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Methods Of Modern Mathematical Physics

A brief introduction to the development of the homotopy perturbation method is given, and the main milestones are elucidated with more than 90 references. This paper further improves the method by ...

https://journals.sagepub.com/doi/pdf/10.1177/1461348418811028

Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators

For a wide class of two-body energy operators h(k) on the d-dimensional lattice ℤ
 d
 , d≥3, k being the two-particle quasi-momentum, we prove that if the following two assumptions (i) and (ii) are satisfied, then for all nontrivial values k, k≠0, the discrete spectrum of h(k) below its threshold is non-empty. The assumptions are: (i) the two-particle Hamiltonian h(0) corresponding to the zero value of the quasi-momentum has either an eigenvalue or a virtual level at the bottom of its essential spectrum and (ii) the one-particle free Hamiltonians in the coordinate representation generate positivity preserving semi-groups.

https://faculty.missouri.edu/~makarovk/publications/CMP.pdf

The threshold effects for the two-particle Hamiltonians on lattices

We study discrete Schrodinger operators with compactly supported potentials on Z
 d
 . Constructing spectral representations and representing S-matrices by the generalized eigenfunctions, we show that the potential is uniquely reconstructed from the S-matrix of all energies. We also study the spectral shift function $${\xi(\lambda)}$$
 for the trace class potentials, and estimate the discrete spectrum in terms of the moments of $${\xi(\lambda)}$$
 and the potential.

https://link.springer.com/content/pdf/10.1007%2Fs00023-011-0141-0.pdf

Inverse Problems, Trace Formulae for Discrete Schrödinger Operators

The paper is devoted to the study of the essential spectrum of discrete Schr?dinger operators on the lattice by means of the limit operators method This method has been applied by one of the authors to describe the essential spectrum of (continuous) electromagnetic Schr?dinger operators, square-root Klein?Gordon operators and Dirac operators under quite weak assumptions on the behaviour of the magnetic and electric potential at infinity The present paper aims at illustrating the applicability and efficiency of the limit operators method to discrete problems as well We consider the following classes of the discrete Schr?dinger operators: (1) operators with slowly oscillating at infinity potentials, (2) operators with periodic and semi-periodic potentials, (3) Schr?dinger operators which are discrete quantum analogues of the acoustic propagators for waveguides, (4) operators with potentials having an infinite set of discontinuities and (5) three-particle Schr?dinger operators which describe the motion of two particles around a heavy nuclei on the lattice 

The essential spectrum of Schrödinger operators on lattices

The Hamiltonian of a system of three quantum mechanical particles moving
on the three-dimensional lattice 
	$$ \mathbb{Z}^3 $$
	
 
and interacting via zero-range attractive
potentials is considered For the two-particle energy operator h(k), with 
	$$ k \in \mathbb{T}^3 = (-\pi, \pi]^3 $$
	
 
the two-particle quasi-momentum, the existence of a unique positive eigenvalue
below the bottom of the continuous spectrum of h(k) for k ≠ 0 is proven,
provided that h(0) has a zero energy resonance The location of the essential and
discrete spectra of the three-particle discrete Schrodinger operator H(K), 
	$$ k \in \mathbb{T}^3 $$
	
 
being the three-particle quasi-momentum, is studied The existence of infinitely
many eigenvalues of H(0) is proven It is found that for the number N(0, z) of
eigenvalues of H(0) lying below 
	$$ z < 0 $$
	
 
the following limit exists 
	$$ \lim_{z \to 0-}\, {N(0, z)\over |\log|z\|} = {\mathcal U}_0 $$
	
 
with 
	$$ {\mathcal U}_0 > 0 $$
	
 
Moreover, for all sufficiently small nonzero values of the three-particle
quasi-momentum K the finiteness of the number 
	$$ N(K, \tau_{ess}(K))$$
	
 
of eigenvalues of H(K) 
below the essential spectrum is established and the asymptotics for the number
N(K, 0) of eigenvalues lying below zero is given

https://link.springer.com/content/pdf/10.1007/s00023-004-0181-9.pdf

Schrodinger Operators on Lattices. The Efimov Effect and Discrete Spectrum Asymptotics

The Hamiltonian of a system of three quantum mechanical particles moving on the three-dimensional lattice $\Z^3$ and interacting via zero-range attractive potentials is considered. For the two-particle energy operator $h(k),$ with $k\in \T^3=(-\pi,\pi]^3$ the two-particle quasi-momentum, the existence of a unique positive eigenvalue below the bottom of the continuous spectrum of $h(k)$ for $k\neq0$ is proven, provided that $h(0)$ has a zero energy resonance. The location of the essential and discrete spectra of the three-particle discrete Schr\"{o}dinger operator $H(K), K\in \T^3$ being the three-particle quasi-momentum, is studied. The existence of infinitely many eigenvalues of H(0) is proven. It is found that for the number $N(0,z)$ of eigenvalues of H(0) lying below $z 0$. Moreover, for all sufficiently small nonzero values of the three-particle quasi-momentum $K$ the finiteness of the number $ N(K,\tau_{ess}(K))$ of eigenvalues of $H(K)$ below the essential spectrum is established and the asymptotics for the number $N(K,0)$ of eigenvalues lying below zero is given.

https://arxiv.org/pdf/math-ph/0312026.pdf

Schr\"{o}dinger operators on lattices. The Efimov effect and discrete spectrum asymptotics

Spectral and threshold analysis of a small rank perturbation of the discrete Laplacian

A model operator H associated to a system of three particles on the threedimensional lattice ℤ3 that interact via nonlocal pair potentials is studied. The following results are established. (i) The operator H has infinitely many eigenvalues lying below the bottom of the essential spectrum and accumulating at this point if both the Friedrichs model operators 
$$h_{\mu _\alpha } $$

 (0), α = 1, 2, have threshold resonances. (ii) The operator H has finitely many eigenvalues lying outside the essential spectrum if at least one of the operators 
$$h_{\mu _\alpha } $$

 (0), α = 1, 2, has a threshold eigenvalue.

Zahriddin Muminov

Papers

Schrodinger Operators on Lattices. The Efimov Effect and Discrete Spectrum Asymptotics

The threshold effects for the two-particle Hamiltonians on lattices

Schr\"{o}dinger operators on lattices. The Efimov effect and discrete spectrum asymptotics

Spectral and threshold analysis of a small rank perturbation of the discrete Laplacian

On the number of eigenvalues of a model operator associated to a system of three-particles on lattices