Abstract
Let m ∈ \(L_{loc}^1 (\mathbb{R}^N ) \), 0 ≠ m+ in Kato's class. We investigate the spectral function λ ↦ s(Δ + λm) where s(Δ + λm) denotes the upper bound of the spectrum of the Schrödinger operator Δ + λm. In particular, we determine its derivative at 0. If m- is sufficiently large, we show that there exists a unique λ1 > 0 such that s(Δ + λ1m) = 0. Under suitable conditions on m+ it follows that 0 is an eigenvalue of Δ + λ1m with positive eigenfunction.
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Agmon, S.: Lectures on Exponential Decay of Solutions of Second-order Elliptic Equations. Math. Notes 29, Princeton University Press, Princeton N.J., 1992.
Allegretto, W.: Principal eigenvalues for indefinite-weight elliptic problems in ℝn. Proc. Amer. Math. Soc. 116(1992), 701-706.
Arendt, W.: Gaussian estimates and interpolation of the spectrum in Lp. Differential and Integral Equations 7(1994), 1153-1168.
Arendt, W. and Batty, C. J. K.: Domination and ergodicity for positive semigroups. Proc. Amer. Math. Soc. 114(1992), 743-747.
Arendt, W. and Batty, C. J. K.: Exponential stability of a diffusion equation with absorption. Differential and Integral Equations 6(1993), 1009-1024.
Arendt, W. and Batty, C. J. K.: Absorption semigroups and Dirichlet boundary conditions. Math. Ann. 295(1993), 427-448.
Arendt, W. and Batty, C. J. K.: The spectral bound of Schr¨odinger operators. Potential Analysis 5(1996), 207-230.
Arendt, W. and Batty, C. J. K.: Principal eigenvalues and perturbation. To appear in: Analysis Symposium.Leiden 1993. P. Huijsmans, M.A. Kaashoek, W. Luxemburg, and B. de Pagter (eds.) Birkhäuser.
Batty, C. J. K.: Asymptotic stability of Schrödinger semigroups: path integral methods. Math. Ann. 292(1992), 457-492.
Brown, K. J., Cosner, C., and Fleckinger, J.: Principal eigenvalues for problems with indefinite weight functions on ℝN. Proc. Amer. Math. Soc. 109(1990), 147-155.
Brown, K. J., Daners, D., and López-Gómez, J.: Change of stability for Schrödinger semigroups. Preprint.
Brown, K. J., Lin, S. S., and Tertikas, A.: Existence and nonexistence of steady-state solutions for a selection-migration model in population genetics. J. Math. Biol. 27(1989), 91-104.
Brown, K. J. and Tertikas, A.: The existence of principal eigenvalues for problems with indefinite weight function on ℝk. Proc. Royal Soc. Edinburgh 123A(1993), 561-569.
Daners, D. and Koch Medina, P.: Superconvexity of the evolution operator and parabolic eigenvalue problems on ℝk. Differential and Integral Equations, to appear.
Daners, D. and Koch Medina, P.: Exponential stability, change of stability and eigenvalue problems for linear time periodic parabolic equations on ℝN. Differential and Integral Equations, to appear.
Daners, D.: Principal eigenvalues for some periodic-parabolic operators on ℝN and related topics. Preprint. University of Sydney, 1993.
Davies, E. B.: Spectral Theory and Differential Operators. Cambridge University Press, 1995.
Dautray, R. and Lions, J. L.: Analyse Mathématique et Calcul Numérique.Masson, 1987.
Fefferman, C. L.: The uncertainty principle. Bull. Amer. Math. Soc. 9(1983), 129-206.
Fleming, W. H.: A selection-migration model in population genetics. J. Math. Biol. 2(1975), 219-233.
Hess, P. and Kato, T.: On some linear and nonlinear eigenvalue problems with an indefinite weight function. Comm. Partial Diff. Equ. 5(1980), 999-1030.
Hempel, R. and Voigt, J.: The spectrum of a Schrödinger operator in L p(ℝN ) is p-independent. Comm. Math. Phys. 104(1986), 243-250.
Hempel, R. and Voigt, J.: The spectrum of Schrödinger operators in L p(ℝd ) and in C0(ℝd ). In Mathematical Results in Quantum Mechanics, M. Demuth, P. Exner, H. Neidhardt, and V. Zagrebnov, eds., Birkhäuser, Basel, 1994.
Kato, T.: Perturbation Theory. Springer, 1980.
Nagel, R. (ed.): One-Parameter Semigroups of Positive Operators. Springer LNM 1184, 1986.
Nussbaum, R. D. and Pinchover, Y.: On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications. Journal d'Analyse Mathématique 59(1992), 161-177.
Ouhabaz, E., Stollmann, P., Sturm, T., and Voigt, J.: The Feller property for absorption semigroups. Preprint.
Pinchover, Y.: Criticality and ground states for second-order elliptic equations. J. Diff. Equ. 80(1989), 237-250.
Pinchover, Y.: On criticality and groung states of second order elliptic equations II. J. Diff. Equ. 87(1990), 353-364.
Reed, R. and Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlargeed., Academic Press, New York, 1980.
Reed, M. and Simon, B.: Methods of Modern Mathematical Physics IV: Analysis of Operators. Academic Press, New York, 1978.
Schaefer, H. H.: Banach Lattices and Positice Operators. Springer, Berlin, 1974.
Schechter, M.: The spectrum of the Schrödinger operator. Trans. Amer. Math. Soc. 312(1989), 115-128.
Simon, B.: Functional Integration and Quantum Physics. Academic Press, New York, 1979.
Simon, B.: Brownian motion, L p-properties of Schrödinger operators and the localization of binding. J. Functional Anal. 35(1980), 215-229.
Simon, B.: Large time behaviour of the L p-norm of Schrödinger semigroups. J. Functional Anal. 40(1981), 66–83.
Simon, B.: Schrödinger semigroups. Bull. Amer. Math. Soc. 7(1982), 447-526.
Stollmann, P. and Voigt, J.: A regular potential which is nowhere in L1. Lett. Math. Phys. 9(1985), 227-230.
Voigt, J.: On the perturbation theory for strongly continuous semigroups. Math. Ann. 229(1977), 163-171.
Voigt, J.: Absorption semigroups, their generators, and Schrödinger semigroups. J. Functional Anal. 67(1986), 167-205.
Voigt, J.: Absorption semigroups. J. Operator Theory 20(1988), 117-131.
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Arendt, W., Batty, C.J.K. The Spectral Function and Principal Eigenvalues for Schrödinger Operators. Potential Analysis 7, 415–436 (1997). https://doi.org/10.1023/A:1017928532615
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DOI: https://doi.org/10.1023/A:1017928532615