Essential and discrete spectrum of the Schrödinger operator of a system of two particles on a lattice

Khalkhuzhaev, A., Makhmudov, Kh., Miyassarov, A.

O‘zbekiston matematika jurnali · 2026-yil

Annotatsiya

We consider the Hamiltonian of a system of two fermions on a two-dimensional lattice ${\mathbb{Z}}^2$ with a certain type potential. It is proved that the subspace of odd functions $L^o_{ {2}} {(}{\mathbb{T}}^{ {2}} {)}$ is represented as a direct sum of the subspaces $L^{eo}_{ {2}} {(}{\mathbb{T}}^{ {2}} {)}$ and $L^{oe}_{ {2}} {(}{\mathbb{T}}^{ {2}} {)}$, which are invariant under the operator $H(\mathbf {k}) {,\ }\mathbf {k}=(k_1, k_2) \in {\mathbb{T}}^2,$ associated with this Hamiltonian. For any $k_1 \in {({-\pi,}\pi ]}$, it is proved that the operator $H^{eo}(k_1,\pi)=H(k_1,\pi)\big|_{L^{eo}_{ {2}} {(}{\mathbb{T}}^{ {2}} {)}}$ has an infinite number of eigenvalues and for any $k_1 \in ({-\pi,}\pi {)}$, the operator $H^{oe}(k_1,\pi)=H(k_1,\pi)\big|_{L^{oe}_{ {2}} {(}{\mathbb{T}}^{ {2}} {)}}$ has a finite number eigenvalues lying to the left of the essential spectrum. An asymptotic formula is obtained for the number of eigenvalues of the operator $H^{oe}(k_1,\pi)$ as $k_1\rightarrow \pi.$

Maqola ma’lumotlari
MualliflarKhalkhuzhaev, A., Makhmudov, Kh., Miyassarov, A.
JurnalO‘zbekiston matematika jurnali
NashriyotЎзбекистон Республикаси ФА Математика институти
ISSN2010-7269
Nashr sanasi2026-03-25
Jild70
Son1
TilIngliz
Ilmiy sohaFizika-matematika
DOI10.29229/uzmj.2026-1-11
Indeksga qo‘shilgan2026-08-26

DOI: 10.29229/uzmj.2026-1-11 · Maqolaning asl sahifasi

Kalit so‘zlar

Schr\"{o}dinger operator, lattice, fermion, quasi-momentum, invariant subspaces, essential spectrum, eigenvalue

Ilmiy soha

Fizika-matematika

Iqtibos keltirish

APAKhalkhuzhaev, A., Makhmudov, Kh., & Miyassarov, A. (2026). Essential and discrete spectrum of the Schrödinger operator of a system of two particles on a lattice. O‘zbekiston matematika jurnali, 70(1). https://doi.org/10.29229/uzmj.2026-1-11
GOST (OAK)Khalkhuzhaev, A., Makhmudov, Kh., Miyassarov, A. Essential and discrete spectrum of the Schrödinger operator of a system of two particles on a lattice // O‘zbekiston matematika jurnali. – 2026. – T. 70, № 1. – DOI: 10.29229/uzmj.2026-1-11.
MLAKhalkhuzhaev, A., Makhmudov, Kh., and Miyassarov, A. “Essential and discrete spectrum of the Schrödinger operator of a system of two particles on a lattice.” O‘zbekiston matematika jurnali, 70(1), 2026. https://doi.org/10.29229/uzmj.2026-1-11
ChicagoKhalkhuzhaev, A., Makhmudov, Kh., and Miyassarov, A. “Essential and discrete spectrum of the Schrödinger operator of a system of two particles on a lattice.” O‘zbekiston matematika jurnali, 70(1) (2026). https://doi.org/10.29229/uzmj.2026-1-11
HarvardKhalkhuzhaev, A., Makhmudov, Kh., and Miyassarov, A (2026) ‘Essential and discrete spectrum of the Schrödinger operator of a system of two particles on a lattice’, O‘zbekiston matematika jurnali, 70(1). https://doi.org/10.29229/uzmj.2026-1-11
BibTeX
@article{khalkhuzhaeva2026,
  title = {Essential and discrete spectrum of the Schrödinger operator of a system of two particles on a lattice},
  author = {Khalkhuzhaev, A. and Makhmudov, Kh. and Miyassarov, A.},
  journal = {O‘zbekiston matematika jurnali},
  year = {2026},
  volume = {70},
  number = {1},
  doi = {10.29229/uzmj.2026-1-11},
  url = {https://doi.org/10.29229/uzmj.2026-1-11}
}

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