Spectral properties of a two-particle Hamiltonian on a lattice

We consider a system of two arbitrary quantum particles on a three-dimensional lattice with some dispersion functions (describing particle transport from a site to a neighboring site). The particles interact via an attractive potential at only the nearest-neighbor sites. We study how the number of eigenvalues of a family of operators h(k) depends on the particle interaction energy and the total quasimomentum $k \in \mathbb{T}^3$ , where $\mathbb{T}^3$ is a three-dimensional torus. We find the conditions under which the operator h(0) has a double or triple virtual level at zero depending on the particle interaction energy.     

Spectral properties of singular discrete linear Hamiltonian systems

This paper is concerned with spectral properties of singular discrete linear Hamiltonian systems. It is shown that properties of the essential spectrum of each self-adjoint subspace extension (SSE) of the corresponding minimal subspace are independent of the values of the coefficients of the system on any finite subinterval. The analyticity of the Weyl function is studied by employing the Schwarz reflection principle for the system in the limit point case. Based on the above work, several sufficient conditions are obtained for each SSE to have no essential spectrum points in an interval of the real line in the strong limit point case, and then a sufficient condition for the essential spectrum to be bounded from below (above) and a sufficient condition for the pure discrete spectrum are presented, respectively. As a direct consequence, the related spectral properties of singular higher order symmetric vector difference expressions are given.     

Positivity of the two-particle Hamiltonian on a lattice

We consider a two-particle Hamiltonian on the d-dimensional lattice ℤ^d. We find a sufficient condition for the positivity of a family of operators h(k) appearing after the “separation of the center of mass” of a system of two particles depending on the values of the total quasimomentum k ∈ T^d (where T^d is a d-dimensional torus). We use the obtained result to show that the operator h(k) has positive eigenvalues for nonzero k ∈ T^d. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 3, pp. 381–387, December, 2007.     

Spectral properties of a differential system

The spectrum of the differential operators associated with the differential system$\text{u′=}\text{r}\text{D}\text{D}\text{p}\text{u,}\text{D}\text{y=−y″+qy}\text{on}\text{L}{}^2\text{([0,∞))}$is determined. For this the coefficients are assumed to satisfy rather general properties which combine smoothness and decay. With this the asymptotics of the eigenfunctions can be determined. This in turn leads to properties of the spectra with the aid of the M-matrix.     

Spectral results on Hamiltonian problem

Let $\alpha$ be a non-negative real number, and let $\Theta (G,\alpha )$ be the largest eigenvalue of $A\left(G\right)+\alpha D\left(G\right)$. Specially, $\Theta (G,0)$ and $\Theta (G,1)$ are called the spectral radius and signless Laplacian spectral radius of $G$, respectively. A graph $G$ is said to be Hamiltonian (traceable) if it contains a Hamiltonian cycle (path), and a graph $G$ is called Hamilton-connected if any two vertices are connected by a Hamiltonian path in $G$. The number of edges of $G$ is denoted by $e\left(G\right)$. Recently, the (signless Laplacian) spectral property of Hamiltonian (traceable, Hamilton-connected) graphs received much attention. In this paper, we shall give a general result for all these existed results. To do this, we first generalize the concept of Hamiltonian, traceable, and Hamilton-connected to $s$-suitable, and we secondly present a lower bound for $e\left(G\right)$ to confirm the existence of $s$-suitable graphs. Thirdly, when $0\le \alpha \le 1$, we obtain a lower bound for $\Theta (G,\alpha )$ to confirm the existence of $s$-suitable graphs. Consequently, our results generalize and improve all these existed results in this field, including the main results of Chen et al. (2018), Feng et al. (2017), Füredi et al. (2017), Ge et al. (2016), Li et al. (2016), Nikiforov et al. (2016), Wei et al. (2019) and Yu et al. (2013, 2014).     

A Hunziker-van Winter-Zhislin theorem for a four-particle lattice Schrödinger operator

We consider a discrete Schrödinger operator for four arbitrary particles with short-range pair potentials. We describe the position and the structure of its essential spectrum and prove the Hunziker-van Winter-Zhislin theorem.     

Spectral properties and numerical range of off-diagonal infinite-dimensional Hamiltonian operators

The spectral properties of the off-diagonal infinite-dimensional Hamiltonian operator are studied in this article. Characterizations of the spectrum, the point spectrum and the residual spectrum are given. A spectral inclusion property of numerical range is proved.     

The number of bound states of a one-particle Hamiltonian on a three-dimensional lattice

We consider the Hamiltonian , describing the motion of one quantum particle on a three-dimensional lattice in an external field. We investigate the number of eigenvalues and their arrangement depending on the value of the interaction energy for μ ≥ 0 and λ ≥ 0. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 425–443, March, 2009.     

Two-body problem on spaces of constant curvature: II. Spectral properties of the Hamiltonian

We consider the problem of two bodies with a central interaction on simply connected constant-curvature spaces of arbitrary dimension. We construct the self-adjoint extension of the quantum Hamiltonian, which was explicitly expressed through the radial differential operator and the generators of the isometry group of a configuration space in Part I of this paper. Exact spectral series are constructed for several potentials in the space . Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 481–489, September, 2000.     

Spectral radius and Hamiltonian graphs

In the paper, we will present two sufficient conditions for a bipartite graph to be Hamiltonian and a graph to be traceable, respectively.     

Spectral Properties of an Energy-Dependent Hamiltonian

We study spectral properties of a quantum Hamiltonian with a complex-valued energy-dependent potential related to a model introduced in physics of nuclear reactions[30]and we prove that the principle of limiting absorption holds at any point of a large subset of the essential spectrum.When an additional dissipative or smallness hypothesis is assumed on the potential,we show that the principle of limiting absorption holds at any point of the essential spectrum.     

Spectral theory of discrete linear Hamiltonian systems

This paper is concerned with spectral problems for a class of discrete linear Hamiltonian systems with self-adjoint boundary conditions, where the existence and uniqueness of solutions of initial value problems may not hold. A suitable admissible function space and a difference operator are constructed so that the operator is self-adjoint in the space. Then a series of spectral results are obtained: the reality of eigenvalues, the completeness of the orthogonal normalized eigenfunction system, Rayleigh's principle, the minimax theorem and the dual orthogonality. Especially, the number of eigenvalues including multiplicities and the number of linearly independent eigenfunctions are calculated.     

The Iorio-O'Carroll theorem for anN-particle lattice Hamiltonian

The unitary equivalence of total and free Hamiltonians for the N-particle quantum lattice system with a small coupling constant is proved. This result is obtained by means of the mathematical theory of scattering. The existence and asymptotic completeness of the wave operators is proved. A special representation for the exponent of the Hamiltonian is constructed.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 1, pp. 75–85, April, 1996.Translated by V. I. Serdobolskii.     

Spectral Inclusion Properties of Unbounded Hamiltonian Operators

In this paper, the authors investigate the spectral inclusion properties of the quadratic numerical range for unbounded Hamiltonian operators. Moreover, some examples are presented to illustrate the main results.     

一类Hamiltion算子的谱分布及其可逆性

本文从Hamilton算子的特征函数与Hamilton算子的内部斜对角块算子之间的关系出发,证明了一类Hamilton算子的谱分布,进而得到这类Hamilton算子可逆的充要条件.最后应用具体的例子说明了所得结论的合理性.