Eigenvalues below the essential spectra of singular elliptic operators
Authors:
W. D. Evans and Roger T. Lewis
Journal:
Trans. Amer. Math. Soc. 297 (1986), 197-222
MSC:
Primary 35P15; Secondary 35J25, 47F05
DOI:
https://doi.org/10.1090/S0002-9947-1986-0849475-1
MathSciNet review:
849475
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Abstract: A new technique is developed for determining if the number of eigenvalues below the essential spectrum of a singular elliptic differential operator is finite. A method is given for establishing lower bounds for the least point of the essential spectrum in terms of the behavior of the coefficients and weight near the singularities. Higher-order operators are included in these results as well as second-order Schrödinger operators.
- [1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
- [2] Shmuel Agmon, Lectures on elliptic boundary value problems, Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. MR 0178246
- [3] W. Allegretto, On the equivalence of two types of oscillation for elliptic operators, Pacific J. Math. 55 (1974), 319–328. MR 374628
- [4] W. Allegretto, Nonoscillation theory of elliptic equations of order 2𝑛, Pacific J. Math. 64 (1976), no. 1, 1–16. MR 415044
- [5] W. Allegretto, Finiteness of lower spectra of a class of higher order elliptic operators, Pacific J. Math. 83 (1979), no. 2, 303–309. MR 557930
- [6] Walter Allegretto, Spectral estimates and oscillations of singular differential operators, Proc. Amer. Math. Soc. 73 (1979), no. 1, 51–56. MR 512057, https://doi.org/10.1090/S0002-9939-1979-0512057-7
- [7] W. Allegretto, Positive solutions and spectral properties of second order elliptic operators, Pacific J. Math. 92 (1981), no. 1, 15–25. MR 618041
- [8] D. E. Edmunds and W. D. Evans, Spectral theory and embeddings of Sobolev spaces, Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 120, 431–453. MR 559049, https://doi.org/10.1093/qmath/30.4.431
- [9] W. D. Evans, R. T. Lewis, and A. Zettl, Nonselfadjoint operators and their essential spectra, Ordinary differential equations and operators (Dundee, 1982) Lecture Notes in Math., vol. 1032, Springer, Berlin, 1983, pp. 123–160. MR 742638, https://doi.org/10.1007/BFb0076796
- [10] L. E. Fraenkel, On regularity of the boundary in the theory of Sobolev spaces, Proc. London Math. Soc. (3) 39 (1979), no. 3, 385–427. MR 550077, https://doi.org/10.1112/plms/s3-39.3.385
- [11] Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
- [12] I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators, Translated from the Russian by the IPST staff, Israel Program for Scientific Translations, Jerusalem, 1965; Daniel Davey & Co., Inc., New York, 1966. MR 0190800
- [13] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
- [14] Roger T. Lewis, Singular elliptic operators of second order with purely discrete spectra, Trans. Amer. Math. Soc. 271 (1982), no. 2, 653–666. MR 654855, https://doi.org/10.1090/S0002-9947-1982-0654855-X
- [15] Roger T. Lewis, A Friedrichs inequality and an application, Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 185–191. MR 751190, https://doi.org/10.1017/S0308210500031966
- [16] William F. Moss and John Piepenbrink, Positive solutions of elliptic equations, Pacific J. Math. 75 (1978), no. 1, 219–226. MR 500041
- [17] J. Něcas, Les méthodes directes en théorie des équations elliptiques, Masson, Paris, 1967.
- [18] Arne Persson, Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator, Math. Scand. 8 (1960), 143–153. MR 133586, https://doi.org/10.7146/math.scand.a-10602
- [19] John Piepenbrink, Finiteness of the lower spectrum of Schrödinger operators, Math. Z. 140 (1974), 29–40. MR 355368, https://doi.org/10.1007/BF01218644
- [20] John Piepenbrink, Nonoscillatory elliptic equations, J. Differential Equations 15 (1974), 541–550. MR 342829, https://doi.org/10.1016/0022-0396(74)90072-2
- [21] John Piepenbrink, A conjecture of Glazman, J. Differential Equations 24 (1977), no. 2, 173–177. MR 454400, https://doi.org/10.1016/0022-0396(77)90142-5
- [22] Raymond Redheffer, On the inequality Δ𝑢≧𝑓(𝑢,|𝑔𝑟𝑎𝑑𝑢|), J. Math. Anal. Appl. 1 (1960), 277–299. MR 139754, https://doi.org/10.1016/0022-247X(60)90002-0
- [23] Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
- [24] -, Methods of mathematical physics. Vol. IV, Analysis of Operators, Academic Press, New York, 1978.
- [25] Franz Rellich, Perturbation theory of eigenvalue problems, Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz, Gordon and Breach Science Publishers, New York-London-Paris, 1969. MR 0240668
- [26] Martin Schechter, Spectra of partial differential operators, 2nd ed., North-Holland Series in Applied Mathematics and Mechanics, vol. 14, North-Holland Publishing Co., Amsterdam, 1986. MR 869254
- [27] Martin Schechter, On the spectra of singular elliptic operators, Mathematika 23 (1976), no. 1, 107–115. MR 410115, https://doi.org/10.1112/S0025579300006203
- [28] Martin Schechter, Operator methods in quantum mechanics, North-Holland Publishing Co., New York-Amsterdam, 1981. MR 597895
- [29] Upke-Walther Schmincke, Essential selfadjointness of a Schrödinger operator with strongly singular potential, Math. Z. 124 (1972), 47–50. MR 298254, https://doi.org/10.1007/BF01142581
- [30] U.-W. Schmincke, The lower spectrum of Schrödinger operators, Arch. Rational Mech. Anal. 75 (1980/81), no. 2, 147–155. MR 605526, https://doi.org/10.1007/BF00250476
- [31] Barry Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526. MR 670130, https://doi.org/10.1090/S0273-0979-1982-15041-8
- [32] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1986-0849475-1
Article copyright:
© Copyright 1986
American Mathematical Society