Geodesic
On discreteness of spectrum of a functional differential operator
Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 213-229

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
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.
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.
DOI : 10.21136/MB.2014.143850
Classification : 34K06, 34K08, 34L05
Keywords: spectrum; functional differential operator 
Labovskiy, Sergey; Getimane, Mário Frengue. On discreteness of spectrum of a functional differential operator. Mathematica Bohemica, Tome 139 (2014) no. 2, pp. 213-229. doi: 10.21136/MB.2014.143850
@article{10_21136_MB_2014_143850,
     author = {Labovskiy, Sergey and Getimane, M\'ario Frengue},
     title = {On discreteness of spectrum of a functional differential operator},
     journal = {Mathematica Bohemica},
     pages = {213--229},
     year = {2014},
     volume = {139},
     number = {2},
     doi = {10.21136/MB.2014.143850},
     mrnumber = {3238835},
     zbl = {06362254},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2014.143850/}
}
TY  - JOUR
AU  - Labovskiy, Sergey
AU  - Getimane, Mário Frengue
TI  - On discreteness of spectrum of a functional differential operator
JO  - Mathematica Bohemica
PY  - 2014
SP  - 213
EP  - 229
VL  - 139
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2014.143850/
DO  - 10.21136/MB.2014.143850
LA  - en
ID  - 10_21136_MB_2014_143850
ER  - 
%0 Journal Article
%A Labovskiy, Sergey
%A Getimane, Mário Frengue
%T On discreteness of spectrum of a functional differential operator
%J Mathematica Bohemica
%D 2014
%P 213-229
%V 139
%N 2
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2014.143850/
%R 10.21136/MB.2014.143850
%G en
%F 10_21136_MB_2014_143850

[1] Akhiezer, N. I., Glazman, I. M.: Theory of Linear Operators in Hilbert Space. Translated from the Russian. Dover Publications, New York (1993). | MR

[2] Birman, M. S., Solomjak, M. Z.: Spectral Theory of Self-Adjoint Operators in Hilbert Space. Translated from the Russian. Mathematics and Its Applications. Soviet Series 5 D. Reidel Publishing Company, Dordrecht (1987). | MR

[3] Cohn, D. L.: Measure Theory. Birkhäuser, Boston (1993). | MR | Zbl

[4] Friedrichs, K.: Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren I. Math. Ann. 109 (1934), 465-487. | DOI | MR | Zbl

[5] Friedrichs, K.: Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren II. Math. Ann. 109 (1934), 685-713. | DOI | MR | Zbl

[6] Gelfand, I. M., Fomin, S. V.: Calculus of Variations. Translated from the Russian. Prentice-Hall, Englewood Cliffs (1963). | MR

[7] Getimane, M., Labovskiy, S.: On discreteness of spectrum of a functional-differential operator. Funct. Differ. Equ. 20 (2013), 109-121. | MR

[8] Ismagilov, R. S.: Conditions for semiboundedness and discreteness of the spectrum for one-dimensional differential equations. Dokl. Akad. Nauk SSSR 140 (1961), 33-36. | MR

[9] Kantorovich, L. V., Akilov, G. P.: Functional Analysis. Nauka, Moskva (1984), Russian. | MR | Zbl

[10] Labovskii, S.: Little vibrations of an abstract mechanical system and corresponding eigenvalue problem. Funct. Differ. Equ. 6 (1999), 155-167. | MR | Zbl

[11] Labovskij, S. M.: On the Sturm-Liouville problem for a linear singular functional-differential equation. Translated from the Russian. Russ. Math. 40 (1996), 50-56. | MR

[12] Labovskiy, S.: Small vibrations of mechanical system. Funct. Differ. Equ. 16 (2009), 447-468. | MR

[13] Maz'ya, V., Shubin, M.: Discreteness of spectrum and positivity criteria for Schrödinger operators. Ann. Math. 162 (2005), 919-942. | DOI | MR | Zbl

[14] Molchanov, A. M.: On conditions for discreteness of the spectrum of self-adjoint differential equations of the second order. Tr. Mosk. Mat. Obshch. 2 (1953), 169-199 Russian. | MR

Cité par Sources :