Abstract
In this paper we show how spectral theory for Herglotz functions and differential operators is related to and dependent on the geometrical properties of the complex upper half-plane, viewed as a hyperbolic space. We establish a theory of value distribution for Lebesgue measurable functions f: R→R and introduce the value distribution function associated with any given Herglotz function F. We relate the theory of value distribution for boundary values of Herglotz functions to the description of asymptotics for solutions of the Schrödinger equation on the half-line. We establish two results which play a key role in understanding asymptotic value distribution for Schrödinger operators with sparse potentials, and its implications for spectral theory.
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Breimesser, S. V. and Pearson, D. B.: Asymptotic value distribution for solutions of the Schrödinger equation, Math. Phys. Anal. Geom. 3 (2000), 385–403.
Coddington, E. A. and Levinson, N.: Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.
Eastham, M. S. P. and Kalf, H.: Schrödinger-type Operators with Continuous Spectra, Pitman, Boston, 1982.
Akhiezer, N. I. and Glazman, I. M.: Theory of Linear Operators in Hilbert Space, Pitman, London, 1981.
Pommerenke, C.: Boundary Behaviour of Conformal Maps, Springer, Berlin, 1992.
Pearson, D. B.: Value distribution and spectral theory, Proc. London Math. Soc. 68(3) (1993), 127–144.
Oxtoby, J. C.: Measure and Category, Springer, New York, 1971.
Saks, S.: Theory of the Integral, Hafner Publ., New York, 1933.
Pearson, D. B.: Quantum Scattering and Spectral Theory, Academic Press, London, 1988.
Simon, B. and Stolz, G.: Operators with singular continuous spectrum, V. Sparse potentials, Proc. Amer. Math. Soc. 124(7) (1996), 2073–2080.
McGillivray, I., Stollmann, P. and Stolz, G.: Absence of absolutely continuous spectra for multidimensional Schrödinger operators with high barriers, Bull. London Math. Soc. 27 (1995), 162–168.
Simon, B. and Spencer, T.: Trace class perturbations and the absence of absolutely continuous spectrum, Comm. Math. Phys. 125 (1989), 113–126.
Stolz, G.: Spectal theory for slowly oscillating potentials, II. Schrödinger operators, Math. Nachr. 183 (1997), 275–294.
Deift, P. and Killip, R.: On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials, Comm. Math. Phys. 203 (1999), 341–347.
Christodoulides, Y.: Spectral theory of Herglotz functions and their compositions and the Schrödinger equation, PhD thesis (2002), University of Hull.
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Breimesser, S.V., Pearson, D.B. Geometrical Aspects of Spectral Theory and Value Distribution for Herglotz Functions. Mathematical Physics, Analysis and Geometry 6, 29–57 (2003). https://doi.org/10.1023/A:1022410108020
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DOI: https://doi.org/10.1023/A:1022410108020