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Absorption semigroups and Dirichlet boundary conditions. (English) Zbl 0788.47031
Given a positive \(C_ 0\)-semigroup \(T\) on \(L^ p(X)\) and an arbitrary non-negative potential \(V\), an absorption semigroup \(T_ V\) is constructed as a \(C_ 0\)-semigroup on \(L^ p(X_ V)\) for a certain subset \(X_ V\) of \(X\). Information is obtained about \(X_ V\). When \(p=1\) and \(T\) is contractive and holomorphic, \(T_ V\) is also holomorphic. It follows that Schrödinger semigroups for arbitrary non- negative potentials are holomorphic on the appropriate part of \(L^ 1({\mathbf R}^ N)\), and the semigroup on \(L^ 1(\Omega)\) generated by the Laplacian with Dirichlet boundary conditions is holomorphic, for arbitrary open sets \(\Omega\).

MSC:
47D06 One-parameter semigroups and linear evolution equations
35K05 Heat equation
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[1] Albeverio, S., Ma, Z.-M. R?ckner, M.: Non-symmetric Dirichlet forms and Markov processes on general state spaces. C.R. Acad. Sci. Paris S?r. A314, 77-82 (1992) · Zbl 0745.60076
[2] Aliprantis, C.D., Burkinshaw, O.: Positive operators. New York: Academic Press 1985 · Zbl 0608.47039
[3] Amann, H.: Dual semigroups and second order linear elliptic boundary value problems. Isr. J. Math.45, 225-254 (1983) · Zbl 0535.35017
[4] Arendt, W.: Vector-valued Laplace transforms and Cauchy problems. Isr. J. Math.59, 327-352 (1987) · Zbl 0637.44001
[5] Arendt, W., Batty, C.J.K.: Exponential stability of a diffusion equation with absorption. J. Diff. Int. Eqn., to appear · Zbl 0817.35007
[6] Arendt, W., Batty, C.J.K., B?nilan Ph.: Asymptotic stability of Schr?dinger semigroups on L1(R N ). Math. Z.209, 511-518 (1992) · Zbl 0761.47019
[7] Arendt, W., Batty, C.J.K., Robinson, D.W.: Positive semigroups generated by elliptic operators on Lie groups. J. Oper. Theor.23, 369-407 (1990) · Zbl 0748.47034
[8] Arendt, W., B?nilan, Ph.: In?galit?s de Kato et semi-groupes sous-markoviens. Rev. Mat. Univ. Compl. Madrid, to appear
[9] Batty, C.J.K.: Asymptotic stability of Schr?dinger semigroups: path integral methods. Math. Ann.292, 457-492 (1992) · Zbl 0736.35025
[10] Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. New York: Academic Press 1968 · Zbl 0169.49204
[11] Cannon, J.T.: Convergence criteria for a sequence of semigroups. Appl. Anal.5, 23-31 (1975) · Zbl 0366.47017
[12] Ciesielski, Z.: Lectures on Brownian motion, heat conduction and potential theory. Math. Inst., Aarhus Univ., 1965 · Zbl 0137.11601
[13] Davies, E.B.: One-parameter semigroups. London: Academic Press 1980 · Zbl 0457.47030
[14] Davies, E.B.: Heat kernels and spectral theory. Cambridge: Cambridge Univ. Press 1989 · Zbl 0699.35006
[15] Demuth, M., Casteren van, J.A.: On spectral theory of selfadjoint Feller generators. Rep. Math. Phys.1, 325-414 (1989) · Zbl 0715.60093
[16] Ethier, S.N., Kurtz, T.G.: Markov processes. New York: Wiley 1986 · Zbl 0592.60049
[17] Fukushima, M.: Dirichlet forms and Markov processes. Tokyo: Kodansha 1980 · Zbl 0422.31007
[18] Herbst, I.W., Zhao, Z.: Sobolev spaces Kac regularity and the Feynman-Kac formula. Sem. Stochastic Processes (Princeton, 1987), Prog. Prob. Stat.154, 171-191, Basel: Birkh?user, 1988 · Zbl 0656.60089
[19] Hille, E., Phillips, R.S.: Functional analysis and semigroups. Am. Math. Soc. Coll. Publ., Providence, R.I., 1957 · Zbl 0078.10004
[20] Kato, T.: Perturbation theory for linear opeators. 2nd ed. Berlin Heidelberg New York: Springer 1976 · Zbl 0342.47009
[21] Kato, T.: Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups, in: Topics in functional analysis Gohberg I., Kac M. (eds.) pp. 185-195. New York Academic Press 1978
[22] Kato, T.: L p -theory of Schr?dinger operators in: Aspects of positivity in functional analysis. Nagel R., Schlotterbeck U., Wolff M. (eds.) pp. 63-78. North-Holland: Amsterdam 1986
[23] Krengel, U.: Ergodic theorems. Berlin: de Gruyter 1985 · Zbl 0575.28009
[24] Lumer, G., Paquet, L.: Semi-groupes holomorphes, produit tensoriel de semi-groupes et ?quations d’?volution, in: S?m de Th?orie du Potentiel, No. 4 (Lect. Notes Math. vol. 713) Berlin Heidelberg New York: Springer 1977/78
[25] Ma, Z.M., R?ckner, M.: An introduction to the theory of non-symmetric Dirichlet forms (to appear)
[26] McKean, H.P.: ? ? plus a bad potential. J. Math. Phys.18, 1277-1279 (1977) · Zbl 0357.47025
[27] Nagel, R. (ed.): One-parameter semigroups of positive operators. (Lect. Notes Math. vol. 1184) Berlin Heidelberg New York: Springer 1986 · Zbl 0585.47030
[28] Ouhabaz, E.-M.: L?-contractivity of semigroups generated by sectorial forms. J. Lond. Math. Soc. (to appear) · Zbl 0788.47034
[29] Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Berlin Heidelberg New York: Springer 1983 · Zbl 0516.47023
[30] Reed, M., Simon B.: Methods of modern mathematical physics I: Functional analysis, revised ed. New York: Academic Press 1980 · Zbl 0459.46001
[31] Reed, M., Simon, B.: Methods of modern mathematical physics II: Fourier analysis, selfadjointness. New York: Academic Press 1985
[32] Reed, M., Simon, B.: Methods of modern mathematical physics IV: Analysis of operators. New York: Academic Press 1978 · Zbl 0401.47001
[33] Robinson, D.W.: Elliptic operators and Lie groups. Oxford Univ. Press 1991 · Zbl 0747.47030
[34] Sato, R.: A note on a local ergodic theorem, Comm. Math. Univ. Carolinae16, 1-11 (1975) · Zbl 0296.28019
[35] Schaefer, H.H.: Banach lattices and positive operators, Berlin Heidelberg New York: Springer 1974 · Zbl 0296.47023
[36] Simon, B.: Functional integration and quantum physics, New York: Acasdemic Press, 1979 · Zbl 0434.28013
[37] Simon, B.: Brownian motion, L p properties of Schr?dinger operators, and the localization of binding. J. Funct. Anal.35, 215-229 (1980) · Zbl 0446.47041
[38] Simon, B.: Schr?dinger semigroups. Bull. Am. Math. Soc.7, 447-526 (1982) · Zbl 0524.35002
[39] Stewart, H.B.: Generation of analytic semigroups by strongly elliptic operators. Trans. Am. Math. Soc.199, 141-162 (1974) · Zbl 0264.35043
[40] Stewart, H. B.: Generation of analytic ssemigroups by strongly elliptic operators under general boundary conditions. Trans. Am. Math. Soc.259, 229-310 (1980) · Zbl 0451.35033
[41] Stollmann, P., Voigt, J.: A regular potential which is nowhere in L1. Lett. Math. Phys.9, 227-230 (1985) · Zbl 0588.47055
[42] Stroock, D.: The Kac approach to potential theory: Part I. J. Math. Mech.16, 820-852 (1967) · Zbl 0148.36202
[43] Tanabe, H.: On semilinear operators of elliptic and parabolic type, in: Functional analysis and numerical analysis, Japan-France Seminar, Tokyo and Kyoto, 1976, H. Fujita (ed.) pp. 455-463. Japan Soc. Promotion Science, Tokyo, 1978
[44] Van Casteren, J.A.: Generators of strongly continuous semigroups, (Res. Notes Math. 115) Boston: Pitman 1985 · Zbl 0576.47023
[45] Voigt, J.: Absorption semigroups, their generators, and Schr?dinger semigroups. J. Funct. Anal.68, 167-205 (1986) · Zbl 0628.47027
[46] Voigt, J.: Absorption semigroups. J. Oper. Theory20, 117-131 (1988) · Zbl 0702.47024
[47] Yosida, K.: Functional analysis 6th ed. Berlin Heidelberg New York: Springer 1980 · Zbl 0435.46002
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