scholarly journals The existence of eigenvalues of Schrödinger operator on a lattice in the gap of the essential spectrum

2021 ◽  
Vol 2070 (1) ◽  
pp. 012017
Author(s):  
J.I. Abdullaev ◽  
A.M. Khalkhuzhaev
Keyword(s):  
Essential Spectrum ◽  
Schrödinger Operator ◽  
Three Dimensional ◽  
Schrodinger Operator ◽  
Discrete Schrödinger Operator ◽  
Dimensional Lattice ◽  
Pairwise Potential ◽  
Zero Range

Abstract We consider a three-particle discrete Schrödinger operator Hμγ (K), K 2 T3 associated to a system of three particles (two fermions and one different particle) interacting through zero range pairwise potential μ > 0 on the three-dimensional lattice Z 3. It is proved that the operator Hμγ (K), ||K|| < δ, for γ > γ0 has at least two eigenvalues in the gap of the essential spectrum for sufficiently large μ > 0.

scholarly journals Bound states of a system of two bosons with a spherically potential on a lattice

2021 ◽  
Vol 2070 (1) ◽  
pp. 012023
Author(s):  
J.I. Abdullaev ◽  
Sh.H. Ergashova ◽  
Y.S. Shotemirov
Keyword(s):  
Invariant Subspace ◽  
Schrödinger Operator ◽  
Bound States ◽  
Three Dimensional ◽  
Invariant Subspaces ◽  
Bound State ◽  
Schrodinger Operator ◽  
Dimensional Lattice

Abstract We consider a Hamiltonian of a system of two bosons on a three-dimensional lattice Z 3 with a spherically simmetric potential. The corresponding Schrödinger operator H(k) this system has four invariant subspaces L(123), L(1), L(2) and L(3). The Hamiltonian of this system has a unique bound state over each invariant subspace L(1), L(2) and L(3). The corresponding energy values of these bound states are calculated exactly.

THE FINITENESS OF THE NUMBER OF EIGENVALUES OF THE FOUR-PARTICLE SCHRÖDINGER OPERATOR WITH THREE-PARTICLE CONTACT INTERACTION

2021 ◽  
Vol 10 (7) ◽  
pp. 2933-2946
Author(s):  
I. Bozorov ◽  
U. Shadiev ◽  
G. Yodgorov
Keyword(s):  
Essential Spectrum ◽  
Contact Interaction ◽  
Interaction Potential ◽  
Three Dimensional ◽  
Schrodinger Operator ◽  
Dimensional Lattice ◽  
Quasi Momentum ◽  
Particle Contact ◽  
Quantum Particles ◽  
Arbitrary Quantum

In this paper, we consider the four-particle Schr\"{o}dinger operator corresponding to the Hamiltonian of a system of four arbitrary quantum particles via a three-particle contact interaction potential on a three-dimensional lattice. The finiteness of the number of eigenvalues of the Schr\"{o}dinger operator lying to the left of the essential spectrum for zero value of the total quasi-momentum is proved.

ON THE SPECTRAL AND SCATTERING THEORY OF THE SCHRÖDINGER OPERATOR WITH SURFACE POTENTIAL

2000 ◽  
Vol 12 (04) ◽  
pp. 561-573 ◽  
Author(s):  
AYHAM CHAHROUR ◽  
JAOUAD SAHBANI
Keyword(s):  
Continuous Spectrum ◽  
Surface Potential ◽  
Schrödinger Operator ◽  
Scattering Theory ◽  
Schrodinger Operator ◽  
Absolutely Continuous Spectrum ◽  
Discrete Schrödinger Operator ◽  
Absolutely Continuous ◽  
Spectral And Scattering Theory

We consider a discrete Schrödinger operator H=-Δ+V acting in ℓ2 (ℤd+1), with potential V supported by the subspace ℤd×{0}. We prove that σ (-Δ)=[-2 (d+1), 2(d+1)] is contained in the absolutely continuous spectrum of H. For this we develop a scattering theory for H. We emphasize the fact that this result applies to arbitrary potentials, so it depends on the structure of the problem rather than on a particular choice of the potential.

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