the nonrelativistic level. In the spin-& case we propose, inspired by the Dirac equation, explicit reduced Hamiltonians to describe atomic energy levels and present arguments making plausible that their eigenvalues are in closer agreement with the experimental data than their nonrelativistic counterparts. We also consider extensions to arbitrary spin and, in the spin-$ case, coupling of a quantized radiation field. In view of eventual applications to "completely integrable" one-dimensional field theories the case of one space dimension is studied as well, both in quantum mechanics and in classical mechanics.

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“…(2.19) in Ref. [52] in our notation) is easily seen to imply that (/ E,l~"VSE;'f2j, < 'J2S--2r, (5.53) (This bound may be improved presumably.) Thus, if h < 22-297 (5.54) (which will be assumed from now on), we can define H"(h) as the unique operator associated with the form sum of E, and --hV".…”

Section: Spectral Properties Of H"mentioning

confidence: 77%

A positive energy dynamics and scattering theory for directly interacting relativistic particles

Ruijsenaars¹

1980

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Starting from the tensor product of N irreducible positive energy representations of the PoincarC group describing N free relativistic particles with arbitrary spins and positive masses, we construct an interacting positive energy representation by modifying the total 4-momentum operator. We first make a transformation to a Hilbert space on which the free total 4-momentum operator equals the product of a dimensionless center-of-mass 4-vector ((I k I2 + 1)1/2, k) and a free "reduced Hamiltonian"Hro, which is a positive operator acting only on internal variables, and then replace HrO by an interacting reduced Hamiltonian H, = H> + V, where V commutes with the Lorentz group and is such that H, is a positive operator. The resulting product form is shown to imply that the wave operators intertwine the free and interacting representations so that the s-operator is Lorentz invariant. From a physical point of view the scheme is related to the framework first introduced by Bakamjian and Thomas, in which the Hamiltonian and boost generators are modified, but the above procedure makes a mathematically rigorous discussion much simpler. In the spin-zero case we introduce a natural generalization of the pair potentials of nonrelativistic N-particle Schrodinger theory to the present relativistic setting, study its scattering theory, and point out some problems that do not have analogs at the nonrelativistic level. In the spin-& case we propose, inspired by the Dirac equation, explicit reduced Hamiltonians to describe atomic energy levels and present arguments making plausible that their eigenvalues are in closer agreement with the experimental data than their nonrelativistic counterparts. We also consider extensions to arbitrary spin and, in the spin-$ case, coupling of a quantized radiation field. In view of eventual applications to "completely integrable" one-dimensional field theories the case of one space dimension is studied as well, both in quantum mechanics and in classical mechanics.

show abstract

“…where u i denotes the i-th component of u ∈ [H 1/2 (R 3 )] N and t is the quadratic form defined in (7). Similarly we define the quadratic forms v, r γ and k γ , all with…”

Section: Exponential Decay Of Thementioning

confidence: 99%