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Uniqueness in the characteristic Cauchy problem of the Klein-Gordon equation and tame restrictions of generalized functions. (English) Zbl 1133.35393

Summary: We show that every tempered distribution, which is a solution of the (homogeneous) Klein-Gordon equation, admits a “tame” restriction to the characteristic (hyper)surface \(\{x^{0} + x^{n} = 0\}\) in \((1 + n)\)-dimensional Minkowski space and is uniquely determined by this restriction. The restriction belongs to the space \({{\mathcal S}'_{\partial_-}({\mathbb{R}}^n)}\) which we have introduced by P. Ullrich [J. Math. Phys. 45, No. 8, 3109–3145 (2004; Zbl 1071.81083)]. Moreover, we show that every element of \({{\mathcal S}'_{\partial_-}({\mathbb{R}}^n)}\) appears as the “tame” restriction of a solution of the (homogeneous) Klein-Gordon equation.

MSC:

35L15 Initial value problems for second-order hyperbolic equations
46F05 Topological linear spaces of test functions, distributions and ultradistributions
35A05 General existence and uniqueness theorems (PDE) (MSC2000)

Citations:

Zbl 1071.81083
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References:

[1] Atiyah M.F., MacDonald I.G. (1969) Introduction to Commutative Algebra. Addison–Wesley, Reading · Zbl 0175.03601
[2] Bogolubov N.N., Logunov A.A., Oksak A.I., Todorov I. (1990) General Principles of Quantum Field Theory. Kluwer, Dordrecht · Zbl 0732.46040
[3] Brodsky S.J., Pauli H.-C. (1998) Quantum chromodynamics and other field theories on the light cone. Phys. Lett. C 301: 299 hep-ph/9705477
[4] Chang S., Root R.G., Yan T. (1973) Quantum field theories in the infinite-momentum frame. I. Quantization of scalar and Dirac fields. Phys. Rev. D 7: 1133 · doi:10.1103/PhysRevD.7.1133
[5] Dirac P.A.M. (1949) Forms of relativistic dynamics. Rev. Mod. Phys. 21: 392 · Zbl 0035.26803 · doi:10.1103/RevModPhys.21.392
[6] Ehrenpreis L. (1962) Solution of some problems of division, Part IV. Am. J. Math. 82: 522 · Zbl 0098.08401 · doi:10.2307/2372971
[7] Gårding L., Malgrange B. (1958) Opérateurs différentiels partiellement hypoelliptiques. C. R. Acad. Sci. 247: 2083 · Zbl 0142.38101
[8] Gårding L., Malgrange B. (1961) Opérateurs différentiels partiellement hypoelliptiques et partiellement elliptiques. Math. Scand. 9: 5 · Zbl 0108.10101
[9] Hartshorne R. (1977) Algebraic Geometry. Springer, Berlin Heidelberg New York · Zbl 0367.14001
[10] Heinzl Th., Werner E. (1994) Light-front quantization as an initial-boundary-value problem. Z. Phys. C 62: 521 · doi:10.1007/BF01555913
[11] Hörmander L. (1990) The Analysis of Linear Partial Differential Operators I. Springer, Berlin Heidelberg New York
[12] Hörmander L. (1990) The analysis of linear partial differential operators II. Springer, Berlin Heidelberg New York
[13] Leutwyler H., Klauder J.R., Streit L. (1970) Quantum field theory on lightlike slabs. Nuovo Cimento A 66: 536 · Zbl 0191.27103 · doi:10.1007/BF02826338
[14] Rudin W. (1990) Functional Analysis. McGraw Hill, New York, Reprint · Zbl 0698.43001
[15] Schweber S.S. (1962) An Introduction to Relativistic Quantum Field Theory. Harper & Row, New York · Zbl 0111.43102
[16] Ullrich P. (2004) On the restriction of quantum fields to a lightlike surface. J. Math. Phys. 45: 3109 · Zbl 1071.81083 · doi:10.1063/1.1765746
[17] Ullrich, P.: Tame restrictions of distributions and the wave front set (in preparation)
[18] Ullrich, P.: F-tempered distributions (in preparation)
[19] Ullrich P., Werner E. (2006) On the problem of mass-dependence of the two-point function of the real scalar free massive field on the light cone. J. Phys. A: Math. Gen. 39: 6057 hep-th/0503176 · Zbl 1103.81032 · doi:10.1088/0305-4470/39/20/029
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