Davies, E. B.; Simon, B. Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. (English) Zbl 0568.47034 J. Funct. Anal. 59, 335-395 (1984). The authors investigate connections between integral kernels of positivity preserving semigroups and \(L^ p\)-contractivity properties. There are treated essentially four connected topics: (1) Extension properties for \(e^{-tA}\) from \(L^ 2\) to \(L^{\infty}\) where A is a Schrödinger operator generated by its ground state. (2) The same problem for the Dirichlet Laplacian for certain subsets of \({\mathbb{R}}^ n.\) (3) Sobolev estimates up to the boundary. (4) Pointwise bounds for the integral kernel of \(e^{-Nt}\) in terms of the ground state of H. Reviewer: H.Siedentop Cited in 6 ReviewsCited in 224 Documents MSC: 47D03 Groups and semigroups of linear operators 47F05 General theory of partial differential operators 35J10 Schrödinger operator, Schrödinger equation Keywords:ultracontractivity; integral kernels of positivity preserving semigroups; \(L^ p\)-contractivity properties; Schrödinger operator; Dirichlet Laplacian; Sobolev estimates; Pointwise bounds; ground state × Cite Format Result Cite Review PDF Full Text: DOI References: [2] Aizenman, M.; Simon, B., Brownian motion and Harnack’s inequality for Schrödinger operators, Comm. Pure Appl. Math., 35, 209-273 (1982) · Zbl 0459.60069 [3] Bergh, J.; Löfstrom, J., Interpolation Spaces: An Introduction (1976), Springer: Springer New York/Berlin · Zbl 0344.46071 [4] Brossard, J., The problem of Dirichlet for the Schrödinger operator, Ann. Inst. Four. (Grenoble) (1984) [5] Carmona, R., Regularity properties of Schrödinger and Dirichlet semigroups, J. Funct. Anal., 17, 227-237 (1974) [6] Carmona, R., Pointwise bounds for Schrödinger eigenstates, Comm. Math. Phys., 62, 97-106 (1978) · Zbl 0403.47016 [7] Carmona, R.; Simon, B., Pointwise bounds on eigenfunctions and wave packets in \(N\)-body quantum systems. V. Lower bounds and path integrals, Comm. Math. Phys., 80, 59-98 (1981) · Zbl 0464.35085 [8] Davies, E. B., One Parameter Semigroups (1980), Academic Press: Academic Press New York · Zbl 0457.47030 [9] Davies, E. B., Hypercontractive and related bounds for double well Schrödinger operators, Quart. J. Math. Oxford Ser. (2), 34, 407-421 (1983) · Zbl 0546.34021 [10] Davies, E. B., Spectral properties of metastable Markov semigroups, J. Funct. Anal., 52, 315-329 (1983) · Zbl 0525.47030 [12] Davies, E. B., JWKB and related bounds on Schrödinger eigenfunctions, Bull. London Math. Soc., 14, 273-284 (1982) · Zbl 0525.35026 [13] Davies, E. B.; Simon, B., Ultracontractive semigroups and some problems in analysis, (Aspects of Mathematics and Its Applications (1984), North-Holland: North-Holland Amsterdam) · Zbl 0638.47044 [14] Eckman, J. P., Hypercontractivity for anharmonic oscillators, J. Funct. Anal., 16, 388-404 (1974) · Zbl 0285.47032 [15] Gilbarg, D.; Trudinger, N., Elliptic Partial Differential Equations of Second Order (1977), Springer: Springer New York/Berlin · Zbl 0361.35003 [16] Glimm, J., Boson fields with nonlinear self-interaction in two dimensions, Comm. Math. Phys., 8, 12-25 (1968) · Zbl 0173.29903 [17] Grisvard, P., Singularities for the problem of limits in polyhedrons, (Sem. Goul.-Schwartz (1981-1982)), Ex. No. 8 · Zbl 0633.73014 [18] Gross, L., Logarithmic Sobolev inequalities, Amer. J. Math., 97, 1061-1083 (1976) · Zbl 0318.46049 [20] Harrell, E., On the rate of asymptotic eigenvalue degeneracy, Comm. Math. Phys., 60, 73-95 (1978) · Zbl 0395.34023 [21] Helffer, B.; Sjostrand, J., Multiple wells in the semi-classical limit I, Comm. Partial Differential Equations, 9, 4, 337-408 (1984) · Zbl 0546.35053 [23] Nelson, E., A quartic interaction in two dimensions, (Mathematical Theory of Elementary Particles (1966), MIT Press: MIT Press Cambridge, Mass), 69-73 [24] Nelson, E., The free Markov field, J. Funct. Anal., 12, 211-227 (1973) · Zbl 0273.60079 [25] Persson, A., Compact linear mappings between interpolation spaces, Ark. Mat., 5, 215-219 (1964) · Zbl 0128.35204 [26] Port, S.; Stone, C., Brownian Motion and Classical Potential Theory (1978), Academic Press: Academic Press New York · Zbl 0413.60067 [27] Reed, M.; Simon, B., Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness (1975), Academic Press: Academic Press New York · Zbl 0308.47002 [28] Reed, M.; Simon, B., Methods of Modern Mathematical Physics III: Scattering Theory (1979), Academic Press: Academic Press New York · Zbl 0405.47007 [29] Reed, M.; Simon, B., Methods of Modern Mathematical Physics IV: Analysis of Operators (1978), Academic Press: Academic Press New York · Zbl 0401.47001 [30] Rosen, J., Sobolev inequalities for weighted spaces and supercontractive estimates, Trans. Amer. Math. Soc., 222, 367-376 (1976) · Zbl 0344.46072 [32] Simon, B., Ergodic semigroups of positivity preserving self-adjoint operators, J. Funct. Anal., 12, 335-339 (1973) · Zbl 0252.47034 [33] Simon, B., Pointwise bounds on eigenfunctions and wave packets in \(N\)-body quantum systems III, Trans. Amer. Math. Soc., 208, 317-329 (1975) · Zbl 0305.35078 [34] Simon, B., Large time behavior of the \(L^p\) norm of Schrödinger semigroups, J. Funct. Anal., 40, 66-83 (1981) · Zbl 0478.47024 [36] Simon, B., Some quantum operators with discrete spectrum but classically continuous spectrum, Ann. Physics, 146, 209-220 (1983) · Zbl 0547.35039 [37] Simon, B., Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.), 7, 447-526 (1982) · Zbl 0524.35002 [38] Simon, B., The \(P(φ)_2\) Euclidean (quantum) field theory, Princeton Series in Physics (1974), Princeton Univ. Press: Princeton Univ. Press Princeton, N.J · Zbl 1175.81146 [39] Simon, B.; Hoegh-Krohn, R., Hypercontractive semigroups and two-dimensional self-coupled Bose fields, J. Funct. Anal., 9, 121-180 (1972) · Zbl 0241.47029 [41] Rosen, J.; Simon, B., Fluctuations of the paths in \(P(φ)_1\) Markov processes, Ann. Probab., 4, 155-174 (1976) · Zbl 0337.60064 [43] Simon, B., Functional Integration and Quantum Physics (1979), Academic Press: Academic Press New York · Zbl 0434.28013 [44] Jerison, D.; Kenig, C., Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math., 46, 80-147 (1982) · Zbl 0514.31003 [45] Miller, K., Barriers on cones for uniformly elliptic operators, Ann. Mat. Pura Appl. (4), 76, 93-105 (1976) · Zbl 0149.32101 [46] Miller, K., Extremal barriers on cones with Phragmèn-Lindlöf theorems and other applications, Ann. Mat. Pura Appl. (4), 90, 297-329 (1971) · Zbl 0231.35004 [47] Oddson, J., On the boundary point principle for elliptic equations in the plane, Bull. Amer. Math. Soc. (N.S.), 74, 666-667 (1978) · Zbl 0157.18102 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.