| Title:
|
On discreteness of spectrum of a functional differential operator (English) |
| Author:
|
Labovskiy, Sergey |
| Author:
|
Getimane, Mário Frengue |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
139 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
213-229 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study conditions of discreteness of spectrum of the functional-differential operator \[ \mathcal {L} u=-u''+p(x)u(x)+\int _{-\infty }^\infty (u(x)-u(s)) {\rm d}_s r(x,s) \] on $(-\infty ,\infty )$. In the absence of the integral term this operator is a one-dimensional Schrödinger operator. In this paper we consider a symmetric operator with real spectrum. Conditions of discreteness are obtained in terms of the first eigenvalue of a truncated operator. We also obtain one simple condition for discreteness of spectrum. (English) |
| Keyword:
|
spectrum |
| Keyword:
|
functional differential operator |
| MSC:
|
34K06 |
| MSC:
|
34K08 |
| MSC:
|
34L05 |
| idZBL:
|
Zbl 06362254 |
| idMR:
|
MR3238835 |
| DOI:
|
10.21136/MB.2014.143850 |
| . |
| Date available:
|
2014-07-14T08:16:22Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143850 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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[4] Friedrichs, K.: Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren I.Math. Ann. 109 (1934), 465-487. Zbl 0008.39203, MR 1512905, 10.1007/BF01449150 |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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[12] Labovskiy, S.: Small vibrations of mechanical system.Funct. Differ. Equ. 16 (2009), 447-468. MR 2573916 |
| Reference:
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[13] Maz'ya, V., Shubin, M.: Discreteness of spectrum and positivity criteria for Schrödinger operators.Ann. Math. 162 (2005), 919-942. Zbl 1106.35043, MR 2183285, 10.4007/annals.2005.162.919 |
| Reference:
|
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| . |
