Spectral and scattering theory for some abstract QFT Hamiltonians. (English) Zbl 1166.81326
Summary: We introduce an abstract class of bosonic QFT Hamiltonians and study their spectral and scattering theories. These Hamiltonians are of the form \(H= d\Gamma(\omega)+V\) acting on the bosonic Fock space \(\Gamma({\mathfrak h})\), where \(\omega\) is a massive one-particle Hamiltonian acting on \({\mathfrak h}\) and \(V\) is a Wick polynomial \(\text{Wick}(w)\) for a kernel \(w\) satisfying some decay properties at infinity.
We describe the essential spectrum of \(H\), prove a Mourre estimate outside a set of thresholds and prove the existence of asymptotic fields. Our main result is the asymptotic completeness of the scattering theory, which means that the CCR representations given by the asymptotic fields are of Fock type, with the asymptotic vacua equal to the bound states of \(H\). As a consequence, \(H\) is unitarily equivalent to a collection of second quantized Hamiltonians.
We describe the essential spectrum of \(H\), prove a Mourre estimate outside a set of thresholds and prove the existence of asymptotic fields. Our main result is the asymptotic completeness of the scattering theory, which means that the CCR representations given by the asymptotic fields are of Fock type, with the asymptotic vacua equal to the bound states of \(H\). As a consequence, \(H\) is unitarily equivalent to a collection of second quantized Hamiltonians.
MSC:
81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
81T08 | Constructive quantum field theory |
47N50 | Applications of operator theory in the physical sciences |
81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |
81U05 | \(2\)-body potential quantum scattering theory |
81T10 | Model quantum field theories |
References:
[1] | DOI: 10.1007/978-3-0348-7762-6 · doi:10.1007/978-3-0348-7762-6 |
[2] | DOI: 10.1006/jfan.1999.3472 · Zbl 0997.47061 · doi:10.1006/jfan.1999.3472 |
[3] | Boutet de Monvel A., Astérisque 210 pp 75– |
[4] | DOI: 10.1007/s002200000233 · Zbl 1082.81518 · doi:10.1007/s002200000233 |
[5] | DOI: 10.1142/S0129055X99000155 · Zbl 1044.81556 · doi:10.1142/S0129055X99000155 |
[6] | DOI: 10.1007/s00220-006-0134-x · Zbl 1137.81047 · doi:10.1007/s00220-006-0134-x |
[7] | DOI: 10.1016/j.jfa.2006.12.009 · Zbl 1121.46055 · doi:10.1016/j.jfa.2006.12.009 |
[8] | DOI: 10.1007/s002200050758 · Zbl 0961.81009 · doi:10.1007/s002200050758 |
[9] | DOI: 10.1007/s00220-004-1111-x · Zbl 1091.81059 · doi:10.1007/s00220-004-1111-x |
[10] | Gérard C., J. Math. Kyoto Univ. 38 pp 595– · Zbl 0934.35111 · doi:10.1215/kjm/1250518000 |
[11] | DOI: 10.1007/s00023-008-0396-2 · Zbl 1165.81033 · doi:10.1007/s00023-008-0396-2 |
[12] | Hörmander L., The Analysis of Linear Partial Differential Operators 3 (1985) |
[13] | DOI: 10.1080/03605309908821502 · Zbl 0944.35014 · doi:10.1080/03605309908821502 |
[14] | DOI: 10.4310/ATMP.2003.v7.n4.a3 · Zbl 1058.81080 · doi:10.4310/ATMP.2003.v7.n4.a3 |
[15] | DOI: 10.1007/s00023-003-0136-6 · Zbl 1057.81024 · doi:10.1007/s00023-003-0136-6 |
[16] | DOI: 10.1080/03605307808820077 · Zbl 0392.35056 · doi:10.1080/03605307808820077 |
[17] | DOI: 10.1002/cpa.3160240306 · doi:10.1002/cpa.3160240306 |
[18] | DOI: 10.1016/0022-1236(72)90008-0 · Zbl 0241.47029 · doi:10.1016/0022-1236(72)90008-0 |
[19] | DOI: 10.1017/CBO9780511535178 · Zbl 1078.81004 · doi:10.1017/CBO9780511535178 |
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