zbMATH — the first resource for mathematics

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 \(\Omega\).

47D06 One-parameter semigroups and linear evolution equations
35K05 Heat equation
Full Text: DOI EuDML
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.