Relation of the nonlinear Heisenberg algebras in two dimensions with linear ones

In this paper, we discuss the relation of the nonlinear Heisenberg algebras in two dimensions with linear ones following the Nowicki and Tkachuk’s approach for one-dimensional case. For one-dimensional harmonic oscillator, we obtain the solution explicitly. For the nonlinear Heisenberg algebras in two dimensions, we intro-duce two generators to transform this algebra into the linear one. For the linear version of the nonlinear Heisenberg algebras in two dimensions, we obtain the eigenfunction for the position and angular momentum operator and solve the harmonic oscillator problem in two dimensions.


Introduction
The first form of the Heisenberg algebra giving the minimal length uncertainty was first introduced by Kempf, Mangano, and Mann [1] in the following form: which suggests the existence of the fundamental minimal length In this direction, much development has been accomplished in order to study the effect of minimal length on the quantum physical systems as well as on the classical ones, but only a few problems are shown to be solved exactly. They are one-dimensional harmonic oscillator with minimal length uncertainty in position [1,2] and also with minimal length uncertainty in position and momentum [3,4], D-dimensional isotropic harmonic oscillator [5,6], three-dimensional Dirac oscillator [7], (1 ? 1)-dimensional Dirac oscillator within Lorentz-covariant deformed algebra [8], one-dimensional Coulomb problem [9], and the singular inverse square potential with a minimal length [10,11]. Three-dimensional Coulomb problem with deformed Heisenberg algebra was solved within the perturbation theory [12][13][14][15].
In this paper, we discuss the relation of the nonlinear Heisenberg algebras in two dimensions with linear ones following the Nowicki and Tkachuk's approach [16] for one-dimensional case. For one-dimensional harmonic oscillator, we obtain the solution explicitly. For the nonlinear Heisenberg algebras in two dimensions, we introduce two generators to transform this algebra into the linear one. For the linear version of the nonlinear Heisenberg algebras in two dimensions, we obtain the eigenfunction for the position and angular momentum operator and solve the harmonic oscillator problem in two dimensions.
One-dimensional deformed nonlinear Heisenberg algebra It means that the space has the same properties in two opposite directions. The momentum representation reads and acts on the square integrable functions /ðpÞ 2 L 2 ðÀa; a; f Þ; ða 1Þ where the norm of / is given by More general momentum representation is given in ''Appendix''. Nowicki and Tkachuk extended the algebra (3) into the three generator algebra by one additional generator F ¼ f ðpÞ and obtain a concrete form of f as follows: Now let us consider the harmonic oscillator with the hamiltonian The Schrödinger equation reads Let us change the variable like ffiffiffi b p p ¼ sinh n, which means that n goes to zero when b approaches zero. Then, Eq. (9) becomes When we consider the small value of n, we have Replacing n 2 ¼ z, we get If we set wðzÞ ¼ e À z 2lw yðzÞ; we get This equation can be solved using the Frobenius method. If we adopt yðzÞ ¼ X 1 n¼0 a n z nþk ð16Þ and insert it into Eq. (15), we have From the characteristic Eq. (17), we have two values of k: For k ¼ 0, we have and for k ¼ 1=2, we have where Kummer's function is defined as ðaÞ n z n n!ðbÞ n ð21Þ and is the rising factorial.

Linearization of a two-dimensional deformed nonlinear Heisenberg algebra
Let us consider a two-dimensional deformed nonlinear Heisenberg algebra with deformation function f(P): where f ðPÞ ¼ f ðP 1 ; P 2 Þ is an positive function obeying It means that the space has the same properties in two opposite X 1 -and X 2 -directions. From the commutation relations (23), we have the momentum representation of the operators as follows: This fixes the remaining commutation relations, so the full algebra is then given by Indeed one can easily check that the relation (26) obeys the Jacobi identity. Now we assume that the operators X 1 ; X 2 ; P 1 and P 2 act on the square integrable functions /ðp 1 ; p 2 Þ 2 L 2 ðÀa; a; f Þ; ða 1Þ where the norm of / is given by For the self-adjointness of X 1 and X 2 ; we have /ðÀa; p 2 Þ ¼ AE/ða; p 2 Þ and /ðp 1 ; ÀaÞ ¼ AE/ðp 1 ; aÞ: Now we extend this algebra by two additional operators Thus, the extended algebra E is generated by the six generators. Using representation (25), one can easily find We require that both fX 1 ; P 1 ; Fg and fX 2 ; P 2 ; Fg form a subalgebras of E. Then, one can put where a; b; c; a 0 ; b 0 ; c 0 are real parameters. Note that the linear combination in the right-hand side of (30) (or 31) does not contain X 1 (or X 2 ) because f o P 1 f (or f o P 2 f ) is a function of P 1 ; P 2 only. Using Eq. (24) and changing P 1 , P 2 into ÀP 1 , ÀP 2 , respectively, one find Comparing Eqs. (30) (31) or with Eqs. (32) (or 33), one can see From now on we will restrict our concern to the case of Thus, the algebra E reads This is Lie algebra. One can find the Casimir operator (invariant) for this algebra commuting with all elements of the algebra. Now let us define the following operators: Then, the algebra E can be written as The algebra E possess some subalgebras: 1. subalgebra E 1 generated by A 1 ; A 2 ; A 3 2. subalgebra E 2 generated by A 2 ; A 4 ; A 6 3. subalgebra E 3 generated by A 1 ; A 5 ; A 6 4. subalgebra E 4 generated by A 3 ; A 4 ; A 5 : It is convenient to use two pairs of commuting hermitian operators P AE and Q AE defined as follows: J Theor Appl Phys (2015) 9:201-206 203 Indeed one can easily check that In this case, we have the algebra A generated by P AE ; Q AE ; A 3 ; A 4 ; A 5 . Algebra E has six generators, while A has seven ones. It seems to be nonsense because two algebras should be isomorphic. To cure this problem, let us consider the inverse relations of Eq. (40) We know that A 2 can be expressed in terms of both P AE and Q AE , which gives a constraint This constraint decreases the number of generators of the algebra A , so two algebras are isomorphic. Besides Eq. (41), the remaining commutation relations of the algebra A are ½A 3 ; P AE ¼ AEiP AE ; ½P þ ; P À ¼ 0 The Casimir operator is then given by This algebra has two subalgebras (namely ISO(1, 1)) generated by A 3 ; P AE and A 4 ; Q AE . Two sets of the ladder operators can be expressed in terms of the momentum operators as follows: If we set mP 1 ¼ sinh n cos g; mP 2 ¼ sinh n sin g; we can express P AE and Q AE as The n and g can be expressed in terms of the momentum operators as Then, A 3 ; A 4 ; A 5 can be written as Eigenvectors of the position operator and angular momentum operator In this section, we discuss the eigenvalue equation for the position operator and angular momentum operator. The eigenvalues for the position operators read we have If we adopt we have /ðn; gÞ ¼ ½sinh nðl 1 cos g þ l 2 sin gÞ À1=m : ð56Þ Now let us investigate the eigenvalue of the angular momentum operator L defined as The eigenvalue equation reads Solving Eq. (58), we have Two-dimensional Harmonic oscillator Now let us consider the isotropic harmonic Hamiltonian Using Eq. (47) and Eq. (50), we obtain the expression of H: Then, the Schrödinger equation reads À lw 2 m 2 2 ðo 2 n þcothno n þcoth 2 no 2 g Þþ 1 2lm 2 sinh 2 n ! w ¼ Ew: If we set w ¼ RðnÞe img , we have R 00 þ coth nR 0 þ ð À A sinh 2 n À m 2 coth 2 nÞR ¼ 0; ð63Þ Now consider the case that n is sufficiently small. In this case, we have sinh n % n; coth n % 1 n : Then, Eq. (63) reduces to If we set n 2 ¼ lwm 2 z, we have Solving Eq. (65), we get and E nm ¼ hwð2n þ m þ 1Þ; n ¼ 0; 1; . . .; m ¼ 0; 1; . . .; n: The ground state energy is given by which corresponds to the classical result.

Conclusion
Recently, Nowicki and Tkachuk [16] considered a onedimensional deformed nonlinear Heisenberg algebra with function of deformation f(P), namely ½X; P ¼ if ðPÞ. They discussed the relation of the nonlinear Heisenberg algebras with linear ones. We introduced the variable n ¼ sinh À1 ð ffiffiffi b p pÞ to solve the one-dimensional harmonic problem for the small value of n. We extended Nowicki and Tkachuk's work to the two-dimensional case. We obtained the linearized algebra E from the two-dimensional nonlinear Heisenberg algebras by adding two generators. Introducing two variables n ¼ sinh À1 ðm 2 p 2 1 þ m 2 p 2 2 Þ g ¼ tan À1 p 2 p 1 ; we expressed all generators of the algebra E in terms of n and g. We also solved the eigenvalue equation for the position and angular momentum operator. Finally, we discussed twodimensional isotropic harmonic oscillator problem and obtained the corresponding energy eigenvalue and wave function for the small value of n. We found that the ground state energy for this model corresponds to the classical result.