C. R. Handy
The Eigenvalue Moment Method (EMM), Handy (2001), Handy and Wang (2001)) is applied to the $H_α\equiv P^2 + iX^3 + iαX$ Hamiltonian, enabling the algebraic/numerical generation of converging bounds to the complex energies of the $L^2$ states, as argued (through asymptotic methods) by Delabaere and Trinh (J. Phys. A: Math. Gen. {\bf 33} 8771 (2000)).
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Recent investigations by Bender and Boettcher (Phys. Rev. Lett 80, 5243 (1998)) and Mezincescu (J. Phys. A. 33, 4911 (2000)) have argued that the discrete spectrum of the non-hermitian potential $V(x) = -ix^3$ should be real. We give further evidence for this through a novel formulation which transforms the general one dimensional Schrodinger equation (with complex potential) into a fourth order linear differential equation for $|Ψ(x)|^2$. This permits the application of the Eigenvalue Moment Method, developed by Handy, Bessis, and coworkers (Phys. Rev. Lett. 55, 931 (1985);60, 253 (1988a,b)),…
We study the Schroedinger equation of a class of two-level systems under the action of a periodic time-dependent external field in the situation where the energy difference 2epsilon between the free energy levels is sufficiently small with respect to the strength of the external interaction. Under suitable conditions we show that this equation has a solution in terms of converging power series expansions in epsilon. In contrast to other expansion methods, like in the Dyson expansion, the method we present is not plagued by the presence of ``secular terms''. Due to this feature we were able to …
For low dimension systems admitting a moment equation representation (MER), the development of an effective eigenenergy bounding theory applicable to all discrete states had remained elusive, until now. Whereas Handy et al (1988 Phys. Rev. Lett. 60 253) demonstrated the effectiveness of the {\it Moment Problem} based, Eigenvalue Moment Method (EMM), for generating arbitrarily tight bounds to the multidimensional, positive, bosonic ground state, its extension to arbitrary excited states seemed intractable. We have discovered a new, moment representation based, quantization formalism that achiev…
Converging Periodic Boundary Conditions and Detection of Topological Gaps on Regular Hyperbolic Tessellations
本文针对一般 {p,q}-双曲镶嵌提出收敛周期边界条件的实用方案，据此识别体哈密顿量真实谱隙，并构造出与格点 K-理论预言一致的拓扑模型，进而展示拓扑谱流和软界面诱导的拓扑通道。
General characterisation of Hamiltonians generating velocity-independent forces
本文放宽了动量为多项式的假设，将哈密顿量视为状态空间函数并发展出强一般性约束，发现更广泛的生成非保守速度无关力的哈密顿量类别及其二分性，并完成了二维与一维空间中此类力的分类。
Spectrum-generating algebra and intertwiners of the resonant Pais-Uhlenbeck oscillator
共振Pais-Uhlenbeck振子在经典等价哈密顿量下产生不同量子理论，通过谱生成代数和Jordan链结构揭示量子化歧义。
We study the properties of Hamiltonians defined as the generators of transfer matrices in quasi- one-dimensional waveguides. For single- or multi-mode waveguides obeying flux conservation and time-reversal invariance, the Hamiltonians defined in this way are non-Hermitian, but satisfy symmetry properties that have previously been identified in the literature as "pseudo Hermiticity" and "anti-PT symmetry". We show how simple one-channel and two-channel models exhibit transitions between real, imaginary, and complex eigenvalue pairs.
Generating functions for irreversible Hamiltonian systems
该研究将保守-不可逆函数推广至光滑流形，分析其局部表示与双二次函数，指出其不必由几乎 Poisson 括号加权积表示，并为任意代数特别是复代数上的代数框架提供可能。
The transitionless tracking (TT) algorithm enables the exact tracking of quantum adiabatic dynamics in an arbitrary short time by adding a counterdiabatic Hamiltonian to the original adiabatic Hamiltonian. By applying Husimi's method originally developed for a quantum parametric oscillator (QPO) to the transitionless QPO achieved using the TT algorithm, we obtain the transition probability generating function with a time-dependent parameter constituted with solutions of the corresponding classical parametric oscillator (CPO). By obtaining the explicit solutions of this CPO using the phase-ampl…
The purpose of this paper is to demonstrate a new method of generating exact solutions to the Einstein's equations obtained by the Hamiltonian reduction. The key element to the successful Hamiltonian reduction is finding the privileged spacetime coordinates in which physical degrees of freedom manifestly reside in the conformal two-metric, and all the other metric components are determined by the conformal two-metric. In the privileged coordinates the Einstein's constraint equations become trivial; the Hamiltonian and momentum constraints are simply the defining equations of a non-vanishing gr…
The family F of all potentials V(x) for which the Hamiltonian H in one space dimension possesses a high order Lie symmetry is determined. A sub-family F', which contains a class of potentials allowing a realization of so(2,1) as spectrum generating algebra of H through differential operators of finite order, is identified. Furthermore and surprisingly, the families F and F' are shown to be related to the stationary KdV hierarchy. Hence, the "harmless" Hamiltonian H connects different mathematical objects, high order Lie symmetry, realization of so(2,1)-spectrum generating algebra and families …
We use a Hamiltonian (transition matrix) description of height-restricted Dyck paths on the plane in which generating functions for the paths arise as matrix elements of the propagator to evaluate the length and area generating function for paths with arbitrary starting and ending points, expressing it as a rational combination of determinants. Exploiting a connection between random walks and quantum exclusion statistics that we previously established, we express this generating function in terms of grand partition functions for exclusion particles in a finite harmonic spectrum and present an …
When discussing consequences of symmetries of dynamical systems based on Noether's first theorem, most standard textbooks on classical or quantum mechanics present a conclusion stating that a global continuous Lie symmetry implies the existence of a time independent conserved Noether charge which is the generator of the action on phase space of that symmetry, and which necessarily must as well commute with the Hamiltonian. However this need not be so, nor does that statement do justice to the complete scope and reach of Noether's first theorem. Rather a much less restrictive statement applies,…
Moment based methods have produced efficient multiscale quantization algorithms for solving singular perturbation/strong coupling problems. One of these, the Eigenvalue Moment Method (EMM), developed by Handy et al (Phys. Rev. Lett.{\bf 55}, 931 (1985); ibid, {\bf 60}, 253 (1988b)), generates converging lower and upper bounds to a specific discrete state energy, once the signature property of the associated wavefunction is known. This method is particularly effective for multidimensional, bosonic ground state problems, since the corresponding wavefunction must be of uniform signature, and can …
Empirical adequacy of the time operator canonically conjugate to a Hamiltonian generating translations
本文探讨了允许存在正则共轭时间算子的无界哈密顿量系统，给出了多个与真实世界同样复杂且允许时间算子的量子世界示例，证明其不会更易衰变到无穷负能量，并指出这些系统是希尔伯特空间基础主义的具体反例，为恢复量子力学时空对称性及Page-Wootters形式提供了动机。
In the framework of SUSYQM extended to deal with non-Hermitian Hamiltonians, we analyze three sets of complex potentials with real spectra, recently derived by a potential algebraic approach based upon the complex Lie algebra sl(2, C). This extends to the complex domain the well-known relationship between SUSYQM and potential algebras for Hermitian Hamiltonians, resulting from their common link with the factorization method and Darboux transformations. In the same framework, we also generate for the first time a pair of elliptic partner potentials of Weierstrass $\wp$ type, one of them being r…
Generating twisted Cherednik eigenfunctions
本文通过作用某些（生成和置换）操作，递归地构造了与Ding-Iohara-Miki代数相关的扭曲Cherednik本征函数，类似于Kirillov-Noumi算子的推广。
We express discrete Painlevé equations as discrete Hamiltonian systems. The discrete Hamiltonian systems here mean the canonical transformations defined by generating functions. Our construction relies on the classification of the discrete Painlevé equations based on the surface-type. The discrete Hamiltonians we obtain are written in the logarithm and dilogarithm functions.
We analyze the dynamical equations obeyed by a classical system with position-dependent mass. It is shown that there is a non-conservative force quadratic in the velocity associated to the variable mass. We construct the Lagrangian and the Hamiltonian for this system and find the modifications required in the Euler-Lagrange and Hamilton's equations to reproduce the appropriate Newton's dynamical law. Since the Hamiltonian is not time invariant, we get a constant of motion suited to write the dynamical equations in the form of the Hamilton's ones. The time-dependent first integrals of motion ar…
In a recent paper the so-called Spectrum Generating Algebra (SGA) technique has been applied to the N-dimensional Taub-NUT system, a maximally superintegrable Hamiltonian system which can be interpreted as a one-parameter deformation of the Kepler-Coulomb system. Such a Hamiltonian is associated to a specific Bertrand space of non-constant curvature. The SGA procedure unveils the symmetry algebra underlying the Hamiltonian system and, moreover, enables one to solve the equations of motion. Here we will follow the same path to tackle the Darboux III system, another maximally superintegrable sys…
Following Sklyanin's proposal in the periodic case, we derive the generating function of the Gaudin Hamiltonians with boundary terms. Our derivation is based on the quasi-classical expansion of the linear combination of the transfer matrix of the XXX Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function. By defining the appropriate Bethe vectors which yield strikingly simple off shell action of the generating function, we fully implement the algebraic B…
In this paper, we construct a Spectrum Generating Algebra (SGA) for a quantum system with purely continuous spectrum: the quantum free particle in a Lobachevski space with constant negative curvature. The SGA contains the geometrical symmetry algebra of the system plus a subalgebra of operators that give the spectrum of the system and connects the eigenfunctions of the Hamiltonian among themselves. In our case, the geometrical symmetry algebra is $\frak{so}(3,1)$ and the SGA is $\frak{so}(4,2)$. We start with a representation of $\frak{so}(4,2)$ by functions on a realization of the Lobachevski…
Integrals of motion are constructed from noncommutative (NC) Kepler dynamics, generating $SO(3),$ $SO(4),$ and $SO(1,3)$ dynamical symmetry groups. The Hamiltonian vector field is derived in action-angle coordinates, and the existence of a hierarchy of bi-Hamiltonian structures is highlighted. Then, a family of Nijenhuis recursion operators is computed and discussed.
In the derivation of the generating function of the Gaudin Hamiltonians with boundary terms, we follow the same approach used previously in the rational case, which in turn was based on Sklyanin's method in the periodic case. Our derivation is centered on the quasi-classical expansion of the linear combination of the transfer matrix of the XXZ Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function. By defining the appropriate Bethe vectors which yield st…
Topological constraints on a dynamical system often manifest themselves as breaking of the Hamiltonian structure; well-known examples are non-holonomic constraints on Lagrangian mechanics. The statistical mechanics under such topological constraints is the subject of the present study. Conventional arguments based on phase spaces, Jacobi identity, invariant measure, or the H theorem are no longer applicable, since all these notions stem from the symplectic geometry underlying canonical Hamiltonian systems. Remembering that Hamiltonian systems are endowed with field tensors (canonical 2-forms) …
We show that a recently developed method for generating bounds for the discrete energy states of the non-hermitian $-ix^3$ potential (Handy 2001) is applicable to complex rotated versions of the Hamiltonian. This has important implications for extension of the method in the analysis of resonant states, Regge poles, and general bound states in the complex plane (Bender and Boettcher (1998)).
Variational integrators are derived for structure-preserving simulation of stochastic forced Hamiltonian systems. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and considering a stochastic generalization of the deterministic Lagrange-d'Alembert principle. Our approach presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators…
Moment generating function of the tacnode process
研究tacnode过程在多个区间上的计数函数的生成函数，通过哈密顿量积分表示得到大间隙渐近及中心极限定理。
We study bound states of a 3--particle system in $\mathbb{R}^3$ described by the Hamiltonian $H(λ_n) = H_0 + v_{12} + λ_n (v_{13} + v_{23})$, where the particle pair $\{1,2\}$ has a zero energy resonance and no bound states, while other particle pairs have neither bound states nor zero energy resonances. It is assumed that for a converging sequence of coupling constants $λ_n \to λ_{cr}$ the Hamiltonian $H(λ_n)$ has a sequence of levels with negative energies $E_n$ and wave functions $ψ_n$, where the sequence $ψ_n$ totally spreads in the sense that $\lim_{n \to \infty}\int_{|ζ| \leq R} |ψ_n (ζ)…
Stochastic Discrete Hamiltonian Variational Integrators
本文提出了一种随机哈密顿系统的结构保持数值方法，通过构造随机离散哈密顿量来近似II型随机生成函数，统一了多种已有积分器并提出新的结构保持数值方案，所得积分器具有辛性、保持李群对称的守恒量，并包含随机辛龙格-库塔方法作为特例。
The group theoretical treatment of bound and scattering state problems is extended to include band structure. We show that one can realize Hamiltonians with periodic potentials as dynamical symmetries, where representation theory provides analytic solutions, or which can be treated with more general spectrum generating algebraic methods. We find dynamical symmetries for which we derive the transfer matrices and dispersion relations. Both compact and non-compact groups are found to play a role.
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