Geodesic
On the Discrete Spectrum of a Family of Differential Operators
Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 3, pp. 70-78 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider a family $\mathbf{A}_\alpha$ of differential operators in $L^2(\mathbb{R}^2)$ depending on a parameter $\alpha\ge0$. The operator $\mathbf{A}_\alpha$ formally corresponds to the quadratic form $$ \mathbf{a}_\alpha[U]=\int_{\mathbb{R}^2}\biggl(|U_x|^2+\frac{1}{2}(|U_y|^2 +y^2|U|^2)\biggr)\,dx\,dy +\alpha\int_\mathbb{R}y|U(0,y)|^2\,dy. $$ The perturbation determined by the second term in this sum is only relatively bounded but not relatively compact with respect to the unperturbed quadratic form $\mathbf{a}_0$. The spectral properties of $\mathbf{A}_\alpha$ strongly depend on $\alpha$. In particular, $\sigma(\mathbf{A}_0)=[1/2,\infty)$; for $0\alpha\sqrt 2$, finitely many eigenvalues $l_n1/2$ are added to the spectrum; and for $\alpha>\sqrt2$ (where the quadratic form approach does not apply), the spectrum is purely continuous and coincides with $\mathbb{R}$. We study the asymptotic behavior of the number of eigenvalues as $\alpha\nearrow\sqrt 2$ and reduce this problem to the problem on the spectral asymptotics for a certain Jacobi matrix.
Keywords: discrete spectrum
Mots-clés : perturbation, Jacobi matrix. 
@article{FAA_2004_38_3_a5,
     author = {M. Z. Solomyak},
     title = {On the {Discrete} {Spectrum} of a {Family} of {Differential} {Operators}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {70--78},
     year = {2004},
     volume = {38},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2004_38_3_a5/}
}
TY  - JOUR
AU  - M. Z. Solomyak
TI  - On the Discrete Spectrum of a Family of Differential Operators
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2004
SP  - 70
EP  - 78
VL  - 38
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/FAA_2004_38_3_a5/
LA  - ru
ID  - FAA_2004_38_3_a5
ER  - 
%0 Journal Article
%A M. Z. Solomyak
%T On the Discrete Spectrum of a Family of Differential Operators
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2004
%P 70-78
%V 38
%N 3
%U http://geodesic.mathdoc.fr/item/FAA_2004_38_3_a5/
%G ru
%F FAA_2004_38_3_a5
M. Z. Solomyak. On the Discrete Spectrum of a Family of Differential Operators. Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 3, pp. 70-78. http://geodesic.mathdoc.fr/item/FAA_2004_38_3_a5/

[1] Van Assche W., “Pollaczek polynomials and summability methods”, J. Math. Anal. Appl., 147 (1990), 498–505 | DOI | MR | Zbl

[2] Birman M. Sh., Solomyak M. Z., “Schrödinger operator. Estimates for number of bound states as function-theoretical problem”, Amer. Math. Soc. Transl. Ser. 2, 150, Amer. Math. Soc., Providence, R.I., 1992, 1–54 | MR

[3] Chihara T. S., An introduction to orthogonal polynomials, Gordon and Breach, 1978 | MR | Zbl

[4] Elaydi S. N., An introduction to difference equations, Springer-Verlag, New York, 1999 | MR | Zbl

[5] Fort T., Finite difference equations in the real domain, Oxford University Press, 1948 | MR | Zbl

[6] Geronimo J. S., “On the spectra of infinite-dimensional Jacobi matrices”, J. Approx. Theory, 53 (1988), 251–256 | DOI | MR

[7] Naboko S., Solomyak M., “On the absolutely continuous spectrum in a model of an irreversible quantum graph”, Proc. London Math. Soc. (3), 92:1 (2006), 251–272 | DOI | MR | Zbl

[8] Smilansky U., “Irreversible quantum graphs”, Waves Random Media, 14 (2004), 143–153 | DOI | MR

[9] Solomyak M., “On a differential operator appearing in the theory of irreversible quantum graphs”, Waves Random Media, 14 (2004), 173–185 | DOI | MR