Geodesic
On continuity of the spectrum of a singular quasi-differential operator with respect to a parameter
Eurasian mathematical journal, Tome 2 (2011) no. 3, pp. 67-81 
Kh. K. Ishkin. On continuity of the spectrum of a singular quasi-differential operator with respect to a parameter. Eurasian mathematical journal, Tome 2 (2011) no. 3, pp. 67-81. http://geodesic.mathdoc.fr/item/EMJ_2011_2_3_a3/
@article{EMJ_2011_2_3_a3,
     author = {Kh. K. Ishkin},
     title = {On continuity of the spectrum of a~singular quasi-differential operator with respect to a~parameter},
     journal = {Eurasian mathematical journal},
     pages = {67--81},
     year = {2011},
     volume = {2},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2011_2_3_a3/}
}
TY  - JOUR
AU  - Kh. K. Ishkin
TI  - On continuity of the spectrum of a singular quasi-differential operator with respect to a parameter
JO  - Eurasian mathematical journal
PY  - 2011
SP  - 67
EP  - 81
VL  - 2
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/EMJ_2011_2_3_a3/
LA  - en
ID  - EMJ_2011_2_3_a3
ER  - 
%0 Journal Article
%A Kh. K. Ishkin
%T On continuity of the spectrum of a singular quasi-differential operator with respect to a parameter
%J Eurasian mathematical journal
%D 2011
%P 67-81
%V 2
%N 3
%U http://geodesic.mathdoc.fr/item/EMJ_2011_2_3_a3/
%G en
%F EMJ_2011_2_3_a3

Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain sufficient conditions for continuity of the eigenvalues of semibounded quasi-differential operators of order $2n$ on the half-axis with respect to the parameters that appear in the corresponding differential expression. In addition we obtain a generalization of the well-known result of M. G. Krein [9] concerning description of the quadratic form of a regular quasi-differential operator in the singular case, when the deficiency indices of the minimal operator are equal to $(n,n)$.

[1] P. B. Bailey, M. K. Gordon, L. F. Shampine, “Automatic solution of the Sturm–Lioville problem”, ACM Trans. Math. Software, 4 (1978), 193–208 | DOI | MR | Zbl

[2] P. B. Bailey, W. N. Everitt, A. Zettl, “SLEIGN2: An eigenfunction – eigenvalue code for singular Sturm–Lioville problems”, ACM Trans. Math. Software, 21 (2001), 142–196

[3] V. I. Burenkov, Sobolev spaces on domains, Texts in Mathematics, 137, B. G. Teubner, Stuttgart–Leipzig, 1998 | MR | Zbl

[4] M. Dauge, B. Helffer, “Eigenvalues variation. II. Multidimensional problems”, J. Diff. Equations, 104 (1993), 263–297 | DOI | MR | Zbl

[5] C. T. Fulton, S. Pruess, “Mathematical software for Sturm–Lioville problems”, ACM Trans. Math. Software, 19 (1993), 360–376 | DOI | Zbl

[6] T. Kato, Perturbation theory for linear operators, Springer Verlag, Berlin–Heidelberg–New York, 1966 | MR | Zbl

[7] Q. Kong, A. Zettl, “Eigenvalues of regular Sturm–Liouville problems”, J. Diff. Equations, 131 (1996), 1–19 | DOI | MR | Zbl

[8] Q. Kong, H. Wu, A. Zettl, “Dependence of eigenvalues on the problem”, Math. Nachr., 188 (1997), 173–201 | DOI | MR | Zbl

[9] M. G. Krein, “The theory of self-adjoint extensions of semi-bounded Hermitian operators and its applications, II”, Mat. sb., 21(63):3 (1947), 365–404 (in Russian) | MR | Zbl

[10] M. Möller, A. Zettl, “Differentiable dependence of simple eigenvalues of operators in Banach spaces”, J. Operator Theory, 36 (1996), 335–355 | MR | Zbl

[11] M. A. Naimark, Linear differential operators, v. II, Ungar, New York, 1968 | MR | Zbl

[12] J. D. Pryce, “The NAG Sturm–Lioville codes and some applications”, NAG Newsl., 3 (1986), 4–26

[13] M. Reed, B. Simon, Methods of Modern Mathematical Physics, v. I, Functional Analysis, Academic Press, New York–London, 1972 | MR | Zbl

[14] M. Reed, B. Simon, Methods of modern mathematical physics, v. IV, Analysis of operators, Academic Press, New York–London, 1979 | MR | Zbl