Well-posedness of non-autonomous linear evolution equations for generators whose commutators are scalar

详细信息    查看全文

• 作者：Jochen Schmid
• 关键词：Well ; posedness ; Non ; autonomous linear evolution equations ; First or higher commutators of the generators are complex scalars
• 刊名：Journal of Evolution Equations
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：16
• 期：1
• 页码：21-50
• 全文大小：713 KB
• 参考文献：1.W. Arendt, C. Batty, M. Hieber, F. Neubrander: Vector-valued Laplace transforms and Cauchy problems. 2nd edition, Birkhäuser, 2012.
  2.O. Bratteli, D. W. Robinson: Operator algebras and quantum statistical mechanics 1 and 2. 2nd edition, Springer, 1987, 1997.
  3.F. Casas, A. Murua, M. Nadinic: Efficient computation of the Zassenhaus formula. arXiv:​1204.​0389 (2012).
  4.D. Cohn: Measure theory. 2nd edition, Birkhäuser, 2013.
  5.Constantin A.: The construction of an evolution system in the hyperbolic case and applications. Math. Nachr. 224, 49–73 (2001)MathSciNet CrossRef MATH
  6.Dereziński J., Gérard C.: Asymptotic completeness in quantum field theory. Massive Pauli–Fierz Hamiltonians. Rev. Math. Phys. 11, 383–450 (1999)MathSciNet CrossRef MATH
  7.J. Dereziński: Van Hove Hamiltonians – exactly solvable models of the infrared and ultraviolet problem. Ann. H. Poincaré 713–738 (2003)
  8.J. Dimock: Quantum mechanics and quantum field theory. A mathematical primer. Cambridge University Press, 2011.
  9.J. R. Dorroh: A simplified proof of a theorem of Kato on linear evolution equations J. Math. Soc. Japan 27 (1975), 474–478.
  10.K.-J. Engel, R. Nagel: One-parameter semigroups for linear evolution equations. Springer, 2000.
  11.F. Fer: Résolution de l’équation matricielle dU/dt = pU par produit infini d’exponentielles matricielles. Bull. Cl. Sci. 44 (1958), 818–829.
  12.Goldstein J.A.: Abstract evolution equations. Trans. Amer. Math. Soc. 141, 159–184 (1969)MathSciNet CrossRef MATH
  13.S. J. Gustafson, I. M. Sigal: Mathematical concepts of quantum mechanics. 2nd edition, Springer, 2011.
  14.S. Ishii: Linear evolution equations du/dt + A(t)u = 0: a case where A(t) is strongly uniform-measurable. J. Math. Soc. Japan 34 (1982), 413–424.
  15.Kato T.: Integration of the equation of evolution in a Banach space. J. Math. Soc. Japan 5, 208–234 (1953)MathSciNet CrossRef MATH
  16.Kato T.: Linear evolution equations of “hyperbolic” type. J. Fac. Sci. Univ. Tokyo 17, 241–258 (1970)MathSciNet MATH
  17.Kato T.: Linear evolution equations of “hyperbolic” type II. J. Math. Soc. Japan 25, 648–666 (1973)MathSciNet CrossRef MATH
  18.T. Kato: Abstract differential equations and nonlinear mixed problems. Lezioni Fermiane, Accademia Nazionale dei Lincei, Scuola Normale Superiore, Pisa (1985), 1–89.
  19.Kato T.: Abstract evolution equations, linear and quasilinear, revisited. In Lecture Notes in Math. 1540, 241–258 (1993)MathSciNet MATH
  20.J. V. Keler, S. Teufel: Non-adiabatic transitions in a massless scalar field. arXiv:​1204.​0344 (2012).
  21.Kobayasi K.: On a theorem for linear evolution equations of hyperbolic type. J. Math. Soc. Japan 31, 647–654 (1979)MathSciNet CrossRef
  22.Magnus W.: On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7, 649–673 (1954)MathSciNet CrossRef MATH
  23.M. Merkli: The ideal quantum gas. In Lecture Notes in Math. 1880 (2006), 183–233.
  24.R. Nagel, G. Nickel: Well-posedness for nonautonomous abstract Cauchy problems. Progr. Nonlinear Differential Equations Appl. 50 (2002), 279–293.
  25.R. Nagel, G. Nickel, R. Schnaubelt: private communication, 2014.
  26.H. Neidhardt, V. A. Zagrebnov: Linear non-autonomous Cauchy problems and evolution semigroups. Adv. Diff. Eq. 14 (2009), 289–340.
  27.H. Neidhardt, V. A. Zagrebnov: private communication, 2014.
  28.G. Nickel: On evolution semigroups and wellposedness of nonautonomous Cauchy problems. PhD thesis, 1996.
  29.G. Nickel, R. Schnaubelt: An extension of Kato’s stability condition for non-autonomous Cauchy problems. Taiw. J. Math. 2 (1998), 483–496.
  30.Nickel G.: Evolution semigroups and product formulas for nonautonomous Cauchy problems. Math Nachr. 212, 101–115 (2000)MathSciNet CrossRef MATH
  31.A. Pazy: Semigroups of linear operators and applications to partial differential equations. Springer, 1983.
  32.Phillips R.S.: Perturbation theory for semi-groups of linear operators. Trans. Amer. Math. Soc. 74, 199–221 (1953)MathSciNet CrossRef
  33.M. Reed, B. Simon: Methods of modern mathematical physics I-IV. Academic Press, 1980, 1975, 1979, 1978.
  34.J. Schmid, M. Griesemer: Well-posedness of non-autonomous linear evolution equations in uniformly convex spaces. arXiv:​1501.​06092 (2015).
  35.R. Schnaubelt: Well-posedness and asymptotic behaviour of non-autonomous linear evolution equations. Progr. Nonlinear Differential Equations Appl. 50 (2002), 311–338.
  36.H. Spohn: Dynamics of charged particles and their radiation field. Cambridge University Press, 2004.
  37.M. Suzuki: On the convergence of exponential operators – the Zassenhaus formula, BCH formula and systematic approximants. Commun. Math. Phys. 57 (1977), 193–200.
  38.Tanaka N.: Generation of linear evolution operators. Proc. Amer. Math. Soc. 128, 2007–2015 (2000)MathSciNet CrossRef MATH
  39.Tanaka N.: A characterization of evolution operators. Studia Math. 146(3), 285–299 (2001)MathSciNet CrossRef MATH
  40.R. M. Wilcox: Exponential operators and parameter differentiation in quantum physics. J. Math. Phys. 3 (1967), 962–982.
  41.A. Yagi: On a class of linear evolution equations of “hyperbolic” type in reflexive Banach spaces. Osaka J. Math. 16 (1979), 301–315.
  42.A. Yagi: Remarks on proof of a theorem of Kato and Kobayasi on linear evolution equations. Osaka J. Math. 17 (1980), 233–244.
  43.Yajima K.: Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys. 110, 415–426 (1987)MathSciNet CrossRef MATH
  
• 作者单位：Jochen Schmid (1)
  
  1. Fachbereich Mathematik, Universität Stuttgart, 70569, Stuttgart, Germany
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1424-3202

文摘

We prove the well-posedness of non-autonomous linear evolution equations for generators \({A(t): D(A(t)) \subset X \to X}\) whose pairwise commutators are complex scalars, and in addition, we establish an explicit representation formula for the evolution. We also prove well-posedness in the more general case where instead of the onefold commutators only the p-fold commutators of the operators A(t) are complex scalars. All these results are furnished with rather mild stability and regularity assumptions: Indeed, stability in X and strong continuity conditions are sufficient. Additionally, we improve a well-posedness result of Kato for group generators A(t) by showing that the original norm continuity condition can be relaxed to strong continuity. Applications include Segal field operators and Schrödinger operators for particles in external electric fields.