Abstract
In Chap. 12, we investigate discrete spectra of self-adjoint operators. The first main result is the Courant–Fischer–Weyl min–max principle. We state two versions of this theorem, one for operator domains and one for form domains. It allows us to compare the eigenvalues and discrete spectra of lower semibounded self-adjoint operators. Some applications of the min–max principle (Rayleigh–Ritz method, Schrödinger operators) and Temple’s inequality are briefly discussed. Another section contains some results concerning the existence of positive or negative eigenvalues of self-adjoint operators and Schrödinger operators. In the final section, Weyl’s classical asymptotic formula for the eigenvalues of the Dirichlet Laplacian on a bounded open Jordan measurable subset of ℝd is proved.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Classical Articles
Courant, R.: Über die Eigenwerte bei den Differentialgleichungen der Mathematischen Physik. Math. Z. 7, 1–57 (1920)
Fischer, E.: Über quadratische Formen mit reellen Koefficienten. Monatshefte Math. Phys. 16, 234–249 (1905)
Friedrichs, K.O.: Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren. Math. Ann. 109, 465–487 (1934)
Stone, M.H.: Linear Transformations in Hilbert Space. Am. Math. Soc., New York (1932)
Von Neumann, J.: Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren. Math. Ann. 102, 49–131 (1929)
Weyl, H.: Das aymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen. Math. Ann. 71, 441–469 (1912)
Books
Agmon, S.: Lectures on Elliptic Boundary Problems. Van Nostrand, Princeton (1965)
Albeverio, S., Gesztesy, S., Hoegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics. Am. Math. Soc., Providence (2005)
Apostol, T.: Mathematical Analysis. Addison-Wesley, Reading (1960)
Berard, P.: Lectures on Spectral Geometry. Lecture Notes Math., vol. 1207. Springer-Verlag, New York (1986)
Birman, M.S., Solomyak, M.Z.: Spectral Theory of Selfadjoint Operators in Hilbert Space. Kluwer, Dordrecht (1987)
Courant, R., Hilbert, D.: Methods of Mathematical Physics. Interscience Publ., New York (1990)
Davies, E.B.: Spectral Theory and Differential Operators. Cambridge University Press, Cambridge (1994)
Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Clarendon Press, Oxford (1987)
Grubb, G.: Distributions and Operators. Springer-Verlag, Berlin (2009)
Kato, T.: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1966)
Ladyzhenskaya, O.A., Uraltseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968)
Lions, J.L.: Equations différentielles opérationnelles et problèmes aux limites. Springer-Verlag, Berlin (1961)
Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. I. Springer-Verlag, Berlin (1972)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics II. Fourier Analysis and Self-Adjointness. Academic Press, New York (1975)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV. Analysis of Operators. Academic Press, New York (1978)
Thirring, W.: A Course in Mathematical Physics, vol. 3. Springer-Verlag, New York (1981)
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Dordrecht (1978)
Articles
Heinz, E.: Beiträge zur Störungstheorie der Spektralzerlegung. Math. Ann. 123, 415–438 (1951)
Kac, M.: Can you hear the shape of a drum? Am. Math. Monthly 73, 1–23 (1966)
Kato, T.: Quadratic forms in Hilbert spaces and asymptotic perturbations series. Techn. Report No. 9, University of California (1955)
Lax, P.D., Milgram, A.N.: Parabolic equations. In: Contributions to the Theory of Partial Differential Equations. Ann. Math. Stud., vol. 33, pp. 167–190. Princeton (1954)
Nelson, E.: Interactions of nonrelativistic particles with a quantized scalar field. J. Math. Phys. 5, 1190–1197 (1964)
Weyl, H.: Ramifications, old and new, of the eigenvalue problem. Bull. Am. Math. Soc. 56, 115–139 (1950)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Schmüdgen, K. (2012). Discrete Spectra of Self-adjoint Operators. In: Unbounded Self-adjoint Operators on Hilbert Space. Graduate Texts in Mathematics, vol 265. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4753-1_12
Download citation
DOI: https://doi.org/10.1007/978-94-007-4753-1_12
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-4752-4
Online ISBN: 978-94-007-4753-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)