Comment on "A new exactly solvable quantum model in $N$ dimensions" [Phys. Lett. A 375(2011)1431, arXiv:1007.1335]

Commen t on ”A new exactly solv able quan tum mo del in N dimensions” [Ph ys. Lett. A 375(2011)1431] B. L. Moreno Ley a nd Shi-Hai Dong ∗ Departamento de F ´ ısica, Escuela Sup erior de F ´ ısica y Ma tem´ aticas, Ins tituto Polit ´ ecnico Nacional, Edificio 9 , Unidad Pr ofesional Adolfo L´ opez Mateos, Mexic o D. F. 077 3 8, Mexico Abstract W e pin p oin t that the w ork ab out ”a new exactly solv able quan tum mo del in N dimensions” by Ballesteros et al. [Phys. Lett. A 375 (2011) 1431] is not a new exactly solv a ble q u an tum mo del since the fla w of the p osition-dep enden t mass Hamiltonian prop osed by th em mak es it less v a luable in physic s. Keywor ds : P osition-dep endent mass; Arbitrary dimension N ; Solv able quantum mo del In r ecen t w ork [1], t he authors Ballesteros et al. claimed that they ha v e found a new exactly solv able quan tum mo del in N dimensions give n b y H = − ¯ h 2 2(1 + λr 2 ) ∇ 2 + ω 2 r 2 2(1 + λr 2 ) , (1) where w e prefer to use v ariable r instead o f original one q for conv enience. They found that the sp ectrum of this mo del is sho wn to b e h ydrogen-lik e (should b e harmonic oscillator-lik e), and their eigen v alues a nd eigenfunctions are explicitly obta ined b y deforming appropriately the symmetry prop erties of the N - dimensional harmonic oscillator. It should b e p oin ted out that suc h treatmen t approach is incorrect since the kinetic energy term should b e defined as [2] ∇ N 1 m ( r ) ∇ N ψ ( r ) = ∇ N 1 m ( r ) ! · [ ∇ N ψ ( r )] + 1 m ( r ) ∇ 2 N ψ ( r ) . (2) ∗ Corresp onding author. E-mail address: dongsh2@yaho o.com; T el:+ 52-55-5 7296000 ex t 552 55; F ax: +52-5 5-57296 000 ext 55 015. 1 F or N - dimens ional spherical symmetry , w e tak e the w a v efunctions ψ ( r ) as follow s [3]: ψ ( r ) = r − ( N − 1) / 2 R ( r ) Y l l N − 2 ,...,l 1 ( ˆ x ) . (3) Substituting this in to the p osition-dep enden t effectiv e mass Schr¨ odinger equation ∇ N 1 m ( r ) ∇ N ψ ( r ) ! + 2[ E − V ( r )] ψ ( r ) = 0 , (4) allo ws us to obtain the following r adial p osition-dep enden t mass Sc hr¨ odinger equation in arbitrary dimensions ( d 2 dr 2 + m ′ ( r ) m ( r ) N − 1 2 r − d dr ! − η 2 − 1 / 4 r 2 + 2 m ( r )[ E − V ( r )] ) R ( r ) = 0 , (5) where m ( r ) = (1 + λr 2 ) , m ′ ( r ) = dm ( r ) /dr a nd η = | l − 1 + N/ 2 | . Since the op erator ∇ N do es not comm utate with the p osition-dep enden t mass m ( r ), then this system do es not exist exact solutio ns at all. This can also b e pro v ed unsolv able to Eq.(5) if substituting the p osition-dep enden t mass m ( r ) in to it. On the other hand, the c hoice of the p osition-dependent mass m ( r ) has no phy sical meaning since the mass m ( r ) go es to infinit y when r → ∞ . Moreov er, it is sho wn from Eq.(1) that the po sition-dependen t mass m ( r ) in kinetic term is equal to (1 + λr 2 ), but it w as taken a s 1 / (1 + λr 2 ) for the harmonic oscillator term. Accordingly , the wrong expression of the Ha milto nian in p osition-dep enden t mass Sc hr¨ odinger equation in a rbitrary dimensions N , the fla w of the c hosen p osition-dep enden t mass m ( r ) as w ell as its inconsistence b et w een the kinetic term and the harmonic o scillator term make it less v a luable in phy sics. Ac kno wledgmen ts : This w ork w as support ed partia lly b y 20110491-SIP-IPN, COF AA-IPN, Mexico. References [1] A. Ballesteros, A. Enciso, F. J. Herranz, O. Ragnisco, D. Riglioni, Ph ys. Lett. A 375(2011 ) 1 431. [2] G. Chen, Ph ys. Lett. A 329(2 004)22. [3] S. H. Dong and Z. Q. Ma, Ph ys. Rev. A 65(200 2)042717. 2