Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Parametrizations of Titchmarsh's $ m(\lambda )$-functions in the limit circle case


Author: Charles T. Fulton
Journal: Trans. Amer. Math. Soc. 229 (1977), 51-63
MSC: Primary 34B20
DOI: https://doi.org/10.1090/S0002-9947-1977-0450657-6
MathSciNet review: 0450657
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For limit-circle eigenvalue problems the so-called $ 'm(\lambda )'$-functions of Titchmarsh [15] are introduced in such a fashion that their parametrization is built into the definition.


References [Enhancements On Off] (What's this?)

  • [1] E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955. MR 16, 1022. MR 0069338 (16:1022b)
  • [2] N. Dunford and J. T. Schwartz, Linear operators. Part II, Interscience, New York, 1963. MR 32 #6181. MR 0188745 (32:6181)
  • [3] C. Fulton, Parametrizations of Titchmarsh's $ m(\lambda )$-functions in the limit circle case, Dissertation, Rheinisch-Westfälischen Tech. Hochschule Aachen, 1973.
  • [4] P. Hartman, Ordinary differential equations, Wiley, New York, 1964. MR 30 #1270. MR 0171038 (30:1270)
  • [5] G. Hellwig, Differential operators of mathematical physics, Springer-Verlag, Berlin, 1964; English transl., Addison-Wesley, Reading, Mass., 1967. MR 20 #2682; 35 #2174. MR 0211292 (35:2174)
  • [6] E. Hille, Lectures on ordinary differential equations, Addison-Wesley, Reading, Mass., 1969. MR 40 #2939. MR 0249698 (40:2939)
  • [7] K. Jörgens, Spectral theory of 2nd-order ordinary differential operators, Lecture Notes, Matematisk Institut, Aarhus Universitet, Denmark, 1962-63.
  • [8] K. Kodaira, The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S-matrices, Amer. J. Math. 71 (1949), 921-945. MR 11, 438. MR 0033421 (11:438c)
  • [9] M. A. Naĭmark, Lineare differentialoperatoren, GITTL, Moscow, 1954; German transl., Akademie-Verlag, 1960. MR 16, 702. MR 0216049 (35:6884)
  • [10] F. Rellich, Halbbeschränkte gewöhonliche Differentialoperatoren zweiter Ordnung, Math. Ann. 122 (1951), 343-368. MR 13, 240. MR 0043316 (13:240g)
  • [11] -, Spectral theory of a second-order differential equation, Lecture notes, New York Univ., 1951.
  • [12] D. B. Sears, Integral transforms and eigenfunction theory, Quart. J. Math. Oxford Ser. (2) 5 (1954), 47-58. MR 15, 959. MR 0062310 (15:959c)
  • [13] D. B. Sears and E. C. Titchmarsh, Some eigenfunction formulae, Quart. J. Math. Oxford Ser. (2) 1 (1950), 165-175. MR 12, 261. MR 0037436 (12:261f)
  • [14] M. H. Stone, Linear transformations in Hilbert space and their applications to analysis, Amer. Math. Soc. Colloq. Publ., vol. 15, Amer. Math. Soc., Providence, R.I., 1932. MR 1451877 (99k:47001)
  • [15] E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. I, 2nd ed., Clarenden Press, Oxford, 1962. MR 31 #426. MR 0176151 (31:426)
  • [16] -, Eigenfunction expansions associated with second-order differential equations. I, 1st ed., Clarendon Press, Oxford, 1946. MR 8, 458. MR 0019765 (8:458d)
  • [17] -, On expansions in eigenfunctions. IV, Quart. J. Math. Oxford Ser. 12 (1941), 33-50. MR 3, 121. MR 0005232 (3:121c)
  • [18] H. Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910), 220-269. MR 1511560
  • [19] K. Yosida, Lectures on differential and integral equations, Interscience, New York and London, 1960. MR 22 #9638. MR 0118869 (22:9638)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34B20

Retrieve articles in all journals with MSC: 34B20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0450657-6
Keywords: Eigenfunction expansion, selfadjoint operator, boundary value problem, boundary conditions, end conditions, $ m(\lambda )$-function
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society