Abstract
References
- Albeverio, S., Gesztesy, F., Hoegh-Kron, R. and Holden, H. Sovable models in quantum mechanics(Springer, New York, Berlin, 1988).
- Coddington, E. A. Extension theory of formally normal and symmetric subspaces, Mem. Amer. Math. Soc. 134, 1–80, 1973.
- Edmunds, D. E. and Evans, W. D. Spectral Theory and Differential Operators (Clarendon Press, Oxford, 1990).
- Giaquinta, M. and Hildebrand, S. Calculus of Variations I (Springer-Verlang, Berlin, Hei- delberg, 2004).
- Gorbachuk, M. L. Self-adjoint boundary value problems for the differential equations for sec- ond order with the unbounded operator coefficient, Functional Analysis and its Applications (Moscow) 5 (1), 10–21, 1971 (in Russian).
- Gorbachuk, V. I. and Gorbachuk, M. L. Boundary Value Problems for Operator Differential Equations(Kluwer Academic Publisher, Dordrecht, 1991).
- Ismailov, Z. I. On the discreteness of the spectrum of normal differential operators for second order, Doklady NAS of Belarus 49 (3), 5–7, 2005.
- Ismailov, Z. I. Compact inverses of first-order normal differential operators, J. Math. Anal. App. USA 320 (1), 266–278, 2006.
- Rofe-Beketov, F. S. and Kholkin, A. M. Spectral theory of differential operators (World Sci- entific Monograph Series in Matmetics 7, New York, 2005).
- Yakubov, S. and Yakubov, Y. Diffrential Operator Equations Ordinary and Partial Differ
- ential Equations(Chapman&Hall/CRC, USA, 1999).
Abstract
In the Hilbert space of vector-functions L
2
(H,(a, b)), where H is any
separable Hilbert space, the general representation in terms of boundary values of all normal extensions of the formally normal minimal
operator, generated by linear differential-operator expressions of third
order in the form
l(u) = u
′′′(t) + A
3
u(t), A : D(A) ⊂ H → H, A = A
∗ ≥ E,
is obtained in the first part of this study. Then, some spectral properties of these normal extensions are investigated. In particular, the
case of A
−1 ∈ S∞(H), asymptotic estimates of normal extensions of
eigenvalues has been established at infinity.
Keywords
Normal extension Compact operator Eigenvalue Asymptotical behavior of eigenvalues 2000 AMS Classification: 47 A 20
References
- Albeverio, S., Gesztesy, F., Hoegh-Kron, R. and Holden, H. Sovable models in quantum mechanics(Springer, New York, Berlin, 1988).
- Coddington, E. A. Extension theory of formally normal and symmetric subspaces, Mem. Amer. Math. Soc. 134, 1–80, 1973.
- Edmunds, D. E. and Evans, W. D. Spectral Theory and Differential Operators (Clarendon Press, Oxford, 1990).
- Giaquinta, M. and Hildebrand, S. Calculus of Variations I (Springer-Verlang, Berlin, Hei- delberg, 2004).
- Gorbachuk, M. L. Self-adjoint boundary value problems for the differential equations for sec- ond order with the unbounded operator coefficient, Functional Analysis and its Applications (Moscow) 5 (1), 10–21, 1971 (in Russian).
- Gorbachuk, V. I. and Gorbachuk, M. L. Boundary Value Problems for Operator Differential Equations(Kluwer Academic Publisher, Dordrecht, 1991).
- Ismailov, Z. I. On the discreteness of the spectrum of normal differential operators for second order, Doklady NAS of Belarus 49 (3), 5–7, 2005.
- Ismailov, Z. I. Compact inverses of first-order normal differential operators, J. Math. Anal. App. USA 320 (1), 266–278, 2006.
- Rofe-Beketov, F. S. and Kholkin, A. M. Spectral theory of differential operators (World Sci- entific Monograph Series in Matmetics 7, New York, 2005).
- Yakubov, S. and Yakubov, Y. Diffrential Operator Equations Ordinary and Partial Differ
- ential Equations(Chapman&Hall/CRC, USA, 1999).
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Article |
| Authors | |
| Publication Date | May 1, 2012 |
| Published in Issue | Year 2012 Volume: 41 Issue: 5 |