Finite total cross-sections in nonrelativistic quantum mechanics

Enss, Volker; Simon, Barry

doi:10.1007/bf01212825

Cited by 39 publications

(30 citation statements)

References 35 publications

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“…The lemma is verified in almost the same way as in the proof of Theorem A. 1.1 of [4]. We give only a sketch for the proof.…”

Section: If We Writementioning

confidence: 67%

Semi-classical bounds on scattering cross sections in two dimensional magnetic fields

Tamura

1997

Nagoya Mathematical Journal

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Abstract. We prove the uniform boundedness of averaged total cross sections or of quantities related to scattering into cones in the semi-classical limit for scattering by two dimensional magnetic fields. We do not necessarily assume that the energy under consideration is in a non-trapping energy range in the sense of classical dynamics. §1. IntroductionThe present work is a continuation to [20] where we have studied the shadow scattering (the quantum total cross section doubles the classical one in the semi-classical limit) in magnetic fields under the assumption that the energy under consideration is in a non-trapping energy range in the sense of classical dynamics and we have proved that the shadow scattering is in gerenal violated in the case of scattering by magnetic fields. We here study the problem about the uniform boundedness of averaged total cross sections or of quantities related to scattering into cones in the semi-classical limit without assuming such a non-trapping energy condition. In final section (Section 9), we also study the bound on cross sections for scattering by magnetic fields with small support. As a conclusion, we can obtain that such a bound seems to depend on the flux of magnetic fields.Throughout the whole exposition, we work exclusively in the two dimensional space R 2 with generic point is a real smooth function with compact support.The Hamiltonian H(A) describes a quantum particle of unit mass moving in the magnetic field b. The magnetic potential A(x) with dA = b is not uniquely determined and cannot be expected to fall off rapidly at infinity, even if b(x) is assumed to be compactly supported. In fact, A{x) does not decay faster than Odxj"" 1 ), ifdoes not vanish (2πβ being called the flux of magnetic field 6), where the integral with no domain attached is taken over the whole space. We use this abbreviation throughout. Thus the perturbation H(A) -H$ to the free Hamiltonian HQ = -Δ/2 is of long-range class. This implies that the scattering matrix does not necessarily admit the usual decomposition /<ί+ {Hilbert-Schmidt class}, Id being the identity operator, as in the shortrange scattering case and also the total cross section is not necessarily finite. Indeed, it depends on the value β and becomes finite only for integer /3GZ. We shall explain the above matter more precisely. The total cross section is invariant under gauge transformations. We fix one of magnetic potentials A(x) with dA -b and define it as follows:(1.4) a x (x) = -(2τr) 1 d2J\og\x-y\b(y)dy, a 2 {x) = (2π) ι dχ I \og\x-y\b(y)dy.

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“…The lemma is verified in almost the same way as in the proof of Theorem A. 1.1 of [4]. We give only a sketch for the proof.…”

Section: If We Writementioning

confidence: 67%

Semi-classical bounds on scattering cross sections in two dimensional magnetic fields

Tamura

1997

Nagoya Mathematical Journal

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“…These results [7] are summarized under the rubric of dilation analytic operators, of which the N-particle Coulomb problems are a special case. This theory of dilatation analyticity has, since, taken on a life of its own [8], quite independently of the asymptotic completeness question, which was eventually resolved (positively) by Enss [9] using quite different methods. The basic tool of dilatation analyticity is the use of complex scale transformations, wherein interparticle distances are scaled as r * reie, 8 being a (possible complex) rotation angle in the present discussion, leading to the method also being referred to as the method of complex scale transformations, giving a clear connection to the much older use of "real" scale transformations of atomic coordinates as used, say, by Hylleraas.…”

Section: Origins Of Complex Scalingmentioning

confidence: 98%

Exploiting the analyticity of Schrödinger operators: Theory and the computation of partial cross sections

Reinhardt

Han

1996

Int. J. Quantum Chem.

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The idea of dilation, or dilatation, analyticity with respect to complex scaling of the interparticle distances for nonrelativistic atomic or molecular electronic Hamiltonians is now over 20 years old and the first major reviews are just over 10. The method continues to be a fruitful source of new theoretical and computational results. Under the scale transformation r = re", the usual "spectrum" of bound states is exactly preserved, and scattering continua are "rotated" off the real axis by an angle of -2Re(e) about their respective thresholds. Useful features of this transformation are (1) that resonances are exposed, and, thus, (complex) resonance eigenvalues are easily calculated as the wave functions are L2, and standard results of Kato-type perturbation theory can thus be applied to them; (2) that utilization of this technique to study atoms in ac and dc fields was an early, and still evolving extension of the original theory; and (3) the fact that the "continua" are rotated off the real energy axis allows scattering information to be extracted from computations carried out entirely L2 in bases as the usual resolvents of scattering theory are no longer singular on the real axis. After a brief survey of the technique and its applications, these ideas are illustrated by discussion of positive energy-bound states and resonances, the extension of the theory to include the dc Stark effect, and a review of the resolution of the initially perplexing problem of atomic and molecular bound states in continua. These theoretical results are followed by a discussion of some very recent computational results, allowing computation of atomic partial photoionization cross sections with no specific coordinate space enforcement of boundary conditions, a highly advantageous situation for calculation of the partial cross section for three-body breakup, as in the process ho + He * He2++ e-+ e-. 0

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“…On the other hand one can derive a bound like gex for large g, where ex depends on the rate of decay of the potential at infinity [38]. On the other hand one can derive a bound like gex for large g, where ex depends on the rate of decay of the potential at infinity [38].…”

Section: Chaptermentioning

confidence: 99%

Non-Relativistic Quantum Dynamics

Amrein¹

1981

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Finite total cross-sections in nonrelativistic quantum mechanics

Abstract: Abstract. We present a simple geometric method for estimating total crosssections in two-body and more generally two cluster scattering. We discuss a variety of aspects of total cross-sections including large coupling constant behavior.

Cited by 39 publications

References 35 publications

Semi-classical bounds on scattering cross sections in two dimensional magnetic fields

Semi-classical bounds on scattering cross sections in two dimensional magnetic fields

Exploiting the analyticity of Schrödinger operators: Theory and the computation of partial cross sections

Non-Relativistic Quantum Dynamics

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