On the Dirac equation with PT-symmetric potentials in the presence of position-dependent mass

The relativistic problem of fermions subject to a PT-symmetric potential in the presence of position-dependent mass is reinvestigated. The influence of the PT-symmetric potential in the continuity equation and in the orthonormalization condition are analyzed. In addition, a misconception diffused in the literature on the interaction of neutral fermions is clarified.     

On some meaningful inner product for real Klein—Gordon fields with positive semi-definite norm

A simple derivation of a meaningful, manifestly covariant inner product for real Klein—Gordon (KG) fields with positive semi-definite norm is provided, which turns out — assuming a symmetric bilinear form — to be the real-KG-field limit of the inner product for complex KG fields reviewed by A. Mostafazadeh and F. Zamani in December 2003, and February 2006 (quant-ph/0312078, quant-ph/0602151, quant-ph/0602161). It is explicitly shown that the positive semi-definite norm associated with the derived inner product for real KG fields measures the number of active positive and negative energy Fourier-modes of the real KG field on the relativistic mass shell. The very existence of an inner product with positive semi-definite norm for the considered real, i.e. neutral, KG fields shows that the metric operator entering the inner product does not contain the charge-conjugation operator. This observation sheds some additional light on the meaning of the C operator in the CPT inner product of PT-symmetric quantum mechanics defined by C.M. Bender, D.C. Brody and H.F. Jones.     

Exact Solutions of Non-Autonomous Quantum Systems with Quadratic Hamiltonian

Based on algebraic dynamics, we present an algorithm to obtain exact solutions of the Schrodinger equation of non-autonomous quantum systems with Hamiltonian expressed in quadratic function of creation and annihilation operators of bosons. The Hamiltonian is treated as a linear function of generators of a symplectic group. Similar to the canonical transformation of classical dynamics, we employ a set of gauge transformations to gradually transform the Hamiltonian to a linear function of Cartan operators. The exact solutions are obtained by inverse gauge transformations. When the system is autonomous, this algorithm can obtain the normal mode of the Hamiltonian, as well as the eigenstates and eigenvalues.     

Solvability of a class of PT-symmetric non-Hermitian Hamiltonians:Bethe ansatz method

We use the Bethe ansatz method to investigate the Schrdinger equation for a class of PT-symmetric non-Hermitian Hamiltonians. Elementary exact solutions for the eigenvalues and the corresponding wave functions are obtained in terms of the roots of a set of algebraic equations. Also, it is shown that the problems possess sl(2) hidden symmetry and then the exact solutions of the problems are obtained by employing the representation theory of sl(2) Lie algebra. It is found that the results of the two methods are the same.     

Classical Motion of Complex 2-D Non-Hermitian Hamiltonian Systems

Classical motion of complex 2-D non-Hermitian Hamiltonian systems is investigated to identify periodic, unbounded and chaotic trajectories. The caustic curves, the Lyapunov exponents, and spectral analysis have been used to identify periodic and chaotic trajectories. Using classical trajectories, we were able to predict quantum transition frequaencies of pseudo-Hermitian non-PT symmetric systems accurately. This indicates that there exists a correspondence between classical mechanics and quantum mechanics for certain non-Hermitian Hamiltonians as in the case of real Hermitians.     

A Set of Lie Symmetrical Conservation Law for Rotational Relativistic Hamiltonian Systems

For the rotational relativistic Hamiltonian system, a new type of the Lie symmetries and conserved quantitiesare given. On the basis of the theory of invariance of differential equations under infinitesimal transformations, andintroducing infinitesimal transformations for generalized coordinates qs and generalized momentums ps, the determiningequations of Lie symmetrical transformations of the system are constructed, which only depend on the canonical variables.A set of non-Noether conserved quantities are directly obtained from the Lie symmetries of the system. An example isgiven to illustrate the application of the results.     

A Set of Lie Symmetrical Conservation Law for Rotational Relativistic Hamiltonian Systems

For the rotational relativistic Hamiltonian system, a new type of the Lie symmetries and conserved quantities are given. On the basis of the theory of invariance of differential equations under infinitesimal transformations, and introducing infinitesimal transformations for generalized coordinates qs and generalized momentums ps, the determining equations of Lie symmetrical transformations of the system are constructed, which only depend on the canonical variables.A set of non-Noether conserved quantities are directly obtained from the Lie symmetries of the system. An example is given to illustrate the application of the results.     

General Schlesinger Systems and Their Symmetry from the View Point of Twistor theory

Isomonodromic deformation of linear differential equations on ?¹ with regular and irregular singular points is considered from the view point of twistor theory. We give explicit form of isomonodromic deformation using the maximal abelian subgroup H of G = GL_N+1(?) which appeared in the theory of general hypergeometric functions on a Grassmannian manifold. This formulation enables us to obtain a group of symmetry for the nonlinear system which is an Weyl group analogue N_G (H)/H.     

Broer-Kaup Systems from Darboux Transformation Related Symmetry Constraints of Kadomtsev-Petviashvili Equation

Using the Darboux transformation related symmetry constraints of the Kadomtsev-Petviashvili (KP) equation, we get some integrable (1+1)-dimensional and (2+1)-dimensional multi-component Broer-Kaup (BK) systems.     

A New Type of Mei Adiabatic Invariant Induced by Perturbation to Mei Symmetry of Hamiltonian Systems

Considering full perturbation to infinitesimal generators in the Mei structure equation, a new type of Mei adiabatic invariant induced by perturbation to Mei symmetry for Hamiltonian system was reported.     

A New Type of Mei Adiabatic Invariant Induced by Perturbation to Mei Symmetry of Hamiltonian Systems

Considering full perturbation to infinitesimal generators in the Mei structure equation, a new type of Mei adiabatic invariant induced by perturbation to Mei symmetry for Hamiltonian system was reported.     

On the Positivity of the Coefficients of a Certain Polynomial Defined by Two Positive Definite Matrices

It is shown that the polynomialp(t) = Tr[(A+tB)^m]has positive coefficients when m = 6 and A and B are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior, general results. This problem arises from a conjecture raised by Bessis, Moussa, and Villani in connection with a long-standing problem in theoretical physics. The full conjecture, as shown recently by Lieb and Seiringer, is equivalent to p(t) having positive coefficients for any m and any two n-by-n positive definite matrices. We show that, generally, the question in the real case reduces to that of singular A and B, and this is a key part of our proof.     

Fermi Resonances and Vibrational Spectrum of a Polyatomic Molecule XY4: an Algebraic Approach

A U(2) algebraic model is introduced for the spectrum of a molecule XY₄, where the interactions between the stretch and bend modes are described by T_d symmetric Fermi resonance terms. The presented algebraic model in a limit corresponds to another model in recent literature. The vibrational spectrum of methane (CH₄) measured recently with modern spectroscopy techniques is employed to test those models. The obtained standard deviation between the observed and the calculated vibrational energy levels in the algebraic model is smaller than that in the corresponding model.     

Geometry of time-dependent $\mathcal{PT}$-symmetric quantum mechanics

A new type of quantum theory known as time-dependent $\mathcal{PT}$-symmetric quantum mechanics has received much attention recently. It has a conceptually intriguing feature of equipping the Hilbert space of a $\mathcal{PT}$-symmetric system with a time-varying inner product. In this work, we explore the geometry of time-dependent $\mathcal{PT}$-symmetric quantum mechanics. We find that a geometric phase can emerge naturally from the cyclic evolution of a $\mathcal{PT}$-symmetric system, and further formulate a series of related differential-geometry concepts, including connection, curvature, parallel transport, metric tensor, and quantum geometric tensor. These findings constitute a useful, perhaps indispensible, tool to investigate geometric properties of $\mathcal{PT}$-symmetric systems with time-varying system's parameters. To exemplify the application of our findings, we show that the unconventional geometric phase [Phys. Rev. Lett. 91 187902 (2003)], which is the sum of a geometric phase and a dynamical phase proportional to the geometric phase, can be expressed as a single geometric phase unveiled in this work.     

Weak Measurements from the Point of View of Bohmian Mechanics

The theory of weak measurements developed by Aharonov and coworkers has been applied by them and others to several interesting problems in which the system of interest is both pre- and post-selected. When the probability of successful post-selection is very small the prediction for the weak value of the measured quantity is often bizarre and sometimes controversial, lying outside the range of possibility for a classical system or for a quantum system in the absence of post-selection (e.g. negative kinetic energies associated with particles found immediately after the weak measurement deep inside a classically forbidden region). In Bohmian mechanics a quantum particle is postulated to be a point-like particle which is always accompanied by a wave which probes its environment and guides its motion accordingly. Hence, from the point of view of this theory, it is natural to ask whether the measured weak value under consideration is a property of the point-like particle or of the wave (or of both) and what, if anything, it is that is actually being measured. In this paper, weak measurements of position, momentum and kinetic energy are considered for very simple case studies with these questions in mind.     

构造哈密尔顿系统jet辛差分格式的生成函数法

考虑哈密尔顿系统的保结构算法,在经典哈密尔顿系统的jet辛算法的基础上,给出了一般哈密尔顿系统的jet辛差分格式的定义.并利用带有变系数辛矩阵的一般哈密尔顿系统中的构造辛差分格式的生成函数法的思想,来建立由一般的反对称矩阵所确定的微分二形式与生成函数的关系,再利用哈密尔顿-雅可比方程来构造jet辛的差分格式.     

Non-Lie Symmetry Groups of (2+1)-Dimensional Nonlinear Systems

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known (2 1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equation and Nizhnik-Novikov-Vesselov equation, both the Lie point symmetry groups and the non-Lie symmetry groups are obtained. The Lie symmetry groups obtained via traditional Lie approaches are only special cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches.     

Non-Lie Symmetry Groups of （2＋1）-Dimensional Nonlinear Systems

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known （2＋1）-dimensional asymmetric Nizhnik-Novikov-Vesselov equation and Nizhnik Novikov-Vesselov equation, both the Lie point symmetry groups and the non-Lie symmetry groups are obtained. The Lie symmetry groups obtained via traditional Lie approaches are only speciai cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches.     

On the Spectrum of an Hamiltonian in Fock Space. Discrete Spectrum Asymptotics

A model operator H associated with the energy operator of a system describing three particles in interaction, without conservation of the number of particles, is considered. The location of the essential spectrum of H is described. The existence of infinitely many eigenvalues (resp. the finiteness of eigenvalues) below the bottom τ_ess(H) of the essential spectrum of H is proved for the case where the associated Friedrichs model has a threshold energy resonance (resp. a threshold eigenvalue). For the number N(z) of eigenvalues of H lying below z < τ_ess(H) the following asymptotics is found