Eigensolutions of cyclopolyacene graphs

A graph of {X, Y}-cyclopolyacene with n of hexagonal rings has been presented that contains four orbits, of which orbits 1 and 4 are occupied by the X-type of vertex and orbits 2 and 4 are occupied by the Y-type, or vice versa. Eigensolutions for such a graph have been derived in analytical form through the use of rotational symmetry followed by a plane of symmetry. Varying X ( = C, N, B, …) and Y ( = C, N, B, …) several types of cyclopolyacene graph may be obtained. Eigenvalue-expressions for such systems containing C, N and B have been shown in analytical form and their total π-electron energies with 2–6 hexagonal rings have been calculated with the help of the expressions developed.     

Formulas for the characteristic polynomial coefficients of the pendant graphs of linear chains,cycles and stars

Formulas for the characteristic polynomial (CP) coefficients of three classes of (n + p)-vertex graphs, i.e. linear chains, cycles and stars where p pendant vertices are attached to n base vertices in one-to-one correspondence (p = 0, 1, 2, …, n), have been developed. Such pendant graphs become reciprocal graphs for linear chains and cycles if p = n. The n-vertex star graphs follow the same rule as paths and cycles, they become reciprocal on adding a pendant vertex to each of n vertices. The formulas so developed have been expressed in matrix product and in analytical forms for the three classes of graphs that require only the values of n and p for calculation of the respective CP coefficients. Such formulas have the general applicability for a large variety of molecular graphs with varying n and p and have been shown to be reduced to the corresponding formulas for reciprocal graphs that are the special cases of the graphs discussed here.     

Recurrent random walks and the absence of continuous symmetry breaking on graphs

We consider geometrically disordered systems with a continuous symmetry groupG, where the internal degrees of freedom are attached to the vertices of a graph. We show that equilibrium states remainG-invariant at any temperatureT>0 if a random walk on the graph is recurrent. This generalizes a previous result obtained by Cassi.     

Limit theorem for the distribution of eigenvalues of the operator of energy

We prove the central limit theorem for the distribution of eigenvalues of the energy operator of a continuous quantum mechanical system. We consider one-dimensional and multidimensional systems.     

Operator method and eigenvalues of the triple-well anharmonic oscillator

Using an algebraic technique based on the Lie algebra of the group SO(2, 1) the eigenvalues of the triple-well anharmonic oscillator are calculated for all ranges of the coupling constant. Approximate analytic formulas for the eigenvalues up to second order are given. The results obtained are compared with those of other authors.     

无限深方势阱本征值和本征态的三种求解方法

一维无限深方势阱模型是量子力学理想模型,经典教材中势阱的边界一般取得比较特殊.或关于坐标原点具有对称性,或势阱左边界位于坐标原点.本文首先展示了如何利用3种方法求解一维任意边界无限深方势阱能量本征值和对应的本征态,不同方法得到的结果彼此之间等价,讨论分析了这3种方法的推导结果,然后得到关于一维任意边界无限深方势阱能量本征值和本征态的通式,从中比较容易看出这两个物理量均与阱宽有关,并且本征波函数与边界值有关,最后将一维结果拓展到二维和三维任意边界无限深方势阱情况.     

An application of hypervirial perturbation theory to calculate energy eigenvalues of the Morse potential for various diatomic molecules

The Schrödinger equation with the potentialV(r)=D[exp(?2β(r?r _e))?2exp(?β(r?r _e))] is treated in the framework of the hypervirial-renormalization parameter scheme. The energy eigenvalues of various eigenstates for different molecules are calculated.     

Angular momenta and eigenvectors of the weakly coupled T ⊗ t Jahn‐Teller system

The eigenvalues of the weakly coupled T ? t Jahn‐Teller problem are known for several decades, and the same holds also true for the eigenstates. These, however, are only given in the traditional position representation, which proves inconvenient if one attempts to extend the weak‐coupling treatment into the region of stronger coupling. Here the solution of the T ? t eigenvalue problem at weak coupling is derived in terms of creation and annihilation operators. This reformulation of the problem is nontrivial, since the algebraic form of the oscillator eigenvectors, being simultaneous angular‐momentum eigenstates, has been worked out only recently and is probably still widely unknown. The electronic and oscillator eigenstates are then coupled to form eigenvectors of the total angular momentum. Finally, in preparation for an extension of the weak‐coupling treatment, the action of the boson creation and annihilation operators on the oscillator eigenvectors is calculated, thus completing the algebraic approach to the weakly coupled T ? t system.     

Analytical eigenspectra of alternant edge-weighted graphs of linear chains and cycles: some applications

Analytical eigenspectra for the graphs of linear chains and cycles with alternant edge weights has been derived with the use of two independent methods, namely, the characteristic polynomial and the graph squaring. In the former method the rotational symmetry and the trigonometric identity have been exploited. These methods along with the expressions of eigenspectra so obtained have been found to be very useful in expressing analytical eigensolutions of some important as well as novel benzenoids, for example, linear p-methylene poly(p-phenylene), cylindrical poly(p-phenylene), zigzag edge graphene, carbon nanotube and carbon nanotori. Some of these eigensolutions have been analysed in exploring some consequences thereof.     

Hamiltonian structures and their reciprocal transformations for the -KdV–CH hierarchy

The r-KdV–CH hierarchy is a generalization of the Korteweg–de Vries and Camassa–Holm hierarchies parameterized by r+1 constants. In this paper we clarify some properties of its multi-Hamiltonian structures including the explicit expressions of the Hamiltonians, the formulae of the central invariants of the associated bihamiltonian structures and the relationship of these bihamiltonian structures with Frobenius manifolds. By introducing a class of generalized Hamiltonian structures, we present in a natural way the transformation formulae of the Hamiltonian structures of the hierarchy under certain reciprocal transformations, and prove the validity of the formulae at the level of dispersionless limit.     

基于VRML及JavaScript的晶体空间旋转对称性演示程序设计

虚拟现实技术是近些年高速发展的一种新技术,它利用计算机软件构造出一个三维空间的虚拟世界。其中VRML(即虚拟现实建模语言)更是发展迅速,VRML是一种用于建立真实世界场景的模型或人们虚拟的三维世界场景的建模语言,利用计算机虚拟现实技术辅助教学日益发展。本文结合VRML及JavaScript技术,研究了晶体空间点阵旋转对称性的虚拟建模方法,给出了便于操作的具有复杂交互功能的程序软件,VRML技术在教育、教学领域具有广阔的应用前景。     

Mei Symmetry and Lie Symmetry of the Rotational Relativistic Variable Mass System

The Mei symmetry and the Lie symmetry of a rotational relativistic variable masssystem are studied. Thedefinitions and criteria of the Mei symmetry and the Lie symmetry of the rotational relativistic variable mass system aregiven. The relation between the Mei symmetry and the Lie symmetry is found. The conserved quantities which the Meisymmetry and the Lie symmetry lead to are obtained. An example is given to illustrate the application of the result.     

A set of the Lie symmetrical conservation laws for the rotational relativistic Birkhoffian system

For a rotational relativistic Birkhoffian system a set of the Lie symmetries and conservation laws is given under infinitesimal transformation. On the basis of the invariance of rotational relativistic Birkhoffian equations under infinitesimal transformations,Lie symmetrical transformations of the system are constructed, which only depend on the Birkhoffian variables. The determining equations of Lie symmetries are given, and a new type of non-noether conserved quantities are directly obtained from Lie symmetries of the system. An example given to illustrate the application of the results.     

Asymptotically degenerate maximum eigenvalues of the eight-vertex model transfer matrix and interfacial tension

We obtain complex nonlinear integral equations for the two asymptotically degenerate maximum eigenvalues of the transfer matrix of the eight-vertex model. These are exact for a lattice of a finite numberN of columns. Solving the equations recursively gives an expansion of the eigenvalues aboutN = . Thus we can obtain the interfacial tension of the model, as well as rederiving our previous result for the free energy.     

Efficiency of star-like graphs and the Atlanta subway network

The distance d(i,j)$d(i,j)$ between any two vertices i$i$ and j$j$ in a graph is the number of edges in a shortest path between i$i$ and j$j$. If there is no path connecting i$i$ and j$j$, then d(i,j)=∞$d(i,j)=\infty$. In 2001, Latora and Marchiori introduced the measure of efficiency between vertices in a graph (Latora and Marchiori, 2001) [1]. The efficiency between two vertices i$i$ and j$j$ is defined to be _∈i,j=j${\in }_{i,j}=j$. In this paper, we investigate the efficiency of star-like networks, and show that networks of this type have a high level of efficiency. We apply these ideas to an analysis of the Metropolitan Atlanta Rapid Transit Authority (MARTA) Subway system, and show this network is 82% as efficient as a network where there is a direct line between every pair of stations.     

Mei Symmetry and Lie Symmetry of the Rotational Relativistic Variable Mass System

The Mei symmetry and the Lie symmetry of a rotational relativistic variable mass system are studied. The definitions and criteria of the Mei symmetry and the Lie symmetry of the rotational relativistic variable mass system are given. The relation between the Mei symmetry and the Lie symmetry is found. The conserved quantities which the Mei symmetry and the Lie symmetry lead to are obtained. An example is given to illustrate the application of the result.     

Use of symmetry plane fragmentation and graph squaring techniques to express the eigenspectra for some vertex-weighted graphs of linear chains and cycles in analytical form

Three types of graphs of linear chains, viz. linear chains with unit increment or decrement in weight on one terminal vertex, linear chains with unit increment or decrement in weight on both the terminal vertices and linear chains with unit increment in weight on one terminal vertex and decrement in that on the other terminal vertex, have been considered. The symmetry plane fragmentation and graph squaring techniques have been exploited to express the eigenspectra of such graphs of linear chains in analytical form, and have subsequently been used to express the eigenspectra of graphs of linear chains and cycles with alternant vertex weights. The derived expressions for the eigenspectra have been used to obtain the eigenspectra of linear polyacenes, methylene-substituted linear polyacenes and cylindrical polyacene strips in analytical form.     

Rotational symmetry of classical orbits, arbitrary quantization of angular momentum and the role of the gauge field in two-dimensional space

We study quantum–classical correspondence in terms of the coherent wave functions of a charged particle in two- dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 2π-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.     

Dirac equation for the harmonic scalar and vector potentials and linear plus coulomb-like tensor potential; the SUSY approach

The problem of analytical solutions of the 3-dimensional Dirac equation is usually studied via techniques such as The Nikiforov-Uvarov (NU) method. Here, we see that one of the most attractive potentials can be brought into a well-known form of Schrödinger-like problem possessing known solutions via the methodology of supersymmetry (SUSY). Next, using the idea of shape invariance, we calculate exact solutions of Dirac equation for quadratic scalar and vector potentials in the presence of a tensor potential that depends on the radial component either linearly or inversely. The tensor potential itself, besides its applications, removes degeneracy, too.