Orthogonality and Boundary Conditions in Quantum Mechanics

One-dimensional particle states are constructed according to orthogonality conditions, without requiring boundary conditions. Free particle states are constructed using Dirac's delta function orthogonality conditions. The states (doublets) depend on two quantum numbers: energy and parity (+ or —). With the aid of projection operators the particles are confined to a constrained region, in a way similar to the action of an infinite well potential. From the resulting overcomplete basis, only the mutually orthogonal states are selected. Four solutions are found, corresponding to different non-commuting Hamiltonians. Their energy eigenstates are labeled with the main quantum number n and parity + or —. The energy eigenvalues are functions of n only. The four cases correspond to different boundary conditions: (I) The wave function vanishes on the boundary (energy levels: l⁺,2^–,3⁺,4^–,...), (II) the derivative of the wavefunction vanishes on the boundary (energy levels 0⁺,l^–,2⁺,3^–,...), (III) periodic boundary conditions (energy levels: 0⁺,2⁺,2^–,4⁺,4^–6⁺,6^–,...), (IV) periodic boundary conditions (energy levels: l⁺,1^–,3⁺,3^–,5⁺,5^–,...). Among the four cases, only solution (III) forms a complete basis in the sense that any function in the constrained region, can be expanded with it. By extending the boundaries of the constrained region to infinity, only solution (III) converges uniformly to the free particle states. Orthogonality seems to be a more basic requirement than boundary conditions. By using projection operators, confinement of the particle to a definite region can be achieved in a conceptually simple and unambiguous way, and physical operators can be written so that they act only in the confined region.     

The Rigged Hilbert Space of the Free Hamiltonian

We explicitly construct the Rigged Hilbert Space (RHS) of the free Hamiltonian H ₀. The construction of the RHS of H ₀ provides yet another opportunity to see that when continuous spectrum is present, the solutions of the Schrödinger equation lie in a RHS rather than just in a Hilbert space.     

Pseudodifferential Operators of Several Variables and Baker Functions

The KP hierarchy consists of an infinite system of nonlinear partial differential equations and is determined by Lax equations, which can be constructed using pseudodifferential operators. The KP hierarchy and the associated Lax equations can be generalized by using pseudodifferential operators of several variables. We construct Baker functions associated to those generalized Lax equations of several variables and prove some of the properties satisfied by such functions.     

Multi-Fractal Formalism for Quasi-Self-Similar Functions

The study of multi-fractal functions has proved important in several domains of physics. Some physical phenomena such as fully developed turbulence or diffusion limited aggregates seem to exhibit some sort of self-similarity. The validity of the multi-fractal formalism has been proved to be valid for self-similar functions. But, multi-fractals encountered in physics or image processing are not exactly self-similar. For this reason, we extend the validity of the multi-fractal formalism for a class of some non-self-similar functions. Our functions are written as the superposition of similar structures at different scales, reminiscent of some possible modelization of turbulence or cascade models. Their expressions look also like wavelet decompositions. For the computation of their spectrum of singularities, it is unknown how to construct Gibbs measures. However, it suffices to use measures constructed according the Frostman's method. Besides, we compute the box dimension of the graphs.     

ON THE ADM EQUATIONS FOR GENERAL RELATIVITY

The Arnowitt–Deser–Misner (ADM) evolution equations for the induced metric and the extrinsic-curvature tensor of the spacelike surfaces which foliate the space-time manifold in canonical general relativity are a first-order system of quasi-linear partial differential equations, supplemented by the constraint equations. Such equations are here mapped into another first-order system. In particular, an evolution equation for the trace of the extrinsic-curvature tensor K is obtained whose solution is related to a discrete spectral resolution of a three-dimensional elliptic operator of Laplace type. Interestingly, all nonlinearities of the original equations give rise to the potential term in . An example of this construction is given in the case of a closed Friedmann–Lemaitre–Robertson–Walker universe. Eventually, the ADM equations are re-expressed as a coupled first-order system for the induced metric and the trace-free part of K. Such a system is written in a form which clarifies how a set of first-order differential operators and their inverses, jointly with spectral resolutions of operators of Laplace type, contribute to solving, at least in principle, the original ADM system.     

Projection operator approach to the thermodynamic formalism of dynamical systems

An analytical perturbative treatment of characteristic exponents describing the fluctuations of temporal coarse-grained quantities in the context of nonlinear dynamical systems is proposed. It is based on the analysis of the resolvent of the corresponding transfer operator by a projection operator method similar to those used in statistical mechanics. Two different approximation schemes are presented and tested for the case of an exactly solvable but nontrivial model system.     

Magnetic Properties of Weakly Doped Antiferromagnets from the Lagrangian t—J Model

From the Lagrangian formalism for the t-J model previously developed, renormalized magnetic properties in weakly doped antiferromagnets can be evaluated. The renormalization effects essentially appear because of the interaction of particle-hole with the spin wave. For small concentration of holes the self-energies are computed. Taking an approximate form for the particle spectral function, the cuasiparticle peak and the incoherent continuum region are analyzed in order to evaluate the softening and the damping in the spin excitations of antiferromagnets weakly doped. The results can be confronted with previous one obtained by means of the Hamiltonian t-J model in the slave-fermion Schwinger boson representation.     

GG Functions and their Relations to General Hypergeometric Functions

In this Letter, we present a new approach to the notion of hypergeometric functions.     

Convergence of dynamical zeta functions

I study poles and zeros of zeta functions in one-dimensional maps. Numerical and analytical arguments are given to show that the first pole of one such zeta function is given by the first zero ofanother zeta function: this describes convergence of the calculations of the first zero, which is generally the physically interesting quantity. Some remarks on how these results should generalize to zeta functions of dynamical systems with pruned symbolic dynamics and in higher dimensions follow.     

On the Relation Between U q (sl(2)) Vertex Operators and q-Zonal Functions

We show how the states constructed from the action of the modes of bosonized vertex operators that intertwine U modules are related toq -zonal functions.     

One-particle subspaces in the stochasticXY model

We study the spectrum of the generatorH of the Glauber dynamics for a model of planar rotators on a lattice in the case of a high temperature 1/. We construct two so-called one-particle subspacesH _± forH and describe the spectrum of the generator in these subspaces. As a consequence we find time asymptotics of the correlations for the Glauber dynamics.     

Solitons on Noncommutative Orbifold T 2/Z N

Following the construction of the projection operators on T ² presented by Gopakumar, Headrick and Spradlin, we construct a set of projection operators on the integral noncommutative orbifold T ²/G(G=Z _N , N=2, 3, 4, 6) which correspond to a set of solitons on T ²/Z _N in noncommutative field theory. In this way, we derive an explicit form of projector on T ²/Z ₆ as an example. We also construct a complete set of projectors on T ²/Z _N by series expansions for integral case.     

量子论中狄拉克符号积分的意义

量子的引入最先是普朗克在1900年为理论“凑合”黑体辐射实验曲线的无奈之举(曲线拟合),然此举如招幡令旗,呼风唤雨,聚溪成流,乘奔御风,浩浩汤汤,终成今日量子流行的漫山遍野之势,是几个能人的集灵思积广益而相辅相成,还是时势造英雄,还是两者兼而有之！普朗克以能量分离的观点看待微观世界,是他在理论推导拟合实验结果逐渐形成的信仰。物理学家狄拉克指出,伟大的物理学家如牛顿和爱因斯坦是靠基本信仰“从上到下”推导出一些大自然的定律的。狄拉克自己的信仰是相信方程的美有时比实验结果更重要,因为实验会有误差。量子的时髦,自然引来众说纷纭,惟在量子园地里“种过树”的人才可能有较深刻的体会。
作者历经50多年的理论探索,首创了有序算符内的积分理论,对发展量子力学数理基础——狄拉克的符号法略有建树,既能抑制爱因斯坦认为量子力学数学不够完美的抱怨,为爱因斯坦的量子纠缠思想提供纠缠态表象,也从数学上将量子力学几率假说落实到有序算符的正态分布,从而推陈出新、别开生面地丰富量子力学、量子统计力学和量子光学的内容。     

Algebras of Operators on Holomorphic Functions and Applications

We develop the theory of operators defined on a space of holomorphic functions. First, we characterize such operators by a suitable space of holomorphic functions. Next, we show that every operator is a limit of a sequence of convolution and multiplication operators. Finally, we define the exponential of an operator which permits us to solve some quantum stochastic differential equations.     

Two-Particle Spectrum of the Generator for Stochastic Model of Planar Rotators at High Temperatures

We study two-particle spectrum branches of the generator in the stochastic model of planar rotators, using the construction of a special basis in two-particle invariant subspaces. We prove that the branches of the spectrum are in a small neighborhood of the point 2. We prove the existence of two bound states in addition to the continuous part of the spectrum in the one-dimensional case.     

Approximate solutions of the Liouville equation. III. Variational principles and projection operators

Aspects of stationary variational principles for the Laplace-transformed Liouville equation are discussed. Projection techniques are used to derive new stationary principles applicable to the space orthogonal to the space spanned by functions occurring in the conservation laws. As a result, any trial function automatically leads to results satisfying the conservation laws. The procedure is also applied to the parity-even and parity-odd distributions which obey equations governed by the square of the Liouville operator. The technique is extended to eliminate the one-body additive contribution to the solution exactly. Finally, the ideas of the moment method, which leads to the continued-fraction representation of autocorrelation functions, are applied to variational principles. We find continued-fraction variational principles such that a zero trial function yields the usual representation. However, a trial function representing noninteracting particles contains the results of the moment method and in addition yields the exact analytic behavior for free particles.Work supported by a grant from the National Science Foundation.     

Maps of intervals with indifferent fixed points: Thermodynamic formalism and phase transitions

We develop the thermodynamic formalism for a large class of maps of the interval with indifferent fixed points. For such systems the formalism yields onedimensional systems with many-body infinite-range interactions for which the thermodynamics is well defined but Gibbs states are not. (Piecewise linear systems of this kind yield the soluble, in a sense, Fisher models.) We prove that such systems exhibit phase transitions, the order of which depends on the behavior at the indifferent fixed points. We obtain the critical exponent describing the singularity of the pressure and analyze the decay of correlations of the equilibrium states at all temperatures. Our technique relies on establishing and exploiting a relation between the transfer operators of the original map and its suitable (expanding) induced version. The technique allows one also to obtain a version of the Bowen-Ruelle formula for the Hausdorff dimension of repellers for maps with indifferent fixed points, and to generalize Fisher results to some nonsoluble models.Meyerhoff Visiting Professor, on leave from the Center for Transport Theory and Mathematical Physics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061.     

Effects as Functions on Projective Hilbert Space

The set of effect operators in a complex Hilbert space can be injectively embedded into the set of functions from the set of one-dimensional projections to the real interval [0, 1]. Properties of this injection are investigated.