Hierarchy Of Hamiltonians Research Articles

Fractional supersymmetry and hierarchy of shape invariant potentials

Fractional supersymmetric quantum mechanics is developed from a generalized Weyl-Heisenberg algebra. The Hamiltonian and the supercharges of fractional supersymmetric dynamical systems are built in terms of the generators of this algebra. The Hamiltonian gives rise to a hierarchy of isospectral Hamiltonians. Special cases of the algebra lead to dynamical systems for which the isospectral supersymmetric partner Hamiltonians are connected by a (translational or cyclic) shape invariance condition.

Supersymmetric solutions of PT-/non-PT-symmetric and non-Hermitian screened Coulomb potential via Hamiltonian hierarchy inspired variational method

The supersymmetric solutions of PT -symmetric and Hermitian/non-Hermitian forms of quantum systems are obtained by solving the Schrodinger equation for the Exponential-Cosine Screened Coulomb potential. The Hamiltonian hierarchy inspired variational method is used to obtain the approximate energy eigenvalues and corresponding wave functions.

SUSY approach to Pauli Hamiltonians with an axial symmetry

A two-dimensional Pauli Hamiltonian describing the interaction of a neutral spin-1/2 particle with a magnetic field having axial and second-order symmetries is considered. After separation of variables, the one-dimensional matrix Hamiltonian is analysed from the point of view of supersymmetric quantum mechanics. Attention is paid to the discrete symmetries of the Hamiltonian and also to the Hamiltonian hierarchies generated by intertwining operators. The spectrum is studied by means of the associated matrix shape invariance. The relation between the intertwining operators and the second-order symmetries is established, and the full set of ladder operators that complete the dynamical algebra is constructed.

${\mathcal{PT}}$-SYMMETRIC SUPERSYMMETRY IN A SOLVABLE SHORT-RANGE MODEL

The simplest purely imaginary and piecewise constant [Formula: see text]-symmetric potential located inside a larger box is studied. Unless its strength exceeds a certain critical value, all the spectrum of its bound states remains real and discrete. We interpret such a model as an initial element of the generalized non-Hermitian Witten's hierarchy of solvable Hamiltonians and construct its first supersymmetric partner in closed form.

Superintegrable quantum u(3) systems and higher rank factorizations

A class of two-dimensional superintegrable systems on a constant curvature surface is considered as the natural generalization of some well known one-dimensional factorized systems. By using standard methods to find the shape-invariant intertwining operators we arrive at a so(6) dynamical algebra and its Hamiltonian hierarchies. We pay attention to those associated to certain unitary irreducible representations that can be displayed by means of three-dimensional polyhedral lattices. We also discuss the role of superpotentials in this new context.

Supersymmetric Solutions of PT-/non-PT-symmetric and Non-Hermitian Central Potentials via Hamiltonian Hierarchy Method

The supersymmetric solutions of PT-/non-PT-symmetric and non-Hermitian deformed Morse and Pöschl-Teller potentials are obtained by solving the Schrödinger equation. The Hamiltonian hierarchy method is used to get the real energy eigenvalues and corresponding eigenfunctions.

A generalized MKdV hierarchy, tri-Hamiltonian structure, higher-order binary constrained flows and its integrable couplings system

A subalgebra of higher-dimension loop algebra A 2 ˜ is constructed, from which a new 3 × 3 isospectral problem is designed. By making use of Tu’s scheme, an integrable Hamiltonian hierarchy of equations in the sense of Liouville is obtained, which possesses tri-Hamiltonian structure. As reduction cases of the hierarchy presented in this paper, the generalized MKdV equation is engendered. By establishing binary symmetric constraints, three constrained flows of the hierarchy are presented, which are then reduced to Hamiltonian systems. Finally, an integrable coupling system is obtained by constructing a high-dimension loop algebra.

$$\mathcal{P}\mathcal{T}$$ -supersymmetric partner of a short-range square well

In a box of size L, a spatially antisymmetric square-well potential of a purely imaginary strength ig and size l < L is interpreted as an initial element of the SUSY hierarchy of solvable Hamiltonians, the energies of which are all real for g < g c (l). The first partner potential is constructed in closed form and discussed.

A simple method for generating integrable hierarchies with multi-potential functions

Abstract A simple efficient method for obtaining integrable hierarchies of evolution equations with multi-potential functions is proposed. By making use of the concept of cycled numbers, a new loop algebra A ∼ 1 ∗ is constructed. Taking account of applicable convenience, a few subalgebras of the loop algebra A ∼ 1 ∗ are presented again, from one of them, two new integrable Hamiltonian hierarchies with multi-potential functions are obtained. Finally, a 3 × 3 loop algebra G ∼ with five dimensions is established, from which a coupled integrable coupling system of one of the above integrable hierarchies is derived. The approach presented in this paper can be used generally.

SUPERSYMMETRIC SOLUTION OF PT-/NON-PT-SYMMETRIC AND NON-HERMITIAN MORSE POTENTIAL VIA HAMILTONIAN HIERARCHY METHOD

Supersymmetric solution of PT-/non-PT-symmetric and non-Hermitian Morse potential is studied to get real and complex-valued energy eigenvalues and the corresponding wave functions. Hamiltonian hierarchy method is used in the calculations.

On Hamiltonian Hierarchies Associated with Hyperbolic Euler Equations

The canonical coordinate-momenta formalism with internal degrees of freedom for hyperbolic Euler–Lagrange equations is applied to the study of flows of Hamiltonian symmetries. The differential constraint between coordinates and momenta supplies the Hamiltonian structure for the initial hyperbolic system and endows the evolution of coordinates and induced evolution of momenta with the Hamiltonian operators which are mutually inverse. Two illustrative examples are given: the bi-Hamiltonian Korteweg–de Vries hierarchy of Noether symmetries of the wave equation is connected with the multicomponent modified KdV analogs associated with the two-dimensional Toda equations uxy e exp(Ku).

A subalgebra of loop algebra 2 and its applications

A subalgebra of loop algebra A2 and its expanding loop algebra are constructed. It follows that both resulting integrable Hamiltonian hierarchies are obtained. As a reduction case of the first hierarchy, a generalized nonlinear coupled Schrodinger equation, the standard heat-conduction and a formalism of the well known Ablowitz, Kaup, Newell and Segur hierarchy are given, respectively. As a reduction case of the second hierarchy, the nonlinear Schrodinger and modified Korteweg de Vries hierarchy and a new integrable system are presented. Especially, a coupled generalized Burgers equation is generated.

A new subalgebra of the Lie algebra A2 and two types of integrable Hamiltonian hierarchies, expanding integrable models

A new subalgebra G of the Lie algebra A 2 is first constructed. Then two loop algebra G 1 , G 2 are presented in terms of different definitions of gradations. Using G 1 , G 2 designs two isospectral problems, respectively. Again utilizing Tu-pattern obtains two types of various integrable Hamiltonian hierarchies of evolution equations. As reduction cases, the well-known Schrödinger equation and MKdV equation are obtained. At last, we turn the subalgebras G 1 , G 2 of the loop algebra A 2 into equivalent subalgebras of the loop algebra A 1 by making a suitable linear transformation so that the two types of 5-dimensional loop algebras are constructed. Two kinds of integrable couplings of the obtained hierarchies are showed. Specially, the integrable couplings of Schrödinger equation and MKdV equation are obtained, respectively.

Variational method for excited states from supersymmetric techniques

We suggest a method for constructing trial eigenfunctions for excited states to be used in the variational method. This method is a generalization of the one that uses a superpotential to obtain the trial functions for the ground state. The construction of an effective hierarchy of Hamiltonians is used to determine excited variational energies. The first four eigenvalues for a quartic double-well potential are calculated for several values of the potential parameter. The results are in very good agreement with the eigenvalues obtained by numerical integration. PACS Nos.: 11.30.Pb, 03.65.Ge

An integrable Hamiltonian hierarchy, a high-dimensional loop algebra and associated integrable coupling system

A subalgebra of loop algebra A2 is established. Therefore, a new isospectral problem is designed. By making use of Tu's scheme, a new integrable system is obtained, which possesses bi-Hamiltonian structure. As its reductions, a formalism similar to the well-known Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and a generalized standard form of the Schrodinger equation are presented. In addition, in order for a kind of expanding integrable system to be obtained, a proper algebraic transformation is supplied to change loop algebra A2 into loop algebra A1. Furthermore, a high-dimensional loop algebra is constructed, which is different from any previous one. An integrable coupling of the system obtained is given. Finally, the Hamiltonian form of a binary symmetric constrained flow of the system obtained is presented.

A type of new integrable Hamiltonian hierarchy and expanding integrable model of its reduced integrable system associated with a new loop algebra

Designing a new isospectral problem from a loop algebra A ̃ 1 . It follows that a type of new integrable hierarchy of evolution equations, possessing bi-Hamiltonian structure, is obtained. As a reduction case, the well-known TC hierarchy is presented. Then a new seven-dimensional loop algebra G is constructed. From which, expanding integrable model of the TC hierarchy is obtained.

An integrable Hamiltonian hierarchy and its constrained flows with generalized Hamiltonian regular representations, as well as its expanding integrable system

A new subalgebra of loop algebra A ̃ 2 is first constructed. It follows that an isospectral problem is established. Using Tu-pattern gives rise to a new integrable hierarchy, which possesses bi-Hamiltonian structure. As its reduction cases, the well-known standard Schrodinger equation and MKdV equation are presented, respectively. Furthermore, by making use of bi-symmetry constraints, generalized Hamiltonian regular representations for the hierarchy are obtained. At last, we obtain an expanding integrable system of this hierarchy by applying a scalar transformation between two isospectral problems and constructing a five-dimensional loop algebra G̃. In particular, the expanding integrable models of Schrodinger equation and MKdV equation are presented, respectively.

Two types of new Lie algebras and corresponding hierarchies of evolution equations

An extension of Lie algebra A n−1 is proposed. As special cases, two new loop algebras are constructed, respectively. It follows that two types of new integrable Hamiltonian hierarchies are engendered. As their reduction cases, generalized nonlinear Schrödinger equations, coupled Fisher equations, the standard heat-conduction equation are obtained, respectively. This method proposed in this Letter can be used generally.

Liouville integrable N-Hamiltonian structures, involutive solutions and separation of variables associated with Kaup–Newell hierarchy

In this paper, firstly, N-Hamiltonian structures for Kaup–Newell (KN) hierarchy is presented from an isospectral problem via the N pairs of Hamiltonian operators (Lenard operators) and it is shown that the Hamiltonian KN system is Liouville integrable. Secondly, a Lax representation in terms of 2×2 matrices for the completely integrable finite-dimensional Hamiltonian system (CIFHS) ( H) is produced through the non-linearization procedure for the KN hierarchy. By making use of the known r-matrix and matrix trace equality, a system of finite-dimensional involutive functions F m ( m=0,1,2,…, F 0= H) which guarantees the integrability of Hamiltonian system ( H), and the Lax representations in terms of 2×2 matrices for the whole Hamiltonian hierarchies ( F m ) ( m=0,1,2,…) are obtained. Moreover, the involutive solutions of the KN hierarchy are given. Finally, it is found that the Hamilton–Jacobi equation for the Hamiltonian system ( H) can be separated under a group of new coordinates introduced by the 2×2 Lax matrix. In addition, the separation equations are given.

The Hierarchy of Hamiltonians for a Restricted Class of Natanzon Potentials

The restricted class of Natanzon potentials with two free parameters is studied within the context of Supersymmetric Quantum Mechanics. The hierarchy of Hamiltonians and a general form for the superpotential is presented. The first members of the superfamily are explicitly evaluated.