Geodesic
Expansions of eigenvalues of a discrete bilaplacian with two-dimensional perturbation
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2024), pp. 77-89 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper we consider the family of operators $$ \widehat{\mathbf H}_\mu:=\widehat\varDelta\widehat\varDelta-\mu\widehat {\mathbf V}, \mu>0, $$ that is, a bilaplacian with a finite-dimensional perturbation on a one-dimensional lattice $ \mathbb{Z} $, where $ \widehat\varDelta $ is a discrete Laplacian, and $ \widehat {\mathbf V} $ is an operator of rank two. It is proved that for any $ \mu> 0 $ the discrete spectrum $ \widehat {\mathbf H}_\mu$ is two-element $ {e_{1}(\mu)}<0$ and ${e_{2}(\mu)}<0 $. We find convergent expansions of the eigenvalues ${e_{i}(\mu)}$, $i=1,2$ in a small neighborhood of zero for small $ \mu>0$.
Keywords: discrete bilaplacian, discrete Schrödinger operator, essential spectrum, eigenvalue, expansion, asymptotics. 
@article{IVM_2024_10_a7,
     author = {T. H. Rasulov and A. M. Khalkhuzhaev and M. A. Pardabaev and Kh. G. Khayitova},
     title = {Expansions of eigenvalues of a discrete bilaplacian with two-dimensional perturbation},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {77--89},
     year = {2024},
     number = {10},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2024_10_a7/}
}
TY  - JOUR
AU  - T. H. Rasulov
AU  - A. M. Khalkhuzhaev
AU  - M. A. Pardabaev
AU  - Kh. G. Khayitova
TI  - Expansions of eigenvalues of a discrete bilaplacian with two-dimensional perturbation
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2024
SP  - 77
EP  - 89
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/IVM_2024_10_a7/
LA  - ru
ID  - IVM_2024_10_a7
ER  - 
%0 Journal Article
%A T. H. Rasulov
%A A. M. Khalkhuzhaev
%A M. A. Pardabaev
%A Kh. G. Khayitova
%T Expansions of eigenvalues of a discrete bilaplacian with two-dimensional perturbation
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2024
%P 77-89
%N 10
%U http://geodesic.mathdoc.fr/item/IVM_2024_10_a7/
%G ru
%F IVM_2024_10_a7
T. H. Rasulov; A. M. Khalkhuzhaev; M. A. Pardabaev; Kh. G. Khayitova. Expansions of eigenvalues of a discrete bilaplacian with two-dimensional perturbation. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2024), pp. 77-89. http://geodesic.mathdoc.fr/item/IVM_2024_10_a7/

[1] McKenna P.J., Walter W., “Nonlinear oscillations in a suspension bridge”, Arch. Rational Mech. Anal., 98 (1987), 167–177 | DOI | MR

[2] Hoffmann S., Plonka G., Weickert J., “Discrete Green's Functions for Harmonic and Biharmonic Inpainting with Sparse Atoms. In: X. Tai et al (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition”, EMMCVPR, Lect. Notes Computer Sci., 8932, 2015, 169–182 | DOI

[3] Ben-Artzi M., Katriel G., “Spline functions, the biharmonic operator and approximate eigenvalues”, Numer. Math., 141 (2019), 839–879 | DOI | MR | Zbl

[4] Graef J., Heidarkhani Sh., Kong L., Wang M., “Existence of solutions to a discrete fourth order boundary value problem”, J. Diff. Equ. Appl., 24:6 (2018), 849–858 | DOI | MR | Zbl

[5] Tee G.J., “A Novel Finite-Difference Approximation to the Biharmonic Operator”, Computer J., 6 (1963), 177–192 | DOI | MR | Zbl

[6] Andrew A., Paine J., “Correction of finite element estimates for Sturm–Liouville eigenvalues”, Numer. Math., 50 (1986), 205–215 | DOI | MR | Zbl

[7] Boumenir A., “Sampling for the fourth-order Sturm–Liouville differential operator”, J. Math. Anal. Appl., 278:2 (2003), 542–550 | DOI | MR | Zbl

[8] Rattana A., Böckmann C., “Matrix methods for computing eigenvalues of Sturm–Liouville problems of order four”, J. Comput. Appl. Math., 249 (2013), 144–156 | DOI | MR | Zbl

[9] Albeverio S., Lakaev S., Makarov K., Muminov Z., “The Threshold Effects for the Two-Particle Hamiltonians on Lattices”, Commun. Math. Phys., 262 (2006), 91–115 | DOI | MR | Zbl

[10] Graf G., Schenker D., “$2$-magnon scattering in the Heisenberg model”, Ann. Inst. Henri Poincaré, Phys. Théor., 67:1 (1997), 91–107 | MR | Zbl

[11] Lakaev S.N., Khalkhuzhaev A.M., Lakaev Sh.S., “Asymptotic behavior of an eigenvalue of the two-particle discrete Schrödinger operator”, Theoret. and Math. Phys., 171:3 (2012), 800–811 | DOI | MR | Zbl

[12] Lakaev S.N., Kholmatov Sh.Yu., “Asymptotics of eigenvalues of two-particle Schrödinger operators on lattices with zero-range interaction”, J. Phys. A: Math. Theor., 44:13 (2011) | DOI | MR | Zbl

[13] Kholmatov Sh., Khalkhuzhaev A., Pardabaev M., “Expansion of eigenvalues of the perturbed discrete bilaplacian”, Monatshefte fur Math., 197 (2022), 607–633 | DOI | MR

[14] Kholmatov Sh., Pardabaev M., “On Spectrum of the Discrete Bilaplacian with Zero-Range Perturbation”, Lobachevskii J. Math., 42:6 (2021), 1286–1293 | DOI | MR | Zbl

[15] Klaus M., “On the bound states of Schrödinger operators in one dimension”, Ann. Phys., 108:2 (1977), 288–300 | DOI | MR | Zbl

[16] Simon B., “The bound state of weakly coupled Schrödinger operators in one and two dimensions”, Ann. Phys., 97:2 (1976), 279–288 | DOI | MR | Zbl

[17] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki, v. 4, Analiz operatorov, Mir, M., 1982 | MR

[18] Bakhronov B.I., Rasulov T.Kh., Rekhman M., “Usloviya suschestvovaniya sobstvennykh znachenii trekhchastichnogo reshetchatogo modelnogo gamiltoniana”, Izv. vuzov. Matem., 2023, no. 7, 3–12

[19] Abdullaev Zh.I., Khalkhuzhaev A.M., Rasulov T.Kh., “Invariantnye podprostranstva i sobstvennye znacheniya trekhchastichnogo operatora Shredingera”, Izv. vuzov. Matem., 2023, no. 9, 3–19 | Zbl

[20] Rasulov T.Kh., Mukhitdinov R.T., “Konechnost diskretnogo spektra modelnogo operatora, assotsiirovannogo s sistemoi trekh chastits na reshetke”, Izv. vuzov. Matem., 2014, no. 1, 61–70 | Zbl