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Advances in Mathematical Physics
Volumeย 2011ย (2011), Article IDย 750168, 20 pages
Research Article

Study of the Generalized Quantum Isotonic Nonlinear Oscillator Potential

1Department of Mathematics and Statistics, University of Prince Edward Island, 550 University Avenue, Charlottetown, PE, C1A 4P3, Canada
2Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulevard West, Montrรฉal, QC, H3G 1M8, Canada
3Department of Physics, Faculty of Arts and Sciences, Gazi University, 06500 Ankara, Turkey

Received 11 March 2011; Accepted 13 April 2011

Academic Editor: B. G.ย Konopelchenko

Copyright ยฉ 2011 Nasser Saad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study the generalized quantum isotonic oscillator Hamiltonian given by ๐ป=โˆ’๐‘‘2/๐‘‘๐‘Ÿ2+๐‘™(๐‘™+1)/๐‘Ÿ2+๐‘ค2๐‘Ÿ2+2๐‘”(๐‘Ÿ2โˆ’๐‘Ž2)/(๐‘Ÿ2+๐‘Ž2)2, ๐‘”>0. Two approaches are explored. A method for finding the quasipolynomial solutions is presented, and explicit expressions for these polynomials are given, along with the conditions on the potential parameters. By using the asymptotic iteration method, we show how the eigenvalues of this Hamiltonian for arbitrary values of the parameters ๐‘”, ๐‘ค, and ๐‘Ž may be found to high accuracy.

1. Introduction

Recently, Cariรฑena et al. [1] studied a quantum nonlinear oscillator potential whose Schrรถdinger equation reads๎ƒฌโˆ’๐‘‘2๐‘‘๐‘ฅ2+๐‘ฅ2+82๐‘ฅ2โˆ’1๎€ท2๐‘ฅ2๎€ธ+12๎ƒญ๐œ“๐‘›(๐‘ฅ)=๐ธ๐‘›๐œ“(๐‘ฅ).(1.1) The interest in this problem came from the fact that it is exactly solvable in a sense that the exact eigenenergies and eigenfunctions can be obtained explicitly. Indeed, Cariรฑena et al. [1] were able to show that๐œ“๐‘›๐‘ƒ(๐‘ฅ)=๐‘›(๐‘ฅ)๎€ท2๐‘ฅ2๎€ธ๐‘’+1โˆ’๐‘ฅ2/2,๐ธ๐‘›=โˆ’3+2๐‘›,๐‘›=0,3,4,5,โ€ฆ,(1.2) where the polynomials factors ๐‘ƒ๐‘›(๐‘ฅ) are related to the Hermite polynomials by means of๐‘ƒ๐‘›๎ƒฏ๐ป(๐‘ฅ)=1,if๐‘›=0,๐‘›(๐‘ฅ)+4๐‘›๐ป๐‘›โˆ’2(๐‘ฅ)+4๐‘›(๐‘›โˆ’3)๐ป๐‘›โˆ’4(๐‘ฅ),if๐‘›=3,4,5,โ€ฆ.(1.3) In a more recent work, Fellows and Smith [2] showed that the potential ๐‘‰(๐‘ฅ)=๐‘ฅ2+8(2๐‘ฅ2โˆ’1)/(2๐‘ฅ2+1)2 as well as, for certain values of the parameters ๐‘ค, ๐‘”, and ๐‘Ž, the potential ๐‘‰(๐‘ฅ)=๐‘ค2๐‘ฅ2+2๐‘”(๐‘ฅ2โˆ’๐‘Ž2)/(๐‘ฅ2+๐‘Ž2)2 of the Schrรถdinger equation๎ƒฌโˆ’๐‘‘2๐‘‘๐‘ฅ2+๐‘ค2๐‘ฅ2๐‘ฅ+2๐‘”2โˆ’๐‘Ž2๎€ท๐‘ฅ2+๐‘Ž2๎€ธ2๎ƒญ๐œ“๐‘›(๐‘ฅ)=2๐ธ๐‘›๐œ“(๐‘ฅ)(1.4) are indeed supersymmetric partners of the harmonic oscillator potential. Using the supersymmetric approach, the authors were able to construct an infinite set of exact soluble potentials, along with their eigenfunctions and eigenvalues. Very recently, Sesma [3], using a Mรถbius transformation, was able to transform (1.4) into a confluent Heun equation [4], and thereby obtain an efficient algorithm to solve the Schrรถdinger equation (1.4) numerically.

The purpose of the present work is to provide a detailed solution, by means of the quasipolynomial solutions and the application of the asymptotic iteration method [5โ€“8], for the Schrรถdinger equation๎ƒฌโˆ’๐‘‘2๐‘‘๐‘Ÿ2+๐‘™(๐‘™+1)๐‘Ÿ2+๐‘ค2๐‘Ÿ2๐‘Ÿ+2๐‘”2โˆ’๐‘Ž2๎€ท๐‘Ÿ2+๐‘Ž2๎€ธ2๎ƒญ๐œ“(๐‘Ÿ)=2๐ธ๐œ“(๐‘Ÿ),(1.5) where ๐‘™ is the angular momentum number ๐‘™=โˆ’1,0,1,โ€ฆ. Our results show that the quasiexact solutions of Sesma [3] as well the results of Cariรฑena et al. [1] follow as special cases of our general approach. The present paper is organized as follows. In Sectionโ€‰โ€‰2, some preliminary analysis of the Schrรถdinger equation (1.5) is presented. A general approach for finding polynomial solutions of (1.5), for certain values of parameters ๐‘ค and ๐‘”, is presented and is based on a recent work of Ciftci et al. [6] for solving the second-order linear differential equation๎ƒฉ3๎“๐‘–=0๐‘Ž3,๐‘–๐‘ฅ๐‘–๎ƒช๐‘ฆ๎…ž๎…ž+๎ƒฉ2๎“๐‘–=0๐‘Ž2,๐‘–๐‘ฅ๐‘–๎ƒช๐‘ฆ๎…žโˆ’๎ƒฉ1๎“๐‘–=0๐œ1,๐‘–๐‘ฅ๐‘–๎ƒช๐‘ฆ=0.(1.6) More general quasiexact solutions, including the results of Sesma [3], are discussed in Section 3. Unrestricted solutions of (1.5) based on the asymptotic iteration method are discussed in Section 4.

2. Generalized Quantum Isotonic Oscillatorโ€”Preliminary Results

A simple scaling argument, using ๐‘Ÿ=๐‘Ž2๐‘ฅ, allows us to write (1.5) as๎ƒฌโˆ’๐‘‘2๐‘‘๐‘ฅ2+๐‘™(๐‘™+1)๐‘ฅ2+๎€ท๐‘ค๐‘Ž2๎€ธ2๐‘ฅ2๐‘ฅ+2๐‘”2โˆ’1๎€ท๐‘ฅ2๎€ธ+12๎ƒญ๐œ“(๐‘ฅ)=2๐ธ๐‘Ž2๐œ“(๐‘ฅ).(2.1) A further substitution ๐‘ง=๐‘ฅ2+1 yields a differential equation with two regular singular points at ๐‘ง=0,1 and one irregular singular point of rank 2 at ๐‘ง=โˆž. The roots ๐œ‡'s of the indicial equation for the regular singular point ๐‘ง=0 reads ๐œ‡ยฑโˆš=1/2(1ยฑ1+4๐‘”), while the roots of the indicial equation at ๐‘ง=1 are ๐œ‡+=(๐‘™+1)/2 and ๐œ‡โˆ’=โˆ’๐‘™/2. Since the singularity for ๐‘งโ†’โˆž corresponds to that for ๐‘ฅโ†’โˆž, it is necessary that the solution for ๐‘งโ†’โˆž behave as ๐œ“(๐‘ฅ)โˆผexp(โˆ’๐‘ค๐‘Ž2๐‘ฅ2/2). Consequently, we may assume the general solution of (2.1) which vanishes at the origin and at infinity takes the form๐œ“๐‘›(๐‘ฅ)=๐‘ฅ๐‘™+1๎€ท๐‘ฅ2๎€ธ+1๐œ‡๐‘’โˆ’(๐‘ค๐‘Ž2/2)๐‘ฅ2๐‘“๐‘›(๐‘ฅ).(2.2) A straightforward calculation shows that ๐‘“๐‘›(๐‘ฅ) are the solutions of the second-order homogeneous linear differential equation๐‘“๎…ž๎…ž๎‚ต(๐‘ฅ)+2(๐‘™+1)๐‘ฅ+4๐œ‡๐‘ฅ๐‘ฅ2+1โˆ’2๐‘ค๐‘Ž2๐‘ฅ๎‚ถ๐‘“๎…ž+๎ƒฌ(๐‘ฅ)2๐ธ๐‘Ž2โˆ’๐‘ค๐‘Ž2๎€ท(2๐‘™+3+4๐œ‡)+2๐œ‡2๐‘™+3+2๐‘ค๐‘Ž2๎€ธ+4๐œ‡(๐œ‡โˆ’1)โˆ’2๐‘”๐‘ฅ2++14(๐‘”โˆ’๐œ‡(๐œ‡โˆ’1))๎€ท๐‘ฅ2๎€ธ+12๎ƒญร—๐‘“(๐‘ฅ)=0.(2.3) In the next sections, we attempt to give a general solution of this equation. For now, we assume that ๐œ‡ takes the value of the indicial root๐œ‡โ‰ก๐œ‡โˆ’=12๎‚€โˆš1โˆ’๎‚,1+4๐‘”(2.4) which allows us to write (2.3) as๐‘“๐‘›๎…ž๎…ž๎‚ต(๐‘ฅ)+2(๐‘™+1)๐‘ฅ+4๐œ‡๐‘ฅ๐‘ฅ2+1โˆ’2๐‘ค๐‘Ž2๐‘ฅ๎‚ถ๐‘“๎…ž๐‘›+๎ƒฌ(๐‘ฅ)2๐ธ๐‘Ž2โˆ’๐‘ค๐‘Ž2๎€ท(2๐‘™+3+4๐œ‡)+2๐œ‡2๐‘™+3+2๐‘ค๐‘Ž2๎€ธ+2๐œ‡(๐œ‡โˆ’1)๐‘ฅ2๎ƒญ๐‘“+1๐‘›(๐‘ฅ)=0.(2.5) We now consider the cases where the following two equations are satisfied: ๎€ท2๐œ‡2๐‘™+3+2๐‘ค๐‘Ž2๎€ธ+2๐œ‡(๐œ‡โˆ’1)=0,๐‘”=๐œ‡(๐œ‡โˆ’1).(2.6) The solutions of this system, for ๐‘” and ๐œ‡, are given explicitly by๎€ท๐‘”=0,or๐‘”=21+๐‘™+๐‘Ž2๐‘ค๎€ธ๎€ท3+2๐‘™+2๐‘Ž2๐‘ค๎€ธ,๎€ท๐œ‡=0,or๐œ‡=โˆ’21+๐‘™+๐‘Ž2๐‘ค๎€ธ.(2.7) Next, we consider each case of these two sets of solutions.

2.1. Caseโ€‰โ€‰1

The first set of solutions (๐‘”,๐œ‡)=(0,0) reduces the differential equation (2.3) to๐‘ฅ๐‘“๐‘›๎…ž๎…ž๎€บ(๐‘ฅ)+โˆ’2๐‘ค๐‘Ž2๐‘ฅ2๎€ป๐‘“+2(๐‘™+1)๎…ž๐‘›๎€ท(๐‘ฅ)+2๐ธ๐‘Ž2โˆ’๐‘ค๐‘Ž2๎€ธ(2๐‘™+3)๐‘ฅ๐‘“๐‘›(๐‘ฅ)=0,(2.8) which is a special case of the general differential equation๎€ท๐‘Ž3,0๐‘ฅ3+๐‘Ž3,1๐‘ฅ2+๐‘Ž3,2๐‘ฅ+๐‘Ž3,3๎€ธ๐‘ฆ๎…ž๎…ž+๎€ท๐‘Ž2,0๐‘ฅ2+๐‘Ž2,1๐‘ฅ+๐‘Ž2,2๎€ธ๐‘ฆ๎…žโˆ’๎€ท๐œ1,0๐‘ฅ+๐œ1,1๎€ธ๐‘ฆ=0,(2.9) with ๐‘Ž3,0=๐‘Ž3,1=๐‘Ž3,3=๐‘Ž2,1=๐œ1,1=0, ๐‘Ž3,2=1, ๐‘Ž2,0=โˆ’2๐‘ค๐‘Ž2, ๐‘Ž2,2=2(๐‘™+1), and ๐œ1,0=โˆ’2๐ธ๐‘Ž2+๐‘ค๐‘Ž2(2๐‘™+3). The necessary and sufficient conditions for polynomial solutions of (2.9) are given by the following theorem [6].

Theorem 2.1. The second-order linear differential equation (2.9) has a polynomial solution of degree ๐‘› if ๐œ1,0=๐‘›(๐‘›โˆ’1)๐‘Ž3,0+๐‘›๐‘Ž2,0,๐‘›=0,1,2,โ€ฆ,(2.10) along with the vanishing of (๐‘›+1)ร—(๐‘›+1)-determinant ฮ”๐‘›+1 given by ฮ”๐‘›+1=|||||||||||||||||๐›ฝ0๐›ผ1๐œ‚1๐›พ1๐›ฝ1๐›ผ2๐œ‚2๐›พ2๐›ฝ2๐›ผ3๐œ‚3๐›พโ‹ฑโ‹ฑโ‹ฑโ‹ฑ๐‘›โˆ’2๐›ฝ๐‘›โˆ’2๐›ผ๐‘›โˆ’1๐œ‚๐‘›โˆ’1๐›พ๐‘›โˆ’1๐›ฝ๐‘›โˆ’1๐›ผ๐‘›๐›พ๐‘›๐›ฝ๐‘›|||||||||||||||||=0,(2.11) where ๐›ฝ๐‘›=๐œ1,1๎€ทโˆ’๐‘›(๐‘›โˆ’1)๐‘Ž3,1+๐‘Ž2,1๎€ธ,๐›ผ๐‘›๎€ท(=โˆ’๐‘›๐‘›โˆ’1)๐‘Ž3,2+๐‘Ž2,2๎€ธ,๐›พ๐‘›=๐œ1,0โˆ’๎€ท(๐‘›โˆ’1)(๐‘›โˆ’2)๐‘Ž3,0+๐‘Ž2,0๎€ธ,๐œ‚๐‘›=โˆ’๐‘›(๐‘›+1)๐‘Ž3,3,(2.12) and ๐œ1,0 is fixed for a given ๐‘› in the determinant ฮ”๐‘›+1=0.

Thus, the necessary condition for the differential equation (2.8) to have polynomial solutions ๐‘“๐‘›โˆ‘(๐‘ฅ)=๐‘›๐‘–=0๐‘๐‘–๐‘ฅ๐‘– is2๐ธ๐‘›๐‘Ž2=๐‘ค๐‘Ž2๎€ท2๐‘›๎…ž๎€ธ+2๐‘™+3,๐‘›๎…ž=0,1,2,โ€ฆ,(2.13) while the sufficient condition, (2.12), isฮ”๐‘›+1=|||||||||||||||||0๐›ผ1๐›พ0010๐›ผ20๐›พ20๐›ผ30๐›พโ‹ฑโ‹ฑโ‹ฑโ‹ฑ๐‘›โˆ’20๐›ผ๐‘›โˆ’10๐›พ๐‘›โˆ’10๐›ผ๐‘›๐›พ๐‘›0|||||||||||||||||=โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ0,if๐‘›=0,2,4,โ€ฆ,(๐‘›โˆ’1)/2๎‘๐‘—=0(โˆ’1)2๐‘—+1๐›ผ2๐‘—+1๐›พ2๐‘—+1=0,if๐‘›=1,3,5,โ€ฆ.,(2.14) where ๐›ฝ๐‘›=0, ๐›ผ๐‘›=โˆ’๐‘›(๐‘›+2๐‘™+1) and ๐›พ๐‘›=2๐‘ค๐‘Ž2(๐‘›โˆ’๐‘›๎…žโˆ’1).

If ๐‘™=โˆ’1, the determinant ฮ”๐‘›+1 is identically zero for all ๐‘›, which is equivalent to the exact solutions of the one-dimensional harmonic oscillator problem.

For ๐‘™โ‰ โˆ’1, we have for ๐‘›=0,2,4,โ€ฆ, ฮ”๐‘›+1โ‰ก0, and we obtain the exact solutions of the Gol'dman and Krivchenkov (or Isotonic) Hamiltonian ๐ป0, where๐ป0๐œ“๐‘›๐‘™๎‚ธโˆ’๐‘‘(๐‘ฅ)โ‰ก2๐‘‘๐‘ฅ2+๐‘™(๐‘™+1)๐‘ฅ2+๐‘ค2๐‘Ž4๐‘ฅ2๎‚น๐œ“๐‘›๐‘™(๐‘ฅ)=2๐ธ๐‘”=0๐‘›๐‘™๐‘Ž2๐œ“๐‘›๐‘™(๐‘ฅ),0โ‰ค๐‘ฅ<โˆž.(2.15) These exact solutions are given by [9]2๐‘Ž2๐ธ๐‘”=0๐‘›๐‘™=๐‘ค๐‘Ž2๐œ“(4๐‘›+2๐‘™+3),๐‘›=0,1,2,โ€ฆ,๐‘›๐‘™(๐‘ฅ)=๐‘ฅ๐‘™+1๐‘’โˆ’๐‘ค๐‘Ž2๐‘ฅ21/2๐น1๎‚€3โˆ’๐‘›;๐‘™+2;๐‘ค๐‘Ž2๐‘ฅ2๎‚,๐‘›=0,1,2,โ€ฆ,(2.16) where the confluent hypergeometric function 1๐น1(โˆ’๐‘›;๐‘Ž;๐‘ง) defined in terms of the Pochhammer symbol (or Gamma function ฮ“(๐‘Ž)) (๐‘Ž)๐‘˜=ฮ“(๐‘Ž+๐‘˜)=๎ƒฏฮ“(๐‘Ž)1,if(๐‘˜=0,๐‘Žโˆˆโ„‚โงต{0}),๐‘Ž(๐‘Ž+1)(๐‘Ž+2)โ‹ฏ(๐‘Ž+๐‘˜โˆ’1),if(๐‘˜=โ„•,๐‘Žโˆˆโ„‚),(2.17) as1๐น1(โˆ’๐‘›;๐‘Ž;๐‘ง)=๐‘›๎“๐‘˜=0(โˆ’๐‘›)๐‘˜๐‘ง๐‘˜(๐‘Ž)๐‘˜๐‘˜!.(2.18) The polynomial solutions ๐‘“๐‘›(๐‘ฅ)=1๐น1(โˆ’๐‘›;๐‘™+(3/2);๐‘ค๐‘Ž2๐‘ฅ2) are easily obtained by using the asymptotic iteration method (AIM), which is summarized by means of the following theorem.

Theorem 2.2 (Ciftci et al. [7, equations (2.13)-(2.14)]). Given ๐œ†0โ‰ก๐œ†0(๐‘ฅ) and ๐‘ 0โ‰ก๐‘ 0(๐‘ฅ) in ๐ถโˆž, the differential equation ๐‘“๎…ž๎…ž(๐‘ฅ)=๐œ†0(๐‘ฅ)๐‘“๎…ž(๐‘ฅ)+๐‘ 0(๐‘ฅ)๐‘“(๐‘ฅ)(2.19) has the general solution ๎‚ตโˆ’๎€œ๐‘“(๐‘ฅ)=exp๐‘ฅ๐ถ๐›ผ(๐‘ก)๐‘‘๐‘ก๎‚ถ๎‚ธ2+๐ถ1๎€œ๐‘ฅ๎‚ต๎€œexp๐‘ก๎€ท๐œ†0(๎€ธ๎‚ถ๎‚น๐œ)+2๐›ผ(๐œ)๐‘‘๐œ๐‘‘๐‘ก,(2.20) if for some ๐‘›โˆˆโ„•+={1,2,โ€ฆ}๐‘ ๐‘›๐œ†๐‘›=๐‘ ๐‘›โˆ’1๐œ†๐‘›โˆ’1=๐›ผ(๐‘ฅ),or๐›ฟ๐‘›(๐‘ฅ)=๐œ†๐‘›๐‘ ๐‘›โˆ’1โˆ’๐œ†๐‘›โˆ’1๐‘ ๐‘›=0,(2.21) where ๐œ†๐‘›=๐œ†๎…ž๐‘›โˆ’1+๐‘ ๐‘›โˆ’1+๐œ†0๐œ†๐‘›,๐‘ ๐‘›=๐‘ ๎…ž๐‘›โˆ’1+๐‘ 0๐œ†๐‘›.(2.22)

For the differential equation (2.8) with๐œ†0๎€ท(๐‘ฅ)=โˆ’โˆ’2๐‘ค๐‘Ž2๐‘ฅ2๎€ธ+2(๐‘™+1)๐‘ฅ,๐‘ 0๎€ท(๐‘ฅ)=โˆ’2๐ธ๐‘Ž2โˆ’๐‘ค๐‘Ž2๎€ธ,(2๐‘™+3)(2.23) the first few iterations with ๐›ฟ๐‘›=๐œ†๐‘›๐‘ ๐‘›โˆ’1โˆ’๐œ†๐‘›โˆ’1๐‘ ๐‘›=0, using (2.20), imply๐‘“0๐‘“(๐‘ฅ)=1,1(๐‘ฅ)=2๐‘ค๐‘Ž2๐‘ฅ2๐‘“โˆ’(2๐‘™+3),2(๐‘ฅ)=4๐‘ค2๐‘Ž4๐‘ฅ4โˆ’4๐‘ค๐‘Ž2(2๐‘™+5)๐‘ฅ2โ‹ฎ+(2๐‘™+3)(2๐‘™+5),(2.24) which we may easily generalized using the definition of the confluent hypergeometric function, (2.18), as๐‘“๐‘›(๐‘ฅ)=1๐น1๎‚€3โˆ’๐‘›;๐‘™+2;๐‘ค๐‘Ž2๐‘ฅ2๎‚,(2.25) up to a constant.

2.2. Caseโ€‰โ€‰2

The second set of solutions ๎€ท2๎€ท(๐‘”,๐œ‡)=1+๐‘™+๐‘Ž2๐‘ค๎€ธ๎€ท3+2๐‘™+2๐‘Ž2๐‘ค๎€ธ๎€ท,โˆ’21+๐‘™+๐‘Ž2๐‘ค๎€ธ๎€ธ(2.26) allow us to write the differential equation (2.3) as๐‘“๐‘›๎…ž๎…ž๎ƒฉ(๐‘ฅ)+2(๐‘™+1)๐‘ฅโˆ’8๎€ท๐‘™+1+๐‘Ž2๐‘ค๎€ธ๐‘ฅ๐‘ฅ2+1โˆ’2๐‘ค๐‘Ž2๐‘ฅ๎ƒช๐‘“๎…ž๐‘›+๎€ท(๐‘ฅ)2๐ธ๐‘Ž2+๐‘ค๐‘Ž2๎€ท6๐‘™+5+8๐‘ค๐‘Ž2๐‘“๎€ธ๎€ธ๐‘›(๐‘ฅ)=0.(2.27) A further change of variable ๐‘ง=๐‘ฅ2+1 allows us to write the differential equation (2.27) as4๐‘ง(๐‘งโˆ’1)๐‘“๎…ž๎…ž๎€ท(๐‘ง)โˆ’4๐‘Ž2๐‘ค๐‘ง2๎€ท+26๐‘™+5+6๐‘ค๐‘Ž2๎€ธ๎€ท๐‘งโˆ’16๐‘™+1+๐‘ค๐‘Ž2๐‘“๎€ธ๎€ธ๎…ž+๎€ท(๐‘ง)2๐ธ๐‘Ž2+๐‘ค๐‘Ž2๎€ท6๐‘™+5+8๐‘ค๐‘Ž2๎€ธ๎€ธ๐‘ง๐‘“(๐‘ง)=0,(2.28) Again, (2.28) is a special case of the differential equation (2.9) with ๐‘Ž3,0=๐‘Ž3,3=๐œ1,1=0, ๐‘Ž3,1=4, ๐‘Ž3,2=โˆ’4, ๐‘Ž2,0=โˆ’4๐‘ค๐‘Ž2, ๐‘Ž2,1=โˆ’2(6๐‘™+5+6๐‘ค๐‘Ž2), ๐‘Ž2,2=16(๐‘™+1+๐‘ค๐‘Ž2), and ๐œ1,0=โˆ’2๐ธ๐‘Ž2โˆ’๐‘ค๐‘Ž2(6๐‘™+5+8๐‘ค๐‘Ž2). Consequently, the polynomial solutions ๐‘“๐‘›(๐‘ฅ) of (2.28) are subject to the following two conditions: the necessary condition (2.10) reads2๐ธ๐‘›๐‘Ž2=๐‘ค๐‘Ž2๎€ท4๐‘›๎…žโˆ’6๐‘™โˆ’5โˆ’8๐‘ค๐‘Ž2๎€ธ,๐‘›๎…ž=0,1,2,โ€ฆ,(2.29) and the sufficient condition; namely, the vanishing of the tridiagonal determinant (2.12), reads ฮ”๐‘›+1=|||||||||||||||||๐›ฝ0๐›ผ1๐›พ1๐›ฝ1๐›ผ2๐›พ2๐›ฝ2๐›ผ3๐›พโ‹ฑโ‹ฑโ‹ฑ๐‘›โˆ’2๐›ฝ๐‘›โˆ’2๐›ผ๐‘›โˆ’1๐›พ๐‘›โˆ’1๐›ฝ๐‘›โˆ’1๐›ผ๐‘›๐›พ๐‘›๐›ฝ๐‘›|||||||||||||||||=0,(2.30) where๐›ฝ๐‘›๎€ท=โˆ’2๐‘›2๐‘›โˆ’6๐‘™โˆ’7โˆ’6๐‘ค๐‘Ž2๎€ธ,๐›ผ๐‘›๎€ท=4๐‘›๐‘›โˆ’4๐‘™โˆ’5โˆ’4๐‘Ž2๐‘ค๎€ธ,๐›พ๐‘›=4๐‘ค๐‘Ž2๎€ท๐‘›โˆ’๐‘›๎…ž๎€ธ,โˆ’1(2.31) and ๐‘›๎…ž=๐‘› is fixed for the given dimension of the determinant ฮ”๐‘›+1. From the sufficient condition (2.31), we obtain the following conditions on the parameters: ฮ”2=0โŸน๐‘Ž2๐‘ค๎€ท๐‘™+1+๐‘Ž2๐‘ค๎€ธฮ”=0,3=0โŸน๐‘Ž2๐‘ค๎€ท๐‘™+1+๐‘Ž2๐‘ค๎€ธ๎€ท1+2๐‘™+2๐‘Ž2๐‘ค๎€ธฮ”=0,4=0โŸน๐‘Ž2๐‘ค๎€ท๐‘™+1+๐‘Ž2๐‘ค๎€ธ๎€ท1+2๐‘™+2๐‘Ž2๐‘ค๎€ธ๎€ท3(1+6๐‘™)+14๐‘Ž2๐‘ค๎€ธฮ”=0,5=0โŸน๐‘Ž2๐‘ค๎€ท๐‘™+1+๐‘Ž2๐‘ค๎€ธ๎€ท1+2๐‘™+2๐‘Ž2๐‘ค๎€ธ๎€ท3(6๐‘™โˆ’1)(6๐‘™+1)+4(38๐‘™+1)๐‘Ž2๐‘ค+44๐‘Ž4๐‘ค2๎€ธฮ”=0,6=0โŸน๐‘Ž2๐‘ค๎€ท๐‘™+1+๐‘Ž2๐‘ค๎€ธ๎€ท1+2๐‘™+2๐‘Ž2๐‘ค๎€ธร—๎€ท3๎€ท(2๐‘™โˆ’1)(6๐‘™โˆ’1)(6๐‘™+1)+2208๐‘™2๎€ธ๐‘Žโˆ’54๐‘™โˆ’52๐‘ค+200๐‘™๐‘Ž4๐‘ค2๎€ธ=0,โ‹ฎ=โ‹ฎ.(2.32) For a physically meaningful solution, we must have ๐‘Ž2๐‘ค>0. This is possible for a very restricted value of the angular momentum number ๐‘™. Since ๐›ฝ0=0, we may observe that ฮ”๐‘›+1=๎€ท๐‘™+1+๐‘Ž2๐‘ค๎€ธ๎€ท1+2๐‘™+2๐‘Ž2๐‘ค๎€ธร—|||||||||||||||||๐›ฝ2๐›ผ3๐›พ3๐›ฝ3๐›ผ4๐›พ4๐›ฝ4๐›ผ5๐›พโ‹ฑโ‹ฑโ‹ฑ๐‘›โˆ’2๐›ฝ๐‘›โˆ’2๐›ผ๐‘›โˆ’1๐›พ๐‘›โˆ’1๐›ฝ๐‘›โˆ’1๐›ผ๐‘›๐›พ๐‘›๐›ฝ๐‘›|||||||||||||||||=๎€ท๐‘™+1+๐‘Ž2๐‘ค๎€ธ๎€ท1+2๐‘™+2๐‘Ž2๐‘ค๎€ธร—๐‘„๐‘™๐‘›โˆ’1๎€ท๐‘Ž2๐‘ค๎€ธ,(2.33) where ๐‘„๐‘™๐‘›โˆ’1(๐‘Ž2๐‘ค) are polynomials in the parameter product ๐‘Ž2๐‘ค.

For physically acceptable solutions, we must have ๐‘™=โˆ’1 and the factor (๐‘™+1+๐‘Ž2๐‘ค) yields ๐‘Ž2๐‘ค=0, which is not physically acceptable, so we ignore it. The second factor (1+2๐‘™+2๐‘Ž2๐‘ค) implies a special value of ๐‘Ž2๐‘ค=1/2, for all ๐‘›, which we will study shortly in full detail. Meanwhile, the polynomials ๐‘„๐‘™๐‘›(๐‘Ž2๐‘ค)๐‘„๐‘™=โˆ’1๐‘›โˆ’1๎€ท๐‘Ž2๐‘ค๎€ธ=โŽงโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽฉ1,if๐‘›=2,14๐‘Ž2๐‘คโˆ’15,if๐‘›=3,44๐‘Ž4๐‘ค2โˆ’148๐‘Ž2๐‘ค+105,if๐‘›=4,200๐‘Ž4๐‘ค2โˆ’514๐‘Ž2โ‹ฎ๐‘ค+315,if๐‘›=5,(2.34) give new values, not reported before, of ๐‘Ž2๐‘ค that yield quasiexact solutions of the Schrรถdinger equation (with one eigenstate)โˆ’๐œ“๐‘›๎…ž๎…ž๎ƒฌ๎€ท(๐‘ฅ)+๐‘ค๐‘Ž2๎€ธ2๐‘ฅ2+4๐‘Ž2๐‘ค๎€ท1+2a2๐‘ค๎€ธ๎€ท๐‘ฅ2๎€ธโˆ’1๎€ท๐‘ฅ2๎€ธ+12๎ƒญ๐œ“๐‘›(๐‘ฅ)=๐‘ค๐‘Ž2๎€ท4๐‘›+1โˆ’8๐‘Ž2๐‘ค๎€ธ๐œ“๐‘›(๐‘ฅ),(2.35) where ๐œ“๐‘›๎€ท๐‘ฅ(๐‘ฅ)=2๎€ธ+1โˆ’2๐‘Ž2๐‘ค๐‘’โˆ’๐‘ค๐‘Ž2๐‘ฅ2/2๐‘“๐‘›(๐‘ฅ),(2.36) and ๐‘“๐‘›(๐‘ฅ) are the solutions of4๐‘ง(๐‘งโˆ’1)๐‘“๎…ž๎…ž๎€ท(๐‘ง)โˆ’4๐‘Ž2๐‘ค๐‘ง2๎€ท+2โˆ’1+6๐‘ค๐‘Ž2๎€ธ๐‘งโˆ’16๐‘ค๐‘Ž2๎€ธ๐‘“๎…ž(๐‘ง)+4๐‘›๐‘ค๐‘Ž2๐‘ง๐‘“(๐‘ง)=0,๐‘ง=๐‘ฅ2+1.(2.37) For example, ฮ”4=0 implies, using (2.34), that ๐‘Ž2๐‘ค=15/14, and thus, we have forโˆ’๐œ“3๎…ž๎…ž๎ƒฌ(๐‘ฅ)+225๐‘ฅ1962+660๎€ท๐‘ฅ492๎€ธโˆ’1๎€ท๐‘ฅ2๎€ธ+12๎ƒญ๐œ“3(๐‘ฅ)=465๐œ“983(๐‘ฅ),(2.38) the exact solution ๐œ“3๎€ท๐‘ฅ(๐‘ฅ)=2๎€ธ+1โˆ’15/7๐‘’โˆ’(15/28)๐‘ฅ2๎€ท45๐‘ฅ6+225๐‘ฅ4+315๐‘ฅ2๎€ธ,โˆ’49(2.39) with a plot of the wave function and potential given in Figure 1.

Figure 1: Plot of the unnormalized wave function ๐œ“3(๐‘ฅ) and the potential ๐‘‰3=(225/196)๐‘ฅ2+(660/49)(๐‘ฅ2โˆ’1)/(๐‘ฅ2+1)2.

Further, ฮ”5=0, equation (2.34) implies ๐‘Ž2๐‘ค=37ยฑโˆš2221422,(2.40) and we have forโˆ’๐œ“4๎…ž๎…žโŽกโŽขโŽขโŽฃ๎ƒฉ(๐‘ฅ)+37ยฑโˆš22214๎ƒช222๐‘ฅ2๎ƒฉ+237ยฑโˆš1121411๎ƒช๎ƒฉ48ยฑโˆš11214๎ƒช๎€ท๐‘ฅ112๎€ธโˆ’1๎€ท๐‘ฅ2๎€ธ+12โŽคโŽฅโŽฅโŽฆ๐œ“4=๎ƒฉ(๐‘ฅ)37ยฑโˆš2221422๎ƒช๎ƒฉ39โˆ“4โˆš11214๎ƒช๐œ“114(๐‘ฅ),(2.41) the exact solutions ๐œ“ยฑ4(๎€ท๐‘ฅ๐‘ฅ)=2๎€ธ+1โˆšโˆ’((37/11)ยฑ(214/11))๐‘’โˆšโˆ’((37/44)ยฑ(214/44))๐‘ฅ2ร—๎‚€1575๐‘ฅ8+๎‚€โˆš9660ยฑ420๎‚๐‘ฅ2146+๎‚€โˆš26250ยฑ2100๎‚๐‘ฅ2144+๎‚€โˆš29820ยฑ2940๎‚๐‘ฅ2142โˆ’๎‚€โˆš1129ยฑ188.214๎‚๎‚(2.42) Similar results can be obtained for ฮ”๐‘›+1=0, for ๐‘›โ‰ฅ5.

2.3. Exactly Solvable Quantum Isotonic Nonlinear Oscillator

As mentioned above, for ๐‘™=โˆ’1 and ๐‘Ž2๐‘ค=1/2, it clear that ฮ”๐‘›+1=0 for all ๐‘› and the one-dimensional Schrรถdinger equation๎ƒฌโˆ’๐‘‘2๐‘‘๐‘ฅ2+๐‘ฅ24+4๎€ท๐‘ฅ2๎€ธโˆ’1๎€ท๐‘ฅ2๎€ธ+12๎ƒญ๐œ“๐‘›๎‚€3(๐‘ฅ)=2๐‘›โˆ’2๎‚๐œ“๐‘›(๐‘ฅ),๐‘›=0,1,2,โ€ฆ(2.43) has the exact solutions๐œ“๐‘›๎€ท๐‘ฅ(๐‘ฅ)=2๎€ธ+1โˆ’1๐‘’โˆ’๐‘ฅ2/4๐‘“๐‘›(๐‘ฅ),(2.44) where ๐‘“๐‘›(๐‘ฅ) are the polynomial solutions of the following second-order linear differential equation (๐‘ง=๐‘ฅ2+1)4๐‘ง(๐‘งโˆ’1)๐‘“๐‘›๎…ž๎…ž๎€ท(๐‘ง)โˆ’2๐‘ง2๎€ธ๐‘“+4๐‘งโˆ’8๎…ž๐‘›(๐‘ง)+2๐‘›๐‘ง๐‘“๐‘›(๐‘ง)=0.(2.45) By using AIM (Theorem 2.2, (2.20)), we find that the polynomial solutions ๐‘“๐‘›(๐‘ฅ) of (2.45) are given explicitly as๐‘“0๐‘“(๐‘ฅ)=1,1(๐‘ฅ)=๐‘ฅ2๐‘“โˆ’2,2(๐‘ฅ)=๐‘ฅ3โˆ’6๐‘ฅ2๐‘“+8,3(๐‘ฅ)=๐‘ฅ4โˆ’16๐‘ฅ3+52๐‘ฅ2๐‘“โˆ’52,4(๐‘ฅ)=๐‘ฅ5โˆ’30๐‘ฅ4+250๐‘ฅ3โˆ’580๐‘ฅ2โ‹ฎ+464,(2.46) a set of polynomial solutions that can be generated using๐‘“0๐‘“(๐‘ฅ)=1,๐‘›(๐‘ฅ)=โˆ’3๐‘ฅ(2๐‘›+1)1๐น1๎‚€3โˆ’๐‘›;2;12๎‚(๐‘ฅโˆ’1)+6((๐‘›+1)๐‘ฅโˆ’1)1๐น1๎‚€3โˆ’๐‘›+1;2;12๎‚,(๐‘ฅโˆ’1)(2.47) up to a constant factor, where, again, 1๐น1 refers to the confluent hypergeometric function defined by (2.18). Note that the polynomials ๐‘“๐‘›(๐‘ฅ) in (2.47) can be expressed in terms of the associated Laguerre polynomials [10] as๐‘“0๐‘“(๐‘ฅ)=1,๐‘›3(๐‘ฅ)=(โˆ’1)๐‘›โˆš๐œ‹ฮ“(๐‘›)ร—๎‚ƒ๎€ท2ฮ“(๐‘›+3/2)(1+๐‘›)(๐‘ฅโˆ’1)2๎€ธ๐ฟ+๐‘›๐‘›1/2๎‚€๐‘ฅโˆ’12๎‚โˆ’(๐‘ฅโˆ’1)((1+๐‘›)๐‘ฅโˆ’1)๐ฟ๐‘›3/2๎‚€๐‘ฅโˆ’12.๎‚๎‚„(2.48)

3. Quasipolynomial Solutions of the Generalized Quantum Isotonic Oscillator

In this section, we study the quasipolynomial solutions of the differential equation (2.3). We note first, using the change of variable ๐‘ง=๐‘ฅ2, equation (2.3) can be written as๐‘“๐‘›๎…ž๎…ž๎‚ต(๐‘ง)+2๐‘™+3+2๐‘ง2๐œ‡๐‘ง+1โˆ’๐‘ค๐‘Ž2๎‚ถ๐‘“๎…ž๐‘›+๎ƒฌ(๐‘ง)2๐ธ๐‘Ž2โˆ’๐‘ค๐‘Ž2(2๐‘™+3+4๐œ‡)+๐œ‡๎€ท4๐‘ง2๐‘™+3+2๐‘ค๐‘Ž2๎€ธโˆ’๐‘”2๐‘ง(๐‘ง+1)2(๐‘งโˆ’1)๐‘ง(๐‘ง+1)2+๐œ‡(๐œ‡โˆ’1)(๐‘ง+1)2๎ƒญ๐‘“๐‘›(๐‘ง)=0.(3.1) By means of the Mรถbius transformation ๐‘ง=๐‘ก/(1โˆ’๐‘ก) that maps the singular points {โˆ’1,0,โˆž} into {0,1,โˆž}, we obtain๐‘“๐‘›๎…ž๎…ž๎‚ต(๐‘ก)+2๐‘™+3+2๐‘ก(1โˆ’๐‘ก)2(๐œ‡โˆ’1)โˆ’1โˆ’๐‘ก๐‘ค๐‘Ž2(1โˆ’๐‘ก)2๎‚ถ๐‘“๎…ž๐‘›+๎ƒฌ๐œ‡๎€ท(๐‘ก)2๐‘™+3+2๐‘ค๐‘Ž2๎€ธ2๐‘ก(1โˆ’๐‘ก)2โˆ’๐‘”2(2๐‘กโˆ’1)๐‘ก(1โˆ’๐‘ก)2+๐œ‡(๐œ‡โˆ’1)(1โˆ’๐‘ก)2๎ƒญ๐‘“๐‘›(๐‘ก)=0,(3.2) where we assume that2๐ธ๐‘Ž2โˆ’(2๐‘™+3+4๐œ‡)๐‘ค๐‘Ž2=0.(3.3) The differential equation (3.2) can be written as๎€ท๐‘ก3โˆ’2๐‘ก2๎€ธ๐‘“+๐‘ก๐‘›๎…ž๎…ž๎‚ƒ(๐‘ก)+โˆ’2(๐œ‡โˆ’1)๐‘ก2+๎‚€2๐œ‡โˆ’๐‘ค๐‘Ž27โˆ’๐‘™โˆ’2๎‚๎‚€3๐‘ก+๐‘™+2๐‘“๎‚๎‚„๎…ž๐‘›+๎‚ƒ๐‘”(๐‘ก)(๐œ‡(๐œ‡โˆ’1)โˆ’๐‘”)๐‘ก+2๎‚€3+๐œ‡๐‘™+2+๐‘ค๐‘Ž2๐‘“๎‚๎‚„๐‘›(๐‘ก)=0,(3.4) which we may now compare with equation (2.9) in Theorem 2.1 with ๐‘Ž3,0=1, ๐‘Ž3,1=โˆ’2, ๐‘Ž3,2=1, ๐‘Ž3,3=0, ๐‘Ž2,0=โˆ’2(๐œ‡โˆ’1), ๐‘Ž2,1=(2๐œ‡โˆ’๐‘ค๐‘Ž2โˆ’๐‘™โˆ’7/2), ๐‘Ž2,2=(๐‘™+3/2), ๐œ1,0=โˆ’(๐œ‡(๐œ‡โˆ’1)โˆ’๐‘”), ๐œ1,1=โˆ’๐‘”/2โˆ’๐œ‡(๐‘™+3/2+๐‘ค๐‘Ž2). We, thus, conclude that the quasipolynomial solutions ๐‘“๐‘›(๐‘ก) of (3.4) are subject to the following conditions:๐‘”=(๐œ‡โˆ’๐‘˜)(๐œ‡โˆ’๐‘˜โˆ’1),๐‘˜=0,1,2,โ€ฆ,(3.5) along with the vanishing of the tridiagonal determinant ฮ”๐‘›+1=0||||||||||||||๐›ฝ0๐›ผ1๐›พ1๐›ฝ1๐›ผ2๐›พ2๐›ฝ2๐›ผ3๐›พโ‹ฑโ‹ฑโ‹ฑ๐‘›โˆ’1๐›ฝ๐‘›โˆ’1๐›ผ๐‘›๐›พ๐‘›๐›ฝ๐‘›||||||||||||||=0,(3.6) where๐›ฝ๐‘›1=โˆ’2๎€ท๎€ท๐‘”+(๐œ‡โˆ’๐‘›)3+2๐‘™+4๐‘›+2๐‘Ž2๐‘ค,๐›ผ๎€ธ๎€ธ๐‘›๎‚€1=โˆ’๐‘›๐‘›+๐‘™+2๎‚,๐›พ๐‘›=๐‘”โˆ’(๐œ‡โˆ’๐‘›+1)(๐œ‡โˆ’๐‘›).(3.7) Here, again, ๐‘”=(๐œ‡โˆ’๐‘˜)(๐œ‡โˆ’๐‘˜โˆ’1) is fixed for given ๐‘˜=๐‘›, the fixed size of the determinant ฮ”๐‘›+1.

3.1. Particular Case: ๐‘›=0

For ๐‘˜(๐‘“๐‘–๐‘ฅ๐‘’๐‘‘)โ‰ก๐‘›=0, the differential equation (3.4) has the exact solution ๐‘“0(๐‘ก)=1 if ๐‘” and ๐œ‡ satisfy, simultaneously, the following system of equations: ๎€ท๐‘”+๐œ‡3+2๐‘™+2๐‘Ž2๐‘ค๎€ธ=0,๐‘”=๐œ‡(๐œ‡โˆ’1).(3.8) Solving this system of equations for ๐‘” and ๐œ‡, we obtain the following values of๎€ท๐‘”=21+๐‘™+๐‘Ž2๐‘ค๎€ธ๎€ท3+2๐‘™+2๐‘Ž2๐‘ค๎€ธ๎€ท,๐œ‡=โˆ’2๐‘™+1+๐‘ค๐‘Ž2๎€ธ,(3.9) and the ground-state energy, in this case, is given by (3.3); namely,๐ธ๐‘Ž21=โˆ’2๐‘Ž2๐‘ค๎€ท5+6๐‘™+8๐‘Ž2๐‘ค๎€ธ,(3.10) which in complete agreement with the results of Section 2.2.

3.2. Particular Case: ๐‘›=1

For ๐‘˜(๐‘“๐‘–๐‘ฅ๐‘’๐‘‘)โ‰ก๐‘›=1, the determinant ฮ”2=0 of (3.7) yields๐‘”2๎€ท+๐‘”โˆ’1+10๐œ‡+2๐‘™(2๐œ‡+1)+2๐‘Ž2๎€ธ๎€ท๐‘ค(2๐œ‡โˆ’1)+๐œ‡(๐œ‡โˆ’1)15+4๐‘™2๎€ท+8๐‘™2+๐‘Ž2๐‘ค๎€ธ+4๐‘Ž2๐‘ค๎€ท5+๐‘Ž2๐‘ค๎€ธ๎€ธ=0,๐‘”โˆ’(๐œ‡โˆ’1)(๐œ‡โˆ’2)=0,(3.11) where the energy is given by use of (3.3), for the computed values of ๐œ‡ and ๐‘”, by๎‚€3๐ธ=๐‘™+2๎‚+2๐œ‡๐‘ค.(3.12) Further, (3.11) yields the solutions for ๐‘™ as functions of ๐œ‡ and ๐‘Ž2๐‘ค๎€ท๐‘™=2โˆ’5+4๐‘Ž2๐‘ค๎€ธ๐œ‡โˆ’2๐œ‡2ยฑ๎”๎€ท4โˆ’43+8๐‘Ž2๐‘ค๎€ธ๐œ‡+9๐œ‡24๐œ‡โ‰ฅโˆ’1,(3.13) where the energy states are now given by (3.12) along with ๐‘™ given by (3.13). We may also note that for๐‘Ž21๐‘ค=2๐‘Ž(๐‘˜+1),๐‘˜=0,1,2,โ€ฆ,(3.14)2๐ธยฑ1=โˆ’๎‚€8๐œ‡(๐‘˜+1)โˆ’2+(2๐‘˜+1)๐œ‡โˆ’6๐œ‡2ยฑโˆš4โˆ’4(4๐‘˜+7)๐œ‡+9๐œ‡2๎‚.(3.15) Further, for ๐‘”=(๐œ‡โˆ’1)(๐œ‡โˆ’2), we obtain the unnormalized wave function (see (2.2))๐œ“1,๐‘™(๐‘ฅ)=๐‘ฅ๐‘™+1๎€ท1+๐‘ฅ2๎€ธ๐œ‡โˆ’1๐‘’โˆ’๐‘ค๐‘Ž2๐‘ฅ2/2๎‚ต1+1+2๐‘™+๐œ‡+2๐‘Ž2๐‘ค5+2๐‘™+๐œ‡+2๐‘Ž2๐‘ค๐‘ฅ2๎‚ถ.(3.16) Thus, we may summarize these results as follows. The exact solutions of the Schrรถdinger equation (2.1) are given by (3.15) and (3.16) only if ๐‘” and ๐œ‡ are the solutions of the system given by (3.11). In Tables 1 and 2, we report few quasiexact solutions that can be obtained using this approach.

Table 1: Conditions on the value of the parameters ๐‘” and ๐œ‡ for the quasipolynomial solutions in the case of ฮ”2=0 with different values of ๐‘ค๐‘Ž2 and ๐‘™.
Table 2: Conditions on the value of the parameters ๐‘” and ๐œ‡ for the quasipolynomial solutions in the case of ฮ”2=0 with different values of ๐‘ค๐‘Ž2 and ๐‘™.
3.2.1. Particular Case ๐‘›=2

For ๐‘˜(๐‘“๐‘–๐‘ฅ๐‘’๐‘‘)โ‰ก๐‘›=2, the determinant ฮ”3=0 along with the necessary condition (3.7) yields๐‘”3+3๐‘”2๎€ท7๐œ‡โˆ’1+2๐‘™(1+๐œ‡)+2๐‘Ž2๎€ธ๎€บ๐‘ค(๐œ‡โˆ’1)โˆ’๐‘”18+56๐‘™+8๐‘™2+18(7+2๐‘™)๐œ‡โˆ’3(5+2๐‘™)(7+2๐‘™)๐œ‡2โˆ’12๐‘Ž2๐‘ค(๐œ‡โˆ’1)((7+2๐‘™)๐œ‡โˆ’4)โˆ’4๐‘Ž4๐‘ค2๎€ป๎€ท(2+3(๐œ‡โˆ’2)๐œ‡)+๐œ‡(๐œ‡โˆ’2)(๐œ‡โˆ’1)105+142๐‘™+60๐‘™2+8๐‘™3+6๐‘Ž2๐‘ค(5+2๐‘™)(7+2๐‘™)+12๐‘Ž4๐‘ค2(7+2๐‘™)+8๐‘Ž6๐‘ค3๎€ธ=0,๐‘”โˆ’(๐œ‡โˆ’2)(๐œ‡โˆ’3)=0,(3.17) where, again, the energy is given, for the computed values of ๐œ‡ and ๐‘” using (3.3) and (3.17), by ๎‚€3๐ธ=๐‘™+2๎‚+2๐œ‡๐‘ค.(3.18) In Table 3, we report the numerical results for some of the exact solutions of ๐œ‡ and ๐‘” using (3.17) and the values of (l,๐‘ค๐‘Ž2)=(โˆ’1,1/2), (๐‘™,๐‘ค๐‘Ž2)=(โˆ’1,1), (๐‘™,๐‘ค๐‘Ž2)=(โˆ’1,3/2),(๐‘™,๐‘ค๐‘Ž2)=(โˆ’1,2), (๐‘™,๐‘ค๐‘Ž2)=(0,1/2), and (๐‘™,๐‘ค๐‘Ž2)=(0,2), respectively. We have also computed the corresponding eigenvalues ๐ธ๐‘ค๐‘Ž22,๐‘™โ‰ก๐ธ๐‘ค๐‘Ž22,๐‘™(๐œ‡,๐‘”).

Table 3: Exact eigenvalues for different values of ๐‘™ and ๐‘ค๐‘Ž2 in the case ฮ”3=0.

4. Numerical Computation by the Use of the Asymptotic Iteration Method

For the potential parameters ๐‘ค, ๐‘Ž2, and ๐‘”, not necessarily obeying the conditions for quasipolynomial solutions discussed in the previous sections, the asymptotic iteration method can be employed to compute the eigenvalues of Schrรถdinger equation (2.1) for arbitrary values ๐‘ค, ๐‘Ž2, and ๐‘”. The functions ๐œ†0 and ๐‘ 0, using (3.2) and (3.3), are given by๐œ†0๎ƒฉ(๐‘ก)=โˆ’2๐‘™+3+22๐‘ก(1โˆ’๐‘ก)๎€ท๎€ท๐ธ๐‘Ž2/2๐‘ค๐‘Ž2๎€ธ๎€ธโˆ’((2๐‘™+3)/4)โˆ’1โˆ’(1โˆ’๐‘ก)๐‘ค๐‘Ž2(1โˆ’๐‘ก)2๎ƒช,๐‘ 0๎ƒฉ(๐‘ก)=โˆ’๎€ท๎€ท๐ธ๐‘Ž2/2๐‘ค๐‘Ž2๎€ธโˆ’((2๐‘™+3)/4)๎€ธ๎€ท2๐‘™+3+2w๐‘Ž2๎€ธ2๐‘ก(1โˆ’๐‘ก)2โˆ’๐‘”2(2๐‘กโˆ’1)๐‘ก(1โˆ’๐‘ก)2+๎€ท๎€ท๐ธ๐‘Ž2/2๐‘ค๐‘Ž2๎€ธโˆ’((2๐‘™+3)/4)๎€ธ๎€ท๎€ท๐ธ๐‘Ž2/๎€ท2๐‘ค๐‘Ž2๎€ธ๎€ธ๎€ธโˆ’((2๐‘™+3)/4)โˆ’1(1โˆ’๐‘ก)2๎ƒช,(4.1) where ๐‘กโˆˆ(0,1). The AIM sequence ๐œ†๐‘›(๐‘ฅ) and ๐‘ ๐‘›(๐‘ฅ) can be calculated iteratively using the iterative sequences (2.22). The energy eigenvalues of the quantum nonlinear isotonic potential (2.1) are obtained from the roots of the termination condition (2.21). According to the asymptotic iteration method, in particular, the study of Champion et al. [5], unless the differential equation is exactly solvable, the termination condition (2.21) produces for each iteration an expression that depends on both ๐‘ก and ๐ธ (for given values of the parameters ๐‘ค๐‘Ž2, ๐‘”, and ๐‘™). In such a case, one faces the problem of finding the best possible starting value ๐‘ก=๐‘ก0 that stabilizes the AIM process [5]. Fortunately, since ๐‘กโˆˆ(0,1), the starting value ๐‘ก0 does not represent a serious issue in our eigenvalue calculation using (4.1) and the termination condition (2.21) in contrast to the case of computing the eigenvalues using ๐œ†0(๐‘ฅ) and ๐‘ 0(๐‘ฅ) as given by, for example, (2.3), where ๐‘ฅโˆˆ(0,โˆž). In Table 4, we report our numerical results for energies of the four lowest states of the generalized isotonic oscillator of parameters ๐‘ค and ๐‘Ž such that ๐‘ค๐‘Ž2=2 and for different values of ๐‘”. In this table, we set ๐‘™=โˆ’1 for computing the energies ๐ธ0๐‘Ž2 and ๐ธ2๐‘Ž2, while we put ๐‘™=0 for computing the energies ๐ธ1๐‘Ž2 and ๐ธ3๐‘Ž2, respectively. For most of these values, the starting value of ๐‘ก is ๐‘ก0=0.5 and is shifted towards zero as ๐‘” gets larger in value. For the values of ๐‘” that admit a quasipolynomial solution, the number of iteration does not exceed three. For most of the other values of ๐‘”, the total number of iteration did not exceed 65. We found that for ๐‘ค๐‘Ž2=2 and the values of ๐‘” reported in Table 4, the number of iteration is relatively small compared to the case of ๐‘ค๐‘Ž2=1/2 and a large value of the parameter ๐‘”. The numerical computations in the present work were done using Maple version 13 running on an IBM architecture personal computer in a high-precision environment. In order to accelerate our computation, we have written our own code for a root-finding algorithm instead of using the default procedure Solve of Maple 13. These numerical results are accurate to the number of decimals reported.

Table 4: Energies of the four lowest states of the generalized isotonic oscillator of parameters ๐‘ค and ๐‘Ž given for ๐‘™=โˆ’1 as ๐‘ค๐‘Ž2=2 and for different values of the parameter ๐‘”. The subscript numbers represents the number of iterations used by AIM.

5. Conclusion

We have provided a detailed solution of the eigenproblem posed by Schrรถdiger's equation with a generalized nonlinear isotonic oscillator potential. We have presented a method for computing the quasipolynomial solutions in cases, where the potential parameters satisfy certain conditions. In other more general cases we have used the asymptotic iteration method to find accurate numerical solutions for arbitrary values of the potential parameters ๐‘”, ๐‘ค, and ๐‘Ž.


Partial financial support of this work under Grant nos. GP249507 and GP3438 from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged by the authors.


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