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Journal articles on the topic 'Hamiltonian operators'

Author: Grafiati

Published: 4 June 2021

Last updated: 14 March 2023

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1

Liu, Yu, Jin Liu, and Da-jun Zhang. "On New Hamiltonian Structures of Two Integrable Couplings." Symmetry 14, no. 11 (2022): 2259. http://dx.doi.org/10.3390/sym14112259.

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In this paper, we present new Hamiltonian operators for the integrable couplings of the Ablowitz–Kaup–Newell–Segur hierarchy and the Kaup–Newell hierarchy. The corresponding Hamiltonians allow nontrivial degeneration. Multi-Hamiltonian structures are investigated. The involutive property is proven for the new and known Hamiltonians with respect to the two Poisson brackets defined by the new and known Hamiltonian operators.

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2

Konig, R. "Simplifying quantum double Hamiltonians using perturbative gadgets." Quantum Information and Computation 10, no. 3&4 (2010): 292–334. http://dx.doi.org/10.26421/qic10.3-4-9.

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Perturbative gadgets were originally introduced to generate effective $k$-local interactions in the low-energy sector of a $2$-local Hamiltonian. Extending this idea, we present gadgets which are specifically suited for realizing Hamiltonians exhibiting non-abelian anyonic excitations. At the core of our construction is a perturbative analysis of a widely used hopping-term Hamiltonian. We show that in the low-energy limit, this Hamiltonian can be approximated by a certain ordered product of operators. In particular, this provides a simplified realization of Kitaev's quantum double Hamiltonians

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3

Vitolo, R. "Computing with Hamiltonian operators." Computer Physics Communications 244 (November 2019): 228–45. http://dx.doi.org/10.1016/j.cpc.2019.05.012.

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4

GIRARDEAU, M. D., G. KREIN, and D. HADJIMICHEF. "FIELD-THEORETIC APPROACH FOR SYSTEMS OF COMPOSITE HADRONS." Modern Physics Letters A 11, no. 14 (1996): 1121–29. http://dx.doi.org/10.1142/s0217732396001156.

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Systems containing simultaneously hadrons and their constituents are most easily described by treating composite hadron field operators on the same kinematical footing as the constituent ones. Introduction of a unitary transformation allows redescription of hadrons by elementary-particle field operators. Transformation of the microscopic Hamiltonian leads to effective Hamiltonians describing all possible processes involving hadrons and their constituents.

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5

Zharinov, V. V. "Hamiltonian operators in differential algebras." Theoretical and Mathematical Physics 193, no. 3 (2017): 1725–36. http://dx.doi.org/10.1134/s0040577917120017.

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6

Kersten, P., I. Krasil’shchik та A. Verbovetsky. "Hamiltonian operators and ℓ-coverings". Journal of Geometry and Physics 50, № 1-4 (2004): 273–302. http://dx.doi.org/10.1016/j.geomphys.2003.09.010.

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7

Daletskii, Yu L., and B. L. Tsygan. "Hamiltonian operators and Hochschild homologies." Functional Analysis and Its Applications 19, no. 4 (1986): 319–21. http://dx.doi.org/10.1007/bf01077301.

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8

Zhao, Qi, and Xiao Yuan. "Exploiting anticommutation in Hamiltonian simulation." Quantum 5 (August 31, 2021): 534. http://dx.doi.org/10.22331/q-2021-08-31-534.

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Quantum computing can efficiently simulate Hamiltonian dynamics of many-body quantum physics, a task that is generally intractable with classical computers. The hardness lies at the ubiquitous anti-commutative relations of quantum operators, in corresponding with the notorious negative sign problem in classical simulation. Intuitively, Hamiltonians with more commutative terms are also easier to simulate on a quantum computer, and anti-commutative relations generally cause more errors, such as in the product formula method. Here, we theoretically explore the role of anti-commutative relation in

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9

van den Berg, Ewout, and Kristan Temme. "Circuit optimization of Hamiltonian simulation by simultaneous diagonalization of Pauli clusters." Quantum 4 (September 12, 2020): 322. http://dx.doi.org/10.22331/q-2020-09-12-322.

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Many applications of practical interest rely on time evolution of Hamiltonians that are given by a sum of Pauli operators. Quantum circuits for exact time evolution of single Pauli operators are well known, and can be extended trivially to sums of commuting Paulis by concatenating the circuits of individual terms. In this paper we reduce the circuit complexity of Hamiltonian simulation by partitioning the Pauli operators into mutually commuting clusters and exponentiating the elements within each cluster after applying simultaneous diagonalization. We provide a practical algorithm for partitio

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10

MARTÍNEZ, D., R. D. MOTA, and V. D. GRANADOS. "SYMMETRY AND SUPERSYMMETRY OF A NEUTRON IN THE MAGNETIC FIELD OF A LINEAR CURRENT." International Journal of Modern Physics A 21, no. 32 (2006): 6621–28. http://dx.doi.org/10.1142/s0217751x06034446.

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We study a neutron in an external magnetic field in coordinate space and show that the 2 × 2 radial matrix operators that factorize the Hamiltonian are contained within the constants of motion of the problem. Also we show that the 2 × 2 partners Hamiltonians satisfy the shape invariance condition.

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11

Anzaldo-Meneses, A. "Supercanonical coordinates for orthosymplectic evolution operators." Journal of Mathematical Physics 63, no. 9 (2022): 092101. http://dx.doi.org/10.1063/5.0083883.

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A time-dependent self-adjoint even Hamiltonian is defined by a linear combination of generators of the semidirect sum [Formula: see text], of the orthosymplectic plus the even Heisenberg algebra by computing the supercommutator of odd binary forms Π, given as linear combinations of odd bilinear generators of the odd Heisenberg algebra [Formula: see text] elements times [Formula: see text] elements, establishing a relationship between entangled boson systems and entangled fermion systems. This approach leads to the concept of intertwining, defined through the resulting quadratic Hamiltonians of

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12

HALL, RICHARD L., WOLFGANG LUCHA, and FRANZ F. SCHÖBERL. "DISCRETE SPECTRA OF SEMIRELATIVISTIC HAMILTONIANS." International Journal of Modern Physics A 18, no. 15 (2003): 2657–80. http://dx.doi.org/10.1142/s0217751x0301406x.

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We review various attempts to localize the discrete spectra of semirelativistic Hamiltonians of the form [Formula: see text] (defined, without loss of generality but for definiteness, in three spatial dimensions) as entering, for instance, in the spinless Salpeter equation; every Hamiltonian in this class of operators consists of the relativistic kinetic energy [Formula: see text], where β > 0 allows for the possibility of more than one particles of mass m, and a spherically symmetric attractive potential V(r), r ≡ |x|. In general, accurate eigenvalues of a nonlocal Hamiltonian operator can

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13

JING, SI-CONG, and HONG-YI FAN. "INVARIANT EIGEN-OPERATOR METHOD OF DERIVING ENERGY-LEVEL GAP FOR NONCOMMUTATIVE QUANTUM MECHANICS." Modern Physics Letters A 20, no. 09 (2005): 691–98. http://dx.doi.org/10.1142/s0217732305015975.

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We propose a new method to derive energy-level gap for Hamiltonians in the context of noncommutative quantum mechanics (NCQM). This method relies on finding invariant eigen-operators whose commutators with Hamiltonian are still the operators themselves but with some eigenvalue-like coefficients, which correspond to the energy-level gaps of the systems. Based on this method, only after some simple algebra, we derive the energy-level gaps for several important systems in NCQM, and most of these results have not been reported in literature so far.

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14

Hussin, V., I. Marquette, and K. Zelaya. "Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials." Journal of Physics A: Mathematical and Theoretical 55, no. 4 (2022): 045205. http://dx.doi.org/10.1088/1751-8121/ac43cc.

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Abstract We extend and generalize the construction of Sturm–Liouville problems for a family of Hamiltonians constrained to fulfill a third-order shape-invariance condition and focusing on the ‘−2x/3’ hierarchy of solutions to the fourth Painlevé transcendent. Such a construction has been previously addressed in the literature for some particular cases but we realize it here in the most general case. The corresponding potential in the Hamiltonian operator is a rationally extended oscillator defined in terms of the conventional Okamoto polynomials, from which we identify three different zero-mod

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15

Balajany, Hamideh, and Mohammad Mehrafarin. "Geometric phase of cosmological scalar and tensor perturbations in f(R) gravity." Modern Physics Letters A 33, no. 14 (2018): 1850077. http://dx.doi.org/10.1142/s0217732318500773.

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By using the conformal equivalence of f(R) gravity in vacuum and the usual Einstein theory with scalar-field matter, we derive the Hamiltonian of the linear cosmological scalar and tensor perturbations in f(R) gravity in the form of time-dependent harmonic oscillator Hamiltonians. We find the invariant operators of the resulting Hamiltonians and use their eigenstates to calculate the adiabatic Berry phase for sub-horizon modes as a Lewis–Riesenfeld phase.

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16

Ma, Wen-Xiu, and Maxim Pavlov. "Extending Hamiltonian operators to get bi-Hamiltonian coupled KdV systems." Physics Letters A 246, no. 6 (1998): 511–22. http://dx.doi.org/10.1016/s0375-9601(98)00555-6.

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17

Patel, Apoorva, and Anjani Priyadarsini. "Optimization of quantum Hamiltonian evolution: From two projection operators to local Hamiltonians." International Journal of Quantum Information 15, no. 02 (2017): 1650027. http://dx.doi.org/10.1142/s0219749916500271.

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Given a quantum Hamiltonian and its evolution time, the corresponding unitary evolution operator can be constructed in many different ways, corresponding to different trajectories between the desired end-points and different series expansions. A choice among these possibilities can then be made to obtain the best computational complexity and control over errors. It is shown how a construction based on Grover's algorithm scales linearly in time and logarithmically in the error bound, and is exponentially superior in error complexity to the scheme based on the straightforward application of the

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18

BELLISSARD, JEAN, and HERMANN SCHULZ-BALDES. "SCATTERING THEORY FOR LATTICE OPERATORS IN DIMENSION d ≥ 3." Reviews in Mathematical Physics 24, no. 08 (2012): 1250020. http://dx.doi.org/10.1142/s0129055x12500201.

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This paper analyzes the scattering theory for periodic tight-binding Hamiltonians perturbed by a finite range impurity. The classical energy gradient flow is used to construct a conjugate (or dilation) operator to the unperturbed Hamiltonian. For dimension d ≥ 3, the wave operator is given by an explicit formula in terms of this dilation operator, the free resolvent and the perturbation. From this formula, the scattering and time delay operators can be read off. Using the index theorem approach, a Levinson theorem is proved which also holds in the presence of embedded eigenvalues and threshold

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19

Huang, Junjie, Xiang Guo, Yonggang Huang, and Alatancang. "Generalized Inverse of Upper Triangular Infinite Dimensional Hamiltonian Operators." Algebra Colloquium 20, no. 03 (2013): 395–402. http://dx.doi.org/10.1142/s1005386713000369.

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In this paper, we deal with the generalized inverse of upper triangular infinite dimensional Hamiltonian operators. Based on the structure operator matrix J in infinite dimensional symplectic spaces, it is shown that the generalized inverse of an infinite dimensional Hamiltonian operator is also Hamiltonian. Further, using the decomposition of spaces, an upper triangular Hamiltonian operator can be written as a new operator matrix of order 3, and then an explicit expression of the generalized inverse is given.

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20

Marzuola, Jeremy L., and Gideon Simpson. "Spectral analysis for matrix Hamiltonian operators." Nonlinearity 24, no. 2 (2010): 389–429. http://dx.doi.org/10.1088/0951-7715/24/2/003.

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21

Zharinov, V. V. "Hamiltonian Operators with Zero-Divergence Constraints." Theoretical and Mathematical Physics 200, no. 1 (2019): 923–37. http://dx.doi.org/10.1134/s0040577919070018.

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22

Azizov, Tomas Ya, Aad Dijksma, and Irina V. Gridneva. "On the boundedness of Hamiltonian operators." Proceedings of the American Mathematical Society 131, no. 2 (2002): 563–76. http://dx.doi.org/10.1090/s0002-9939-02-06565-6.

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23

Fujimoto, Kenji, Jacquelien M. A. Scherpen, and W. Steven Gray. "Hamiltonian realizations of nonlinear adjoint operators." Automatica 38, no. 10 (2002): 1769–75. http://dx.doi.org/10.1016/s0005-1098(02)00079-1.

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24

Fujimoto, Kenji, Jacquelien M. A. Scherpen, and W. Steven Gray. "Hamiltonian Realizations of Nonlinear Adjoint Operators." IFAC Proceedings Volumes 33, no. 2 (2000): 39–44. http://dx.doi.org/10.1016/s1474-6670(17)35544-1.

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25

Mokhov, O. I. "Hamiltonian differential operators and contact geometry." Functional Analysis and Its Applications 21, no. 3 (1987): 217–23. http://dx.doi.org/10.1007/bf02577136.

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26

Olver, Peter J. "Darboux' theorem for Hamiltonian differential operators." Journal of Differential Equations 71, no. 1 (1988): 10–33. http://dx.doi.org/10.1016/0022-0396(88)90036-8.

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27

Chen, Alatancang, Yaru Qi, and Junjie Huang. "Left invertibility of formal Hamiltonian operators." Linear and Multilinear Algebra 63, no. 2 (2014): 235–43. http://dx.doi.org/10.1080/03081087.2013.860596.

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28

Fedulova, L., and I. Gridneva. "On the boundedness of hamiltonian operators." Актуальные направления научных исследований XXI века: теория и практика 2, no. 4 (2014): 451–53. http://dx.doi.org/10.12737/5210.

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29

Vodová, Jiřina. "Low-order Hamiltonian operators having momentum." Journal of Mathematical Analysis and Applications 401, no. 2 (2013): 724–32. http://dx.doi.org/10.1016/j.jmaa.2012.12.067.

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30

der Lende, E. D. van, and H. G. J. Pijls. "Super Hamiltonian operators and Lie superalgebras." Indagationes Mathematicae 1, no. 2 (1990): 221–42. http://dx.doi.org/10.1016/0019-3577(90)90006-9.

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31

Charles, L. "Toeplitz operators and Hamiltonian torus actions." Journal of Functional Analysis 236, no. 1 (2006): 299–350. http://dx.doi.org/10.1016/j.jfa.2005.10.011.

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32

Bravyi, Sergey, Guillaume Duclos-Cianci, David Poulin, and Martin Suchara. "Subsystem surface codes with three-qubit check operators." Quantum Information and Computation 13, no. 11&12 (2013): 963–85. http://dx.doi.org/10.26421/qic13.11-12-4.

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We propose a simplified version of the Kitaev's surface code in which error correction requires only three-qubit parity measurements for Pauli operators XXX and ZZZ. The new code belongs to the class of subsystem stabilizer codes. It inherits many favorable properties of the standard surface code such as encoding of multiple logical qubits on a planar lattice with punctured holes, efficient decoding by either minimum-weight matching or renormalization group methods, and high error threshold. The new subsystem surface code (SSC) gives rise to an exactly solvable Hamiltonian with 3-qubit interac

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33

Bracken, P. "Article." Canadian Journal of Physics 76, no. 9 (1998): 707–17. http://dx.doi.org/10.1139/p98-048.

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A method for calculating complete secular polynomials is discussedthat is based on the evaluation of matrix elementsof a specific Hamiltonian.Several Hamiltonians are presented and described in detail as well astheir physical significance. It is shown that theycan be transformed into an equivalent form in termsof raising and lowering operators, and the third componentof the spin operator. A basis set is definedand the action of a specific Hamiltonian on thebasis set is described in detail. Several Hamiltoniansare given explicitly and in matrix form. Results in terms of secularpolynomials for a

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34

Neidhardt, Hagen, and Valentin Zagrebnov. "On the Right Hamiltonian for Singular Perturbations: General Theory." Reviews in Mathematical Physics 09, no. 05 (1997): 609–33. http://dx.doi.org/10.1142/s0129055x97000221.

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Let the pair of self-adjoint operators {A≥0,W≤0} be such that: (a) there is a dense domain [Formula: see text] such that [Formula: see text] is semibounded from below (stability domain), (b) the symmetric operator [Formula: see text] is not essentially self-adjoint (singularity of the perturbation), (c) the Friedrichs extension [Formula: see text] of [Formula: see text] is maximal with respect to W, i.e., [Formula: see text]. [Formula: see text]. Let [Formula: see text] be a regularizing sequence of bounded operators which tends in the strong resolvent sense to W. The abstract problem of the r

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35

Gavrilik, Alexandre, and Ivan Kachurik. "Pseudo-Hermitian position and momentum operators, Hermitian Hamiltonian, and deformed oscillators." Modern Physics Letters A 34, no. 01 (2019): 1950007. http://dx.doi.org/10.1142/s021773231950007x.

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The recently introduced by us, two- and three-parameter (p, q)- and (p, q, [Formula: see text])-deformed extensions of the Heisenberg algebra were explored under the condition of their direct link with the respective (nonstandard) deformed quantum oscillator algebras. In this paper, we explore certain Hermitian Hamiltonians build in terms of non-Hermitian position and momentum operators obeying definite [Formula: see text](N)-pseudo-hermiticity properties. A generalized nonlinear (with the coefficients depending on the particle number operator N) one-mode Bogoliubov transformation is developed

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36

Mostafazadeh, Ali. "Parasupersymmetric Quantum Mechanics and Indices of Fredholm Operators." International Journal of Modern Physics A 12, no. 15 (1997): 2725–39. http://dx.doi.org/10.1142/s0217751x9700150x.

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The general features of the degeneracy structure of (p = 2) parasupersymmetric quantum mechanics are employed to yield a classification scheme for the form of the parasupersymmetric Hamiltonians. The method is applied to parasupersymmetric systems whose Hamiltonian is the square root of a fourth order polynomial in the generators of the parasupersymmetry. These systems are interesting to study for they lead to the introduction of a set of topological invariants very similar to the Witten indices of ordinary supersymmetric quantum mechanics. The topological invariants associated with parasupers

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37

Hounkonnou, Mahouton Norbert, Mahougnon Justin Landalidji, and Melanija Mitrović. "Einstein Field Equation, Recursion Operators, Noether and Master Symmetries in Conformable Poisson Manifolds." Universe 8, no. 4 (2022): 247. http://dx.doi.org/10.3390/universe8040247.

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We show that a Minkowski phase space endowed with a bracket relatively to a conformable differential realizes a Poisson algebra, confering a bi-Hamiltonian structure to the resulting manifold. We infer that the related Hamiltonian vector field is an infinitesimal Noether symmetry, and compute the corresponding deformed recursion operator. Besides, using the Hamiltonian–Jacobi separability, we construct recursion operators for Hamiltonian vector fields in conformable Poisson–Schwarzschild and Friedmann–Lemaître–Robertson–Walker (FLRW) manifolds, and derive the related constants of motion, Chris

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38

Bernard, Patrick. "The Lax–Oleinik semi-group: a Hamiltonian point of view." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 142, no. 6 (2012): 1131–77. http://dx.doi.org/10.1017/s0308210511000059.

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The weak KAM theory was developed by Fathi in order to study the dynamics of convex Hamiltonian systems. It somehow makes a bridge between viscosity solutions of the Hamilton–Jacobi equation and Mather invariant sets of Hamiltonian systems, although this was fully understood only a posteriori. These theories converge under the hypothesis of convexity, and the richness of applications mostly comes from this remarkable convergence. In this paper, we provide an elementary exposition of some of the basic concepts of weak KAM theory. In a companion paper, Albert Fathi exposed the aspects of his the

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39

Prorok, Dominik, and Anatolij Prykarpatski. "Quantum Current Algebra Symmetries and Integrable Many-Particle Schrödinger Type Quantum Hamiltonian Operators." Symmetry 11, no. 8 (2019): 975. http://dx.doi.org/10.3390/sym11080975.

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Based on the G. Goldin’s quantum current algebra symmetry representation theory, have succeeded in explaining a hidden relationship between the quantum many-particle Hamiltonian operators, defined in the Fock space, their factorized structure and integrability. Interesting for applications quantum oscillatory Hamiltonian operators are considered, the quantum symmetries of the integrable quantum Calogero-Sutherland model are analyzed in detail.

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40

OKUMURA, KO. "VARIOUS CONDENSED MATTER HAMILTONIANS IN TERMS OF U(2/2) OPERATORS AND THEIR SYMMETRY STRUCTURES." Modern Physics Letters B 07, no. 04 (1993): 251–58. http://dx.doi.org/10.1142/s0217984993000266.

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We rewrite various lattice Hamiltonians in condensed matter physics in terms of U(2/2) operators that we introduce. In this representation the symmetry structure of the models becomes clear. Especially, the Heisenberg, the supersymmetric t-J and a newly proposed high-T c superconducting Hamiltonian reduce to the same form [Formula: see text]. This representation also gives us a systematic way of searching for the symmetries of the system.

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41

Ozorio de Almeida, A. M., and O. Brodier. "Nonlinear semiclassical dynamics of open systems." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 369, no. 1935 (2011): 260–77. http://dx.doi.org/10.1098/rsta.2010.0261.

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A semiclassical approximation for an evolving density operator, driven by a ‘closed’ Hamiltonian and ‘open’ Markovian Lindblad operators, is reviewed. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the Hamiltonian is a quadratic function and the Lindblad operators are linear functions of positions and momenta. The semiclassical formulae are interpreted within a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra ‘open’ term in the double Ha

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42

Hilscher, Roman Šimon, and Petr Zemánek. "Friedrichs extension of operators defined by linear Hamiltonian systems on unbounded interval." Mathematica Bohemica 135, no. 2 (2010): 209–22. http://dx.doi.org/10.21136/mb.2010.140698.

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43

Mostafazadeh, Ali. "Pseudo-Hermitian quantum mechanics with unbounded metric operators." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1989 (2013): 20120050. http://dx.doi.org/10.1098/rsta.2012.0050.

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I extend the formulation of pseudo-Hermitian quantum mechanics to η + -pseudo-Hermitian Hamiltonian operators H with an unbounded metric operator η + . In particular, I give the details of the construction of the physical Hilbert space, observables and equivalent Hermitian Hamiltonian for the case that H has a real and discrete spectrum and its eigenvectors belong to the domain of η + and consequently .

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44

WANG, SHUAI, HONG-CHUN YUAN, and HONG-YI FAN. "FRESNEL OPERATOR, SQUEEZED STATE AND WIGNER FUNCTION FOR CALDIROLA–KANAI HAMILTONIAN." Modern Physics Letters A 26, no. 19 (2011): 1433–42. http://dx.doi.org/10.1142/s0217732311035778.

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Based on the technique of integration within an ordered product (IWOP) of operators, we introduce the Fresnel operator for converting a kind of time-dependent Hamiltonian into the standard harmonic oscillator Hamiltonian. The Fresnel operator with the parameters A, B, C, D corresponds to classical optical Fresnel transformation, these parameters are the solution to a set of coupled partial differential equations set up in the above-mentioned converting process. In this way, the Caldirola–Kanai Hamiltonian has been easily converted into the standard harmonic oscillator Hamiltonian. And then the

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45

ALTINOLUK, TOLGA, CARLOS CONTRERAS, ALEX KOVNER, EUGENE LEVIN, MICHAEL LUBLINSKY, and ARTHUR SHULKIM. "QCD REGGEON CALCULUS FROM JIMWLK EVOLUTION." International Journal of Modern Physics: Conference Series 25 (January 2014): 1460025. http://dx.doi.org/10.1142/s2010194514600258.

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We show explicitly how the high energy QCD evolution generated by the KLWMIJ Hamiltonian can be cast in the form of the QCD Reggeon Field Theory. We show how to reduce the KLWMIJ Hamitonian to physical color singlet degrees of freedom. We suggest a natural way of defining the Pomeron and other Reggeons in the framework of the KLWMIJ evolution and derive the QCD Reggeon Field Theory Hamiltonian which includes several lowest Reggeon operators. This Hamiltonian generates evolution equations for all Reggeons in the case of dilute-dense scattering, including the nonlinear Balitsky-Kovchegov equatio

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46

Chamberlain, S. R., J. G. Tucker, J. M. Conroy, and H. G. Miller. "Waxman’s algorithm for non-Hermitian Hamiltonian operators." Journal of Physics Communications 2, no. 2 (2018): 025026. http://dx.doi.org/10.1088/2399-6528/aaaea3.

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47

Puljić, Krunoslav, and Robert Manger. "Evolutionary operators for the Hamiltonian completion problem." Soft Computing 24, no. 23 (2020): 18073–88. http://dx.doi.org/10.1007/s00500-020-05063-8.

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48

Song, Yanli. "Dirac operators on quasi-Hamiltonian G-spaces." Journal of Geometry and Physics 106 (August 2016): 70–86. http://dx.doi.org/10.1016/j.geomphys.2016.01.012.

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49

Kato, Makoto. "Momentum and Hamiltonian operators in generalized coordinates." International Journal of Theoretical Physics 33, no. 4 (1994): 857–64. http://dx.doi.org/10.1007/bf00672822.

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50

Anfossi, Alberto, and Arianna Montorsi. "Spin–fermion mappings for even Hamiltonian operators." Journal of Physics A: Mathematical and General 38, no. 21 (2005): 4519–27. http://dx.doi.org/10.1088/0305-4470/38/21/001.

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