| Title:
|
Spectral properties of fourth order differential operators (English) |
| Author:
|
Došlý, Ondřej |
| Author:
|
Hilscher, Roman |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
122 |
| Issue:
|
2 |
| Year:
|
1997 |
| Pages:
|
153-168 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Necessary and sufficient conditions for discreteness and boundedness below of the spectrum of the singular differential operator $\ell(y)\equiv{1\over w(t)}\ddif{(r(t)\ddif{y})}$, $t\in[a,\infty)$ are established. These conditions are based on a recently proved relationship between spectral properties of $\ell$ and oscillation of a certain associated second order differential equation. (English) |
| Keyword:
|
singular differential operators |
| Keyword:
|
property BD |
| Keyword:
|
oscillation criteria |
| Keyword:
|
principal solution |
| MSC:
|
34B05 |
| MSC:
|
34C10 |
| MSC:
|
34L05 |
| idZBL:
|
Zbl 0894.34028 |
| idMR:
|
MR1460945 |
| DOI:
|
10.21136/MB.1997.125911 |
| . |
| Date available:
|
2009-09-24T21:24:24Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/125911 |
| . |
| Reference:
|
[1] C. D. Ahlbrandt D. B. Hinton. R. T. Lewis: Necessary and sufficient conditions for the discreteness of the spectrum of certain singular differential operators.Canad. J. Math. 33 (1981), 229-246. MR 0608867, 10.4153/CJM-1981-019-1 |
| Reference:
|
[2] C. D. Ahlbrandt D. B. Hinton. R. T. Lewis: The effect of variable change on oscillation and disconjugacy criteria with applications to spectral theory and asymptotic theory.J. Math. Anal. Appl. 81 (1981), 234-277. MR 0618771, 10.1016/0022-247X(81)90060-3 |
| Reference:
|
[3] W. A. Coppel: Disconjugacy.Lectures Notes in Math., No. 220, Springer-Verlag, Berlin, 1971. Zbl 0224.34003, MR 0460785 |
| Reference:
|
[4] O. Došlý: Spectral properties of a class of singular differential operators.To appear in Math. Nachr. MR 1484668 |
| Reference:
|
[5] O. Došlý J. Komenda: Conjugacy criteria and principal solutions of self-adjoint differential equations.Arch. Math. (Brno) 31 (1995), 217-238. MR 1368260 |
| Reference:
|
[6] F. Fiedler: Oscillation criteria for a special class of 2n-order ordinary differential equations.Math. Nachr. 131 (1987), 205-218. Zbl 0644.34024, MR 0908812, 10.1002/mana.19871310119 |
| Reference:
|
[7] F. Fiedler: About certain fourth-order ordinary differential operators-oscillation and discreteness of their spectrum.Math. Nachr. 142 (1989), 235-250. Zbl 0685.34033, MR 1017382, 10.1002/mana.19891420116 |
| Reference:
|
[8] I. M. Glazman: Direct Methods of Qualitative Analysis of Singular Differential Operators.Jerusalem, 1965. |
| Reference:
|
[9] P. Hartman: Self-adjoint, non-oscillatory systems of ordinary, second order, linear differential equations.Duke J. Math. 24 (1956), 25-35. MR 0082591, 10.1215/S0012-7094-57-02405-5 |
| Reference:
|
[10] D. B. Hinton R. T. Lewis: Discrete spectra criteria for singular differential operators with middle terms.Math. Proc. Cambridge Philos. Soc. 77 (1975), 337-347. MR 0367358 |
| Reference:
|
[11] R. T. Lewis: The discreteness of the spectrum of self-adjoint, even order, differential operators.Proc. Amer. Mat. Soc. 42 (1974), 480-482. MR 0330608, 10.1090/S0002-9939-1974-0330608-8 |
| Reference:
|
[12] M. A. Naimark: Linear Differential Operators.Part II. Ungar, New York, 1968. Zbl 0227.34020 |
| Reference:
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[13] W. T. Reid: Sturmian Theory for Ordinary Differential Equations.Springer-Verlag, New York, 1980. Zbl 0459.34001, MR 0606199 |
| Reference:
|
[14] R. L. Sternberg: Variational methods and nonoscillatory theorems for systems of differential equations.Duke J. Math. 19 (1952), 311-322. MR 0048668 |
| Reference:
|
[15] C. A. Swanson: Comparison and Oscillation Theory of Linear Differential Equations.Acad. Press, New York, 1968. Zbl 0191.09904, MR 0463570 |
| Reference:
|
[16] J. Weidmann: Linear Operators in Hilbert Spaces.Springer-Verlag, New York, 1980. Zbl 0434.47001, MR 0566954 |
| . |
