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Change of stability for Schrödinger semigroups*

  • K. J. Brown (a1), D. Daners (a2) and J. López-Gómez (a2)

Abstract

In this paper we analyse the change of stability of Schrödinger semigroups with indefinite potentials when a coupling parameter varies. Generically, the change of stability takes place at a principal eigenvalue associated with the problem. The uniqueness of the principal eigenvalue is shown for several classes of potentials.

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1Allegretto, W.. Principal eigenvalues for indefinite weight elliptic problems in RN. Proc. Amer. Math. Soc. 116 (1992), 701–6.
2Arendt, W. and Batty, C. J. K.. Exponential stability of a diffusion equation with absorption. Differential Integral Equations 6 (1993), 1009–24.
3Beltramo, A. and Hess, P.. On the principal eigenvalues of a periodic-parabolic operator. Comm. Partial Differential Equations 9 (1984), 919–24.
4Browder, F. E.. On the spectral theory of elliptic differential operators. I. Math. Ann. 142 (1961), 22130.
5Brown, K. J., Cosner, C. and Fleckinger, J.. Principal eigenvalues for problems with indefinite weight function on RN. Proc. Amer. Math. Soc. 109 (1990), 147–55.
6Brown, K. J. and Tertikas, A.. The existence of principal eigenvalues for problems with indefinite weight functions on RN. Proc. Roy. Soc. Edinburgh, Sect. A 123 (1993), 561–9.
7Clément, P., Heijmans, H. J. A. M.et al. One-parameter Semigroups, CWI Monograph 5 (Amsterdam: North-Holland, 1987).
8Daners, D. and Koch Medina, P.. Abstract Evolution Equations, Periodic Problems and Applications, Pitman Research Notes in Mathematics 279 (Harlow: Longman, 1992).
9Daners, D. and Koch Medina, P.. Superconvexity of the evolution operator and parabolic eigenvalue problems on RN. Differential Integral Equations 7 (1994), 235–55.
10Hess, P. and Kato, T.. On some linear and nonlinear eigenvalue problems with an indefinite weight function. Comm. Partial Differential Equations 5 (1980), 9991030.
11Itô, K. and McKean, H. P.. Diffusion Processes and Their Sample Paths (Berlin: Springer, 1965).
12Kato, T.. Perturbation Theory for Linear Operators (New York: Springer, 1966).
13Kato, T.. Superconvexity of the spectral radius and convexity of the spectral bound and the type. Math. Z. 180 (1982), 265–73.
14Li, Y. and Ni, W. M.. On conformal scalar curvature equations in RN. Duke Math. J. 57 (1988), 895924.
15Nagel, R.et al. One Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics 1184 (Berlin: Springer, 1986).
16Protter, M. H. and Weinberger, H. F.. Maximum Principles in Differential Equations (New York: Springer, 1984).
17Schaefer, H. H.. Banach Lattices and Positive Operators (Berlin: Springer, 1974).
18Simon, B.. Large time behaviour of the Lp norm of Schrödinger operators. J. Fund. Anal. 40 (1981), 6683.
19Simon, B.. Schrödinger semigroups. Bull. Amer. Math. Soc. 7 (1982), 447526.

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