Path Integrals in Noncommutative Quantum Mechanics

We consider an extension of the Feynman path integral to the quantum mechanics of noncommuting spatial coordinates and formulate the corresponding formalism for noncommutative classical dynamics related to quadratic Lagrangians (Hamiltonians). The basis of our approach is that a quantum mechanical system with a noncommutative configuration space can be regarded as another effective system with commuting spatial coordinates. Because the path integral for quadratic Lagrangians is exactly solvable and a general formula for the probability amplitude exists, we restrict our research to this class of Lagrangians. We find a general relation between quadratic Lagrangians in their commutative and noncommutative regimes and present the corresponding noncommutative path integral. This method is illustrated with two quantum mechanical systems in the noncommutative plane: a particle in a constant field and a harmonic oscillator.     

A Regularization of Quantum Field Hamiltonians with the Aid of p–adic Numbers

Gaussian distributions on infinite-dimensional p-adic spaces are introduced and the corresponding L₂-spaces of p-adic-valued square integrable functions are constructed. Representations of the infinite-dimensional Weyl group are realized in p-adic L₂-spaces. There is a formal analogy with the usual Segal representation. But there is also a large topological difference: parameters of the p-adic infinite-dimensional Weyl group are defined only on some balls (these balls are additive subgroups). p-adic Hilbert space representations of quantum Hamiltonians for systems with an infinite number of degrees of freedom are constructed. Many Hamiltonians with potentials which are too singular to exist as functions over reals are realized as bounded symmetric operators in L₂-spaces with respect to a p-adic Gaussian distribution.     

Hitchin System on Singular Curves

We study the Hitchin system on singular curves. We consider curves obtainable from the projective line by matching at several points or by inserting cusp singularities. It appears that on such singular curves, all basic ingredients of Hitchin integrable systems (moduli space of vector bundles, dualizing sheaf, Higgs field, etc.) can be explicitly described, which can be interesting in itself. Our main result is explicit formulas for the Hitchin Hamiltonians. We also show how to obtain the Hitchin integrable system on such curves by Hamiltonian reduction from a much simpler system on a finite-dimensional space. We pay special attention to a degenerate curve of genus two for which we find an analogue of the Narasimhan–Ramanan parameterization of the moduli space of SL(2) bundles as well as the explicit expressions for the symplectic structure and Hitchin-system Hamiltonians in these coordinates. We demonstrate the efficiency of our approach by rederiving the rational and trigonometric Calogero–Moser systems, which are obtained from Hitchin systems on curves with a marked point and with the respective cusp and node.     

Level shift operators for open quantum systems

Level shift operators describe the second-order displacement of eigenvalues under perturbation. They play a central role in resonance theory and ergodic theory of open quantum systems at positive temperatures. We exhibit intrinsic properties of level shift operators, properties which stem from the structure of open quantum systems at positive temperatures and which are common to all such systems. They determine the geometry of resonances bifurcating from eigenvalues of positive temperature Hamiltonians and they relate the Gibbs state, the kernel of level shift operators, and zero energy resonances. We show that degeneracy of energy levels of the small part of the open quantum system causes the Fermi Golden Rule Condition to be violated and we analyze ergodic properties of such systems.     

Autonomous Stochastic Perturbations of Dynamical Systems

Long-time effects of autonomous stochastic perturbations of Hamiltonian systems are considered. In particular, these perturbations allow us to obtain the averaging principle for deterministic perturbations in the case of Hamiltonians with many critical points. The limiting slow motion in this case is a stochastic process even when the system and perturbations are purely deterministic.     

浅水横流中异重冲击射流的大尺度涡结构

采用RNG湍流模型对浅水横流中异重冲击射流的大尺度涡结构进行了详细的数值研究．分析了冲击区滞止点上游壁面涡结构和近区Scarf涡结构的尺度、形成机理和演化特征．计算得到了上游壁面涡的特征尺度,结果表明上游壁面涡具有高度的三维性,其特征尺度依赖于流速比和环境水深．近区Sarf涡结构对横流冲击射流的横向浓度分布具有重要的影响．当流速比相对较小时,在底层壁射流与环境横流的横向边界附近出现明显的高浓度聚集现象,计算结果表明Scarf涡结构对这一高浓度聚集区的形成起主导作用．     

VARIATIONS ON A THEME BY EULER

1.IntroductionAHallliltolliansystemofdifferentialequationsonRZnisgivedbyP~~H,(P,q),q=HP(P,q),(1)wherep=(pl,'.,P.),q=(ql,',q.)eR"arethegeneralizedcoordinatesandmolllentarespectivelyandH(P,q)istheellergyofthesystem.Thesystem(1)canberewrittenasthecompactf…     

Self-adjointness of quadratic Hamiltonians and a representation method for their exponents

Solutions to a series of Cauchy problems for an equation of Schrödinger type with the Hamiltonian obtained by Wiener-Segal-Fock quantization of (finite- and infinite-dimensional) Hamiltonian systems with quadratic Hamiltonians are studied. Necessary conditions for essential self-adjointness of quantum Hamiltonians considered in the paper are obtained on special domains as a corollary.     

Symmetric form of the Hudson-Parthasarathy stochastic equation

We prove that the Hudson-Parthasarathy equation corresponds, up to unitary equivalence, to the strong resolvent limit of Schrödinger Hamiltonians in Fock space and that the symmetric form of this equation corresponds to the weak limit of the Schrödinger Hamiltonians.Translated fromMatematicheskie Zametki, Vol. 60, No. 5, pp. 726–750, November, 1996.     

Effective su q(2) Models and Polynomial Algebras for Fermion–Boson Hamiltonians

We show that schematic su(2)h ₃ interaction Hamiltonians, where su(2) plays the role of the pseudospin algebra of fermion operators and h ₃ is the Heisenberg algebra for bosons, are closely related to certain nonlinear models defined on a single quantum algebra su _q(2) of quasifermions. In particular, su _q(2) analogues of the Da Providencia–Schütte and extended Lipkin models are presented. We analyze the connection between q and the physical parameters of the fermion–boson system and, using polynomial algebras, discuss the integrability properties of the interaction Hamiltonians.     

Gaudin models with irregular singularities

We introduce a class of quantum integrable systems generalizing the Gaudin model. The corresponding algebras of quantum Hamiltonians are obtained as quotients of the center of the enveloping algebra of an affine Kac-Moody algebra at the critical level, extending the construction of higher Gaudin Hamiltonians from B. Feigin et al. (1994) [17] to the case of non-highest weight representations of affine algebras. We show that these algebras are isomorphic to algebras of functions on the spaces of opers on P¹ with regular as well as irregular singularities at finitely many points. We construct eigenvectors of these Hamiltonians, using Wakimoto modules of critical level, and show that their spectra on finite-dimensional representations are given by opers with trivial monodromy. We also comment on the connection between the generalized Gaudin models and the geometric Langlands correspondence with ramification.     

Hierarchy of Hamilton equations on Banach Lie-Poisson spaces related to restricted Grassmannian

We consider the Banach Lie-Poisson space and its complexification , where the first one of them contains the restricted Grassmannian Grres as a symplectic leaf. Using the Magri method we define an involutive family of Hamiltonians on these Banach Lie-Poisson spaces. The hierarchy of Hamilton equations given by these Hamiltonians is investigated. The operator equations of Ricatti-type are included in this hierarchy. For a few particular cases we give the explicit solutions.     

The complex Hamiltonian systems and quasi-periodic solutions in the derivative nonlinear Schrödinger equations

The complex Hamiltonian systems with real-valued Hamiltonians are generalized to deduce quasi-periodic solutions for a hierarchy of derivative nonlinear Schrödinger (DNLS) equations. The DNLS hierarchy is decomposed into a family of complex finite-dimensional Hamiltonian systems by separating the temporal and spatial variables, and the complex Hamiltonian systems are then proved to be integrable in the Liouville sense. Due to the commutability of complex Hamiltonian flows, the relationship between the DNLS equations and the complex Hamiltonian systems is specified via the Bargmann map. The Abel-Jacobi variable is elaborated to straighten out the DNLS flows as linear superpositions on the Jacobi variety of an invariant Riemann surface. Finally, by using the technique of Riemann-Jacobi inversion, some quasi-periodic solutions are obtained for the DNLS equations in view of the Riemann theorem and the trace formulas.     

Rearrangement of Energy Bands: Chern Numbers in the Presence of Cubic Symmetry

Formation of energy bands in the system of rotation-vibration quantum states of molecules is described within semi-quantum models under the presence of a symmetry group characterizing the equilibrium molecular configuration. Effective rotation-vibration Hamiltonians are written in two-quantum state models with rotational variables treated as classical ones. Eigen-line bundles associated with eigenvalues of 2×2 Hermitian matrix defined on rotational classical phase space which is a two-dimensional sphere are characterized by the first Chern class. Explicit procedure for the calculation of Chern numbers are suggested and realized for a family of Hamiltonians depending on extra control parameters in the presence of symmetry. Effective Hamiltonians for two vibrational states transforming according to some representations of the cubic symmetry group are studied. Chern numbers are evaluated for respective model Hamiltonians. The iso-Chern diagrams are introduced which split the parameter space into regions with fixed Chern numbers.     

Integrable Systems Obtained by Puncture Fusion from Rational and Elliptic Gaudin Systems

Using the procedure for puncture fusion, we obtain new integrable systems with poles of orders higher than one in the Lax operator matrix and consider the Hamiltonians, symplectic structure, and symmetries of these systems. Using the Inozemtsev limit procedure, we find a Toda-like system in the elliptic case having nontrivial commutation relations between the phase-space variables.     

Necessary optimality conditions for different phase portraits in a neighborhood of a singular arc

An optimal control problem with scalar control is characterized by two Hamiltonians related to boundary values of the control parameter. Intermediate (internal) values of the control and the corresponding singular trajectories (arcs) can be constructed in terms of these two Hamiltonians using Poisson brackets. All multiple Poisson brackets using these Hamiltonians two, three, and four times vanish on a singular arc of the second order and the brackets with five Hamiltonians in general differ from zero. There exist six different multiple Poisson brackets in which Hamiltonians are used five times. A regular arc in the optimal phase portrait is linked with a singular arc after one, several, or infinitely many (Fuller phenomenon) switchings. In the paper it is shown that various collections of the signs for these six quantities—multiple Poisson brackets—correspond to the above-mentioned cases. There exist four different collections of the signs for the set consisting of six Poisson brackets. The singularity including a universal surface is investigated for the general case, whereas two other types of singularities are studied in particular examples.     

Model BCS Hamiltonian and Approximating Hamiltonian in the Case of Infinite Volume. IV. Two Branches of Their Common Spectra and States

We consider the model and approximating Hamiltonians directly in the case of infinite volume. We show that each of these Hamiltonians has two branches of the spectrum and two systems of eigenvectors, which represent excitations of the ground states of the model and approximating Hamiltonians as well as the ground states themselves. On both systems of eigenvectors, the model and approximating Hamiltonians coincide with one another. In both branches of the spectrum, there is a gap between the eigenvalues of the ground and excited states.     

A sample-path approach to the optimality of echelon order-up-to policies in serial inventory systems

We present a new proof of the optimality of echelon order-up-to policies in serial inventory systems, first proved by Clark and Scarf. Our proof is based on a sample-path analysis as opposed to the original proof based on dynamic programming induction.     

Exactly solvable models in supersymmetric quantum mechanics and connection with spectrum-generating algebras

Analytic expressions for the eigenvalues and eigenfunctions of nonrelativistic shape-invariant Hamiltonians can be derived using the well-known method of supersymmetric quantum mechanics. Most of these Hamiltonians also possess spectrum-generating algebras and are hence solvable by an independent group theory method. We demonstrate the equivalence of the two solution methods by developing an algebraic framework for shape-invariant Hamiltonians with a general parameter change involving nonlinear extensions of Lie algebras. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 362–374, March, 1999.     

Conservation laws for polynomial Hamiltonians and for discrete models of the Boltzmann equation

Conservation laws that are linear with respect to the number of particles are constructed for classical and quantum Hamiltonians. A class of relaxation models generalizing discrete models of the Boltzmann equation are also considered. Conservation laws are written for these models in the same form as for the Hamiltonians. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 2, pp. 307–315, November, 1999.