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Algebraic Method for Perturbed Three-Body Systems of \(\mathbf {A}_{\mathbf {2}} \) Solvable Potential

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Abstract

In this paper, we try to solve the Schrödinger equation in a quasi-exact solvable method for a three-body problem with a special interaction and by adding an anharmonic perturbation term. We consider the interaction and perturbation theory in the Calogero model by the roots of algebra \(A_{2}\) and rewrite the Hamiltonian in terms of Lie algebra \(gl_{3} \) and \(g^{2}\) generators. Indeed, we show that the gauge transformed Hamiltonian has infinite invariant flags with finite-dimension. Finally, we obtain a range of eigenvalues and eigenfunctions corresponding to its corrections by using the algebraic framework of the perturbation theory.

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Acknowledgements

The authors are deeply grateful to the referee for their careful reading of the manuscript which significantly improved the original version.

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Correspondence to Hossein Panahi.

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Appendices

Appendix A: Representation of the Algebra \(gl_{3} \)

The algebra \(gl_{3} \) with 9 generators in terms of first-order differential operators acting on the (x, y)-plane is presented in the following form [41]

$$\begin{aligned} \begin{array}{lllll} J_{2}^{-} =\partial _{x} \quad (-1,0); &{}\qquad J_{2,2}^{0} =x\partial _{x} &{}\quad (0,0); &{}\qquad J_{3,2}^{0} =y\partial _{x} &{}\quad (-1,1);\\ J_{3}^{-} =\partial _{y} \quad (0,-1); &{}\qquad J_{3,3}^{0} =y\partial _{y} &{}\quad (0,0); &{}\qquad J_{2,3}^{0} =x\partial _{y} &{}\quad (1,-1);\\ J_{2}^{+} =xJ^{0} \quad (1,0); &{}\qquad J_{3}^{+} =yJ^{0} &{}\quad (0,1); &{}\qquad J^{0}=n-x\partial _{x} -y\partial _{y} &{}\quad (0,0);\\ \end{array} \end{aligned}$$
(A.1)

in which \(n\in \mathbb {R}\). The parameter ncan be each real number. The value of the parameter n plays no role, it just changes the reference point for the energy. It can also be observed by applying the generators (A.1) to the coordinate system \(Jx^{p}y^{q}\propto x^{p+\alpha }y^{q+\beta }\), where \((\alpha ,\beta )\) are understood as a grading. It should be noted that if n be a positive integer, its representation space will be a space with the finite dimension, the same as (2.10). Geometrically, the Newton diagram displays the role of changes n in two directions x and y (Fig. 1).

Fig. 1
figure1

Newton diagram of the algebra \(gl_{3} \); corresponding to finite-dimensional representation space (2.10)

Appendix B: Representation of the Algebra \(g^{2}\)

The geometric demonstration of the representation space of algebra \(g^{2}\)is shown in Fig. 2.

Fig. 2
figure2

Newton diagram of the algebra\(g^{2}\); corresponding to finite-dimensional representation space (3.20)

Indeed, another type of generators are generators of algebra \(g^{2}\). This algebra can be written in terms of first- and second-order differential operators that it operates on a representation space consists of x and y [41]. The algebra \(g^{2}\) contains 8 generators, which are:

$$\begin{aligned} \begin{array}{lllll} L^{1}=\partial _{x} &{}\quad (-1,0); &{}\quad L^{2}=x\partial _{x}-\frac{n}{3}&{}\quad (0,0);\\ L^{3}=y\partial _{y} -\frac{n}{3} &{}\quad (0,0); &{}\quad L^{4}=x^{2}\partial _{x} +2xy\partial _{y} -nx &{}\quad (1,0);\\ L^{5}=\partial _{y} &{}\quad (0,-1); &{}\quad L^{6}=x\partial _{y}&{}\quad (1,-1);\\ L^{7}=x^{2}\partial _{y}&{}\quad (2,-1);&{} \quad T=y\partial _{x}^{2}&{}\quad (-2,1); \end{array} \end{aligned}$$
(B.1)

obviously, with the effect of these generators on the representation space consisting of x and y (in a way similar to that in “Appendix A”):

$$\begin{aligned} L(T)x^{p}y^{q}\propto x^{p+\alpha }y^{q+\beta } \end{aligned}$$
(B.2)

Where \((\alpha ,\beta )\) are a grading of the degree of the terms x and y. It is noteworthy that the first seven generators \(L^{i}\) are due to the first-order differential of x and y or a combination of them, so they belong to algebra \(gl_{2} \times \mathbb {R}^{3}\). Here, it can be said that if n is a positive integer, the representation space is irreducible and these generators will have an invariant subspace, which is true to linear space of polynomials in the Fock space (3.20). In order to obtain the algebraic form of the Calogero model, only generators \(gl_{2} \times \mathbb {R}^{3}\) are sufficient to illustrate Hamiltonian (3.19). Then the corresponding algebraic form for the Hamiltonian (3.19) is given by

$$\begin{aligned} h_{A_{2} } =-2L^{2}L^{1}-12L^{3}L^{1}+\frac{8}{3}L^{7}L^{3}-2\left( {1+3\nu } \right) L^{1}-4\omega L^{2}-12\omega L^{3}-\frac{4}{3}L^{7}-2\lambda V_{p}. \end{aligned}$$
(B.3)

Appendix C: Eigenvalues and Eigenfunctions of the \(A_{2} \) Model

Based on Eqs. (2.8) and (3.19), we are able to obtain the normalized wavefunctions of the \(A_{2}\) model for \(n=1,2,...\) as follows:

$$\begin{aligned} \phi _{0}^{1}= & {} \left( {1+(1+3\nu )^{2}} \right) ^{-\frac{1}{2}}\left( {-(1+3\nu )\left| 1 \right\rangle +\left| 2 \right\rangle } \right) \\ \varepsilon _{1}^{1}= & {} \frac{3(1+3\nu )}{2+2\omega }+\frac{3(2+3\nu )}{(1+\omega )(1+4\omega )}+\frac{a_{1} (1+2\omega )}{2(1+\omega )(1+3\nu )}\sqrt{1+(1+3\nu )^{2}} \\ \phi _{1}^{1}= & {} a_{1} \left| 1 \right\rangle +\frac{1}{(1+2\omega )\sqrt{1+(1+3\nu )^{2}} }\left( {\frac{6(2+3\nu )}{(1+4\omega )}-\varepsilon _{1}^{1} +3(1+3\nu )} \right) \left| 2 \right\rangle -\frac{3}{(1+4\omega )\sqrt{1+(1+3\nu )^{2}} }\left| 4 \right\rangle \\ \phi _{0}^{2}= & {} \left( {1+\frac{1}{4}(1+3\nu )^{2}} \right) ^{-\frac{1}{2}}\left( {-\frac{1}{2}(1+3\nu )\left| 1 \right\rangle +\left| 2 \right\rangle } \right) \\ \phi _{0}^{3}= & {} \left( {1+\frac{1}{9}(1+3\nu )^{2}} \right) ^{-\frac{1}{2}}\left( {-\frac{1}{3}(1+3\nu )\left| 1 \right\rangle +\left| 2 \right\rangle }\right) \end{aligned}$$

where \(a_{1}\) is obtained from its normalization condition. Based on above results, we finally obtain the wavefunction in the nth state without distribution term \((k=0)\) has a form

$$\begin{aligned} \phi _{0}^{n} =\left( {1+\frac{1}{n^{2}}(1+3\nu )^{2}} \right) ^{-\frac{1}{2}}\left( {-\frac{1}{n}(1+3\nu )\left| 1 \right\rangle +\left| 2 \right\rangle } \right) \end{aligned}$$

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Panahi, H., Najafizade, S.A. & Mohammadkazemi Gavabar, M. Algebraic Method for Perturbed Three-Body Systems of \(\mathbf {A}_{\mathbf {2}} \) Solvable Potential. Few-Body Syst 61, 9 (2020). https://doi.org/10.1007/s00601-020-1543-7

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