The action-angle representation in quantum mechanics is conceptually quite different from its classical counterpart and motivates a canonical discretization of the phase space. In this work, a discrete and finite-dimensional phase space formalism, in which the phase space variables are discrete and the time is continuous, is developed and the fundamental properties of the discrete Weyl-Wigner-Moyal quantization are derived. The action-angle Wigner function is shown to exist in the semi-discrete limit of this quantization scheme. A comparison with other formalisms which are not explicitly based on canonical discretization is made. Fundamental properties that an action-angle phase space distribution respects are derived. The dynamical properties of the action-angle Wigner function are analysed for discrete and finite-dimensional model Hamiltonians. The limit of the discrete and finite-dimensional formalism including a discrete analogue of the Gaussian wavefunction spread, viz. the binomial wavepacket, is examined and shown by examples that standard (continuum) quantum mechanical results can be obtained as the dimension of the discrete phase space is extended to infinity.
Schwingers finite (D) dimensional periodic Hilbert space representations are studied on the toroidal lattice ℤD × ℤD with specific emphasis on the deformed oscillator subalgebras and the generalized representations of the Wigner function. These subalgebras are shown to be admissible endowed with the non-negative norm of Hilbert space vectors. Hence, they provide the desired canonical basis for the algebraic formulation of the quantum phase problem. Certain equivalence classes in the space of labels are identified within each subalgebra, and connections with area-preserving canonical transformations are examined. The generalized representations of the Wigner function are examined in the finite-dimensional cyclic Schwinger basis. These representations are shown to conform to all fundamental conditions of the generalized phase space Wigner distribution. As a specific application of the Schwinger basis, the number-phase unitary operator pair in ℤD × ℤD is studied and, based on the admissibility of the underlying q-oscillator subalgebra, an algebraic approach to the unitary quantum phase operator is established. This being the focus of this work, connections with the Susskind-Glogower-Carruthers-Nieto phase operator formalism as well as standard action-angle Wigner function formalisms are examined in the infinite-period limit. The concept of continuously shifted Fock basis is introduced to facilitate the Fock space representations of the Wigner function.
The dispersionless-Boussinesq and Benney-Lax equations are equations of hydrodynamic type which can be obtained as reductions of the dispersionless Kadomtsev-Petviashvili equation. We find that for the three-component reduction, the dispersionless Boussinesq and Benney-Lax equations are the same up to a diffeomorphism. This equivalence becomes manifest when the equations of motion are cast into the form of a triplet of conservation laws. Furthermore, in this form we are able to recognize a non-trivial scaling symmetry of these equations which plays an important role in the construction of their bi-Hamiltonian structure. We exhibit a pair of compatible Hamiltonian operators which belong to a restricted class of Dubrovin and Novikov operators appropriate to a system of conservation laws. The recursion operator for this system generates three infinite sequences of conserved Hamiltonians.
We have investigated the reality of exact bound states of complex and/or PT-symmetric non-Hermitian exponential-type generalized Hulthén potential. The Klein-Gordon equation has been solved by using the Nikiforov-Uvarov method which is based on solving the second-order linear differential equations by reduction to a generalized equation of hypergeometric type. In many cases of interest, negative and positive energy states have been discussed for different types of complex potentials.
The algorithmic method introduced by Fokas and Ablowitz to investigate the transformation properties of Painlevé equations is used to obtain one-to-one correspondence between the Painlevé IV, V and VI equations and the second-order second-degree equations of Painlevé type.
A rigorous method was introduced by Fokas and Zhou (1992) for studying the Riemann-Hilbert problem associated with the Painleve II and IV equations. The same methodology has been applied to the Painleve I, III and V equations. In this paper, we will apply the same methodology to the Painleve VI equation. We will show that the Cauchy problem for the Painleve VI equation admits, in general, a global meromorphic solution in t. Furthermore, the special solution which can be written in terms of a hypergeometric function is obtained via solving the special case of the Riemann-Hilbert problem.
The algebra of generalized linear quantum canonical transformations is examined in the perspective of Schwingers unitary-canonical operator basis. Formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action-angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the canonical transformations in the quantum phase space. The representations of the Wigner function in the generalized action-angle unitary operator pair for certain Hamiltonian systems with dynamical symmetry is examined. This generalized canonical formalism is applied to the quantum harmonic oscillator to examine the properties of the unitary quantum phase operator as well as the action-angle Wigner function.
The phase diagram of the two-dimensional biaxially next-nearest-neighbour king
(BNNNI) model is obtained using the real-space renormalisation group method with the
mean-field approximation. The transitions between disordered and commensurate phases
are observed to be single transitions for the ferromagnetic phase at intermediate and high
temperatures and for the antiphase structure at high temperatures. However, within the
approximation, successive transitions are observed between the antiphase structure and
disordered phase at intermediate and low temperatures.
In this second of four papers on the eponymous topic, pointwise convergence of a discrete state function to a continuum state function is shown to imply the algebraic criterion for convergence that was introduced in the prequel. As examples (and as a prerequisite for the sequels), the normal approximation theorem and the convergence of the Kravchuk functions to the Hermite-Gaussians are expressed in terms of the algebraic notion of convergence.
Nonadmissible, weakly admissible and admissible cyclic representations and other algebraic properties of the generalized homographic oscillator (GHO) are studied in detail. For certain ranges of the deformation parameter, it is shown that this new deformed oscillator is a prototype cyclic oscillator endowed with a non-negative (admissible) spectrum. By changing the deformation parameter, the cyclic spectrum can be tuned to have an arbitrarily large period. It is shown that the standard harmonic oscillator is recovered at the nonadmissible infinite-period limit of the GHO. With these properties, the GHO provides a concrete example of an oscillator rich in a variety of cyclic representations. It is well known that such representations are of relevance to the proper algebraic formulation of the quantum-phase operator. Using a general scheme, it is shown that admissible cyclic algebras permit a well-defined Hermitian phase operator of which properties are studied in detail at finite periods as well as at the infinite-period limit. Fujikawas index approach is applied to admissible cyclic representations and in particular to the phase operator in such algebras. Using the specific example of GHO it is confirmed that the infinite-period limit is distinctively singular. The connection with the Pegg-Barnett phase formalism is established in this singular limit as the period of the cyclic representations tends to infinity The singular behaviour at this limit identifies the algebraic problems, in a concrete example, emerging in the formulation of a standard quantum harmonic-oscillator phase operator.
The problem of constructing boundary conditions for nonlinear equations compatible with higher symmetries is considered. In particular, this problem is discussed for the sine-Gordon, Jiber-Shabat, Liouville and KdV equations. New results are obtained for the last two ones. The boundary condition for the KdV contains two arbitrary constants. The substitution u = qx maps it onto the boundary condition with linear dependence on t for the potentiated KdV.
Non-autonomous Svinolupov-Jordan KdV systems are considered. The integrability criteria of such systems are associated with the existence of recursion operators. A new non-autonomous KdV system and its recursion operator is obtained for all N. The examples for N = 2 and 3 are studied in detail. Some possible transformations which map some systems to autonomous ones are also discussed.
We construct a one-dimensional model with two spins and a unique ground state having infinitely many extreme limit Gibbs states. This model is closely related to uniqueness conditions in one-dimensional models.
Certain solutions to Harpers equation are discrete analogues of (and approximations to) the Hermite-Gaussian functions. They are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator, and they lead to a definition of a discrete fractional Fourier transform (FT). The discrete fractional FT is essentially the time-evolution operator of the discrete harmonic oscillator.
A criterion for the uniqueness of limiting Gibbs states in classical models with unique ground states is formulated. Various applications of this criterion formulated in the terminology of percolation theory are discussed.
The algorithmic method introduced by Fokas and Ablowitz to investigate the transformation properties of Painlevé equations is used to obtain a one-to-one correspondence between the Painlevé I, II and III equations and certain second-order second degree equations of Painlevé type.
Invariant distance on the non-commutative C*-algebra C(SUq(2)) is constructed and the generalized functions on the q-symmetric space M = SUq(2)/U(1) are introduced. The Green function and the kernel on M are derived. A path integration is formulated. The Green function for the free massive scalar field on the non-commutative Einstein space R-1 x M is presented.
DOI : 10.81043/aperta.103629
Sayi :31 issue :27 Sayfa :5741-5754
We have studied bound states of the Schrodinger equation for an attractive potential with any finite number (P) of Dirac delta-functions in R-n where n = 1, 2, 3..... The potential is radially symmetric for n greater than or equal to 2 and is given as V(r) = -(h2)/(2m) Sigma(i=1)(P) sigma(i)delta(r - r(i)) where sigma(i) > 0, r(1) < r(2) < (...) 2l+n-2 and none otherwise. Wehave also proven that there are at most P positive roots for the equation X-22(k) = 0 where X = ((X21) (X11) (X22) (X12)) = MpMp-1...M-1 and M-i is an element of SL (2, R) are the particular where X = G21 X 22 transfer matrices mentioned above.
DOI : 10.81043/aperta.97401
Sayi :36 issue :26 Sayfa :7449-7459
We show that Plebanskis second heavenly equation, when written as a first-order nonlinear evolutionary system, admits multi-Hamiltonian structure. Therefore by Magris theorem it is a completely integrable system. Thus it is an example of a completely integrable system in four dimensions.
DOI : 10.1088/0305-4470/38/39/012
Sayi :38 issue :39 Sayfa :8473-8485
We extend the Mason-Newman Lax pair for the elliptic complex Monge-Ampere equation so that this equation itself emerges as an algebraic consequence. We regard the function in the extended Lax equations as a complex potential. Their differential compatibility condition coincides with the determining equation for the symmetries of the complex Monge-Ampere equation. We shall identify the real and imaginary parts of the potential, which we call partner symmetries, with the translational and dilatational symmetry characteristics, respectively. Then we choose the dilatational symmetry characteristic as the new unknown replacing the Kahler potential. This directly leads to a Legendre transformation. Studying the integrability conditions of the Legendre-transformed system we arrive at a set of linear equations satisfied by a single real potential. This enables us to construct non-invariant solutions of the Legendre transform of the complex Monge-Ampere equation. Using these solutions we obtained explicit Legendre-transformed hyper-Kahler metrics with a anti-self-dual Riemann curvature 2-form that admit no Killing vectors. They satisfy the Einstein field equations with Euclidean signature. We give the detailed derivation of the solution announced earlier and present a new solution with an added parameter. We compare our method of partner symmetries for finding non-invariant solutions to that of Dunajski and Mason who use hidden symmetries for the same purpose.
DOI : 10.81043/aperta.96467
Sayi :36 issue :39 Sayfa :10023-10037
Gurses integrability test consists of the compatibility of the linearized equation with an eigenvalue equation and leads to the recursion operator. This test is applied to quasilinear fifth-order equations and the same classification as the formal symmetry method of Mikhailov et al is obtained. The same classification for polynomial equations is obtained using Fokas test, i.e. the existence of one higher-order symmetry. It is shown that the recursion operators of a specific form can be constructed using symmetries and conserved covariants and the recursion operators for polynomial equations are obtained with this method.
DOI : 10.81043/aperta.103491
Sayi :26 issue :24 Sayfa :7511-7519
Nambus construction of multi-linear brackets for super-integrable systems can be thought of as degenerate Poisson brackets with a maximal set of Casimirs in their kernel. By introducing privileged coordinates in phase space these degenerate Poisson brackets are brought to the form of Heisenbergs equations. We propose a definition for constructing quantum operators for classical functions, which enables us to turn the maximally degenerate Poisson brackets into operators. They pose a set of eigenvalue problems for a new state vector. The requirement of the single-valuedness of this eigenfunction leads to quantization. The example of the harmonic oscillator is used to illustrate this general procedure for quantizing a class of maximally super-integrable systems.
DOI : 10.81043/aperta.94017
Sayi :36 issue :27 Sayfa :7559-7567
We extend our method of partner symmetries to the hyperbolic complex Monge-Ampere equation and the second heavenly equation of Plebanski. We show the existence of partner symmetries and derive the relations between them. For certain simple choices of partner symmetries the resulting differential constraints together with the original heavenly equations are transformed to systems of linear equations by an appropriate Legendre transformation. The solutions of these linear equations are generically non-invariant. As a consequence we obtain explicitly new classes of heavenly metrics without Killing vectors.
DOI : 10.1088/0305-4470/37/30/010
Sayi :37 issue :30 Sayfa :7527-7545