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On the spectral analysis of self adjoint operators generated by second order difference equations

  • Dale T. Smith (a1)


In this paper, I shall consider operators generated by difference equations of the form

where Δ is the forward difference operator, and a, p, and r are sequences of real numbers. The connection between the oscillation constant of this equation and the bottom of the essential spectrum of self-adjoint extensions of the operator generated by the equation is given, as well as various other information about the spectrum of such extensions. In particular, I derive conditions for the spectrum to have only countably many eigenvalues below zero, and a simple criterion for the invariance of the essential spectrum.



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1Agmon, S.. Lectures on exponential decay of solutions of second-order elliptic equations bounds on eigenfunctions of N-body Schrödinger operators, Mathematical Notes 29 (Princeton, NJ: Princeton University Press, 1982).
2Atkinson, F. V.. Discrete and continuous boundary value problems, Mathematics in Science and Engineering, vol. 8 (New York: Academic Press, 1964).
3Blumenthal, O.. Über die entwicklung einer willkürlichen Funktion nach den Nennern des Kettenbruches für−∞ ф(x)/(z – x) dx (Dissertation, Gottingen, 1989).
4Case, K. M. and Geronimo, J. S.. Scattering theory and polynomials orthogonal on the real line. Trans. Amer. Math. Soc. 258 (1980), 467493.
5Chen, H.-Y.. Comparison and nonoscillation results for Volterra-Stieltjes integral equations. J. Math. Anal. Appl. 130 (1988), 257270.
6Chen, H.-Y.. A Levin type comparison and oscillation results for Volterra-Stieltjes integral equations. J. Math. Anal. Appl. 141 (1989), 451462.
7Cheng, S. S., Li, H. J. and Patula, W. T.. Bounded and zero convergent solutions of second-order difference equations. J. Math. Anal. Appl. 141 (1989), 463483.
8Chihara, T. S.. An introduction to orthogonal polynomials (New York: Gordon and Breach, 1978).
9Chihara, T. S.. Spectral properties of orthogonal polynomials on unbounded sets. Trans. Amer. Math. Soc. 270 (1982), 623639.
10Chihara, T. S.. Orthogonal polynomials with discrete spectra on the real line. J. Approx. Theory 42 (1984), 97105.
11Chihara, T. S.. On the spectra of certain birth and death processes. SIAM J. Appl. Math. 47 (1987), 662669.
12Chihara, T. S. and Nevai, P.. Orthogonal polynomials and measures with finitely many point masses. J. Approx. Theory 35 (1982), 370380.
13Dombrowski, J.. Orthogonal polynomials and functional analysis. In Orthogonal Polynomials: Theory and practice, ed. Nevai, P., p. 147161 (Norwell, MA: Kluwer Academic Publishers, 1990).
14Dombrowski, J. and Nevai, P.. Orthogonal polynomials, measures, and recurrence relations. SIAM J. Math. Anal. 17 (1986), 752759.
15Edmunds, E. and Evans, W. D.. Spectral Theory and Differential Operators (Oxford: Clarendon Press, 1987).
16Geronimo, J. S. and Nevai, P.. Necessary and sufficient conditions relating the coefficients in the recurrence formula to the spectral function for orthogonal polynomials. SIAM I. Math. Anal. 14 (1983), 622637.
17Glazman, I. M.. Direct Methods of Qualitative Spectral Analysis (Jerusalem: Israel Program for Scientific Translations, 1965).
18Hartman, P. and Putnam, C. R.. The least cluster point of the spectrum of boundary value problems. Amer. J. Math. 70 (1948), 849855.
19Hinton, D. B. and Lewis, R. T.. Spectral analysis of second order linear difference equations. J. Math. Anal. Appl. 63 (1978), 421438.
20Hooker, J. W., Kwong, M. K. and Patula, W. T.. Oscillatory second order linear difference equations and Ricatti equations. SIAM J. Math. Anal. 18 (1987), 5463.
21Mingarelli, A. B.. Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions, Lecture Notes in Mathematics 989 (Berlin: Springer, 1983).
22Patula, W. T.. Growth and oscillation properties of second order linear difference equations. SIAMJ. Math. Anal. 10 (1979), 5561.
23Patula, W. T.. Growth, oscillation, and comparison theorems for second order linear difference equations. SIAM J. Math. Anal. 10 (1979), 12721279.
24Piepenbrink, J.. Nonoscillatory elliptic equations. J. Differential Equations 15 (1974), 541550.
25Reed, M. and Simon, B.. Methods of Modem Mathematical Physics IV: Analysis of Operators (New York: Academic Press, 1978).
26Reid, W. T.. Sturmian Theory for Ordinary Differential Equations, Applied Mathematical Sciences 31 (New York: Springer, 1980).
27Simon, B.. Schrödinger Semigroups. Bull. Amer. Math. Sac. 7 (1982), 447526.
28Smith, D. T.. Exponential decay of resolvents of banded infinite matrices and asymptotics of linear difference equations (Dissertation, Georgia Institute of Technology, March, 1990).
29Stone, M. H.. Linear Transformations in Hilbert Space, American Mathematical Society Colloquium Publications 15 (New York: American Mathematical Society, 1932).
30van Doorn, E. A.. On oscillation properties and the interval of orthogonality of orthogonal polynomials. SIAM J. Math. Anal. 15 (1984), 10311042.
31Weidmann, J.. Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics 68 (New York: Springer, 1980).


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