arXiv:1101.0773v1 [hep-th] 4 Jan 2011
Vestnik Leningradskogo Universiteta (Ser.4 Fiz.Khim.) 4(25) (1988) pp.3-9 (translated from Russian)
FACTORIZATION METHOD IN CURVILINEAR COORDINATES AND PAIRING OF LEVELS FOR MATRIX POTENTIALS A.A.Andrianova , M.V. Iof feb , Tsu Zhun-Pin Saint-Petersburg (Leningrad) State University, 198504 St.-Petersburg, Russia Multidimensional factorization method is formulated in arbitrary curvilinear coordinates. Particular cases of polar and spherical coordinates are considered and matrix potentials with separating variables are constructed. A new class of matrix potentials is obtained which reveals a double degeneracy or equidistant splitting of energy levels (hidden symmetry).
1
Introduction.
The Factorization Method is an effective tool in search and construction of exactly solvable Schr¨odinger problems [1] and of the models with equivalent (almost coinciding) energy spectra [2], [3]. The mainstream of the method was related with one-dimensional problems [1]. Recently, the multidimensional generalization of the Factorization Method was elaborated [4] - [6], and its supersymmetric origin was demonstrated [7], [8]. The main results for multidimensional case were obtained in Cartesian coordinates. Meanwhile, the symmetry properties of original potential mark out the preferred coordinate systems related to orbits of the symmetry group (separation of variables). Usually, these coordinate systems are curvilinear ones. Thus, it would be interesting to reformulate the Multidimensional Factorization Method in curvilinear coordinates and to diagonalize matrix potentials appeared under the multidimensional Darboux transformation (Sect. 2). Thereby, starting from scalar potentials with separated variables a class of matrix potentials will be found which is also amenable to separation of variables . The polar and spherical coordinates and the particular case of Coulomb potential will be considered below in detail. a b
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1
The full variety of Hamiltonians interrelated by the intertwining relations obeys twofold degeneracy of the spectrum. The latter has [2], [3], [7], [8] the supersymmetric interpretation: spectra of systems with even and odd fermionic modes coincide. However, the Factorization Method allows the construction of potentials with additional twofold level degeneracy without a simple supersymmetric interpretation. Together with standard supersymmetry of the full system of equivalent Hamiltonians, this means fourfold degeneracy of levels. Such potentials are built in Sect. 3, and the Factorization Method in curvilinear coordinates is more convenient for their description. More general class of potentials whose energy levels are paired with a given constant (equidistant) splitting is found.
2
Factorization Method in curvilinear coordinates.
We shall start from the case of two-dimensional space with arbitrary curvilinear coordinates (q 1 , q 2 ), where the interval ds2 = gik dq i dq k is given by the corresponding metric tensor gik . The required generalization of Factorization Method includes: 1) constructing the operators Q± l (l = 1, 2), which factorize the initial Hamiltonian: 1 H (0) = − △ + V (0) ; 2 2) constructiing the matrix Hamiltonian 1 (1) (1) Hik = − △ + Vik , 2
(1)
which is connected with H (0) by intertwining relations; 3) constructing the scalar Hamiltonian 1 H (0) = − △ + V (0) , 2 (1)
also intertwined with Hik . These necessary steps of the Factorization Method provide the well known links [4] - [6] (1) between the spectra of operators H (0) , Hik , H (2) and between the corresponding eigenfunctions Ψ(0) , Ψ(1) , Ψ(2) . The expression for the operators Q± can be found by using the representation of the Laplace operator (kinetic terms in the Hamiltonians above) in arbitrary curvilinear coordinates: △ = g ik ∇i ∇k ,
2
where g ik is a contravariant tensor inverse to the metric tensor gik , and the covariant derivative operator ∇i is defined by means of three-index Christoffel symbols Γ : ∂ ϕ ∂q i ≡ ∂i Aj − Γkji Ak
∇i ϕ ≡ ∂i ϕ ≡ ∇i Aj
j
∇i A ≡ ∇i Bjl ≡
∂i A + Γjki Ak ∂i Bjl − Γkji Bkl
(ϕ − scalar), (Aj − covariant vector),
j
−
Γkli Bjk
(Aj − contravariant vector), (Bjl − second rank covariant tensor),
(2)
∂gil + etc. The Christoffel symbols are expressed in terms of the metric tensor = ∂q j ∂gjl ∂g − ∂qijl , and they are introduced in differential calculus of tensor objects to take into ∂q i
Γkij
1 kl g 2
account the change of basic vectors and metrics under parallel transport . In a particular case of Cartesian orthogonal coordinates gij = δij , Γkij = 0, ∇i = ∂i . Let us choose the operators Q± l in the form: 1 1 ± Ql ≡ √ ∓∇l + (∇l χ) = √ ∓∇l + (∂l χ) , (3) 2 2 where exp (−χ)− one of the two solutions of the Schr¨odinger equation H (0) exp (−χ) = E exp (−χ), and + − + + [Q− [Q− l , Qk ] = (∇l ∇k χ), l , Qk ] = [Ql , Qk ] = 0. p The presence of the volume factor det(gik ) in the scalar product and the property ∇i gkj = 0 a are necessary to prove that operators Q± l are mutually conjugate . Then, H (0) can be written in a factorized form: 1 1 l l (0) + l− H = Ql Q + E = − △ + (∇l χ)(∇ χ) − (∇l ∇ χ) + E, (4) 2 2
Ql ± = g lk Q± k,
where the covariant derivative in Q− l acts onto a scalar ∇l Ψ = ∂l Ψ, and covariant derivative in Q+ acts onto a vector in accordance with (2). l √ (1) To build the matrix Hamiltonian H let’s use the operators Pl± ≡ gǫlk Qk ∓ , P l ± = g lk Pk± = √1g ǫlk Q∓ k (g ≡ det gik , with ǫlk being completely antisymmetric unit pseudotensor). k+ k− = 0 = Q− The important property of orthogonality Pk+ Qk − = Pk− Qk + = Q+ kP kP (1) k (0) guarantees, in particular, intertwining the operator H and the operator Hi : (1) k
Hl
(1) k
Hl a
k+ + Pl− P k + + δlk E = δlk H (0) + (∇l ∇k χ) ≡ Q− l Q
− (0) Q− ; k = Ql H
(1) k
Ql + Hl
= H (0) Qk + .
We remind that the flat space only is considered in this paper.
3
(5)
Similarly to the case of Cartesian coordinates [4] - [6], these relations lead to the connection (0) (1) of spectra of H (0) and H (1) : {En } ⊂ {En } and of their wave functions: (1) Ψk (E) = p
4 E−E (1) k
Analogously, the operator Hl H (2)
(0) (0) Q− k Ψ (E), Ψ (E) = p
1 E−E
(1)
Qk + Ψk (E).
is intertwined with the scalar Hamiltonian: 1 1 + l− l l ≡ Pl P + E = − △ + (∇l χ)(∇ χ) + (∇l ∇ χ) + E = H (0) + △χ, 2 2 (2)
(1) k
whose spectrum {En } also lies in the spectrum of Hl (0) (2) of H (1) coincides either with En or with En : (1) k
Hl
(1) k
Pk− = Pl− H (2) ;
P l + Hl
(6)
. An arbitrary point of the spectrum = H (2) P k +
1 1 (1) (1) Pk− Ψ(2) (E); Ψ(2) (E) = p P k + Ψk (E). Ψk (E) = p E−E E−E For illustration we consider the case of polar coordinates (ρ, ϕ) on the plane: 1 1 0 ; Γ122 = −ρ; Γ212 = Γ221 = . gij = 2 0 ρ ρ From (4) - (6) one obtains the explicit form of potentials (∂1 ≡ ∂/∂ρ, ∂2 ≡ ∂/∂ϕ) : 1 1 1 2 1 (0) 2 2 2 V = (∂1 χ) + 2 (∂2 χ) − (∂1 χ) − 2 (∂2 χ) − (∂1 χ) + E, 2 ρ ρ ρ ! 1 1 (0) 2 ∂ ∂ χ − ∂ χ V + ∂ χ 1 2 2 2 3 (1) k 1 ρ ρ , Vi = δik V (0) + (∇i ∇k χ) = ∂1 ∂2 χ − ρ1 ∂2 χ V (0) + ρ12 ∂22 χ + ρ1 ∂1 χ 1 1 1 2 1 2 (2) 2 2 (∂1 χ) + 2 (∂2 χ) + (∂1 χ) + 2 (∂2 χ) + (∂1 χ) + E. V = 2 ρ ρ ρ
(7) (8) (9)
For the particular case of centrally symmetrical χ = χ(ρ), the matrix potential is diagonal just in polar coordinates: (0) V (ρ) + ∂12 χ + E 0 (1) k . Vi = 0 V (0) (ρ) + 1ρ ∂1 χ + E The method can be generalized to a space of arbitrary dimension. For the case of physically interesting three-dimensional space, the operators P ± are second rank tensors: √ Pik± = gǫikl Ql ∓ , Pik+ Qk − = Qk + Pik− = Pik− Qk + = Qk − Pik+ = 0. 4
The intertwining relations are: (1) k
Hi
− (0) Q− ; k = Qi H
(2) k Plk− Hi
=
(1) m − Hl Pim ;
where H
(0)
(1) k
Hi
(2) k
Hi
H (3)
(1) k
Q+ k Hi
=
l− Q+ l Q
H
(3)
= H (0) Q+ i ;
k−
Q
i−
=Q
(2) k
Hi
(2) k Hi ;
(1) m
+ Pkl+ = Pim Hl
(3) Q+ i H
=
;
(2) k Hi Q+ k,
1 1 l + E = − △ + (∇l χ)(∇ χ) − △χ ; 2 2
k+ = Q− + Pli− P lk + + E = δik H (0) + (∇i ∇k χ); i Q
k− = Q+ + Pli+ P kl − + E = δik H (3) − (∇i ∇k χ); i Q 1 1 l − l+ = Ql Q + E = − △ + (∇l χ)(∇ χ) + △χ . 2 2
For the particular case of spherical coordinates (r, θ, ϕ), one has: g11 = 1; Γ122 = −r; and V
(0)
(1) k
Vi
(2) k
Vi
V (3)
Γ133 = −r sin2 θ;
g22 = r 2
g33 = r 2 sin2 θ,
Γ233 = − sin θ cos θ;
1 Γ212 = Γ313 = ; r
Γ323 = cot θ,
2 1 1 1 1 cot θ = (∂1 χ)2 + 2 (∂2 χ)2 + 2 2 (∂3 χ)2 − ∂12 χ − ∂1 χ − 2 ∂22 χ − 2 ∂2 χ − 2 r r r r r sin θ 1 − 2 2 ∂32 χ + E, r sin θ
= δik V (0) + ∇i ∇k χ = 1 1 1 1 (0) 2 ∂1 ∂3 χ − r ∂3 χ ∂1 ∂2 χ − r ∂2 χ r2 V + ∂1 χ r 2 sin2 θ 1 1 1 1 (0) 2 = ∂2 ∂3 χ − cot θ∂3 χ ∂1 ∂2 χ − r ∂2 χ V + r2 ∂2 χ + r ∂1 χ r 2 sin2 θ 1 cot θ 1 2 V (0) + r2 sin ∂1 ∂3 χ − 1r ∂3 χ r12 ∂2 ∂3 χ − cot θ∂3 χ 2 θ ∂3 θ + r ∂1 χ + r 2 ∂2 χ χ → −χ (1) k , = Vi V (0) (χ) → V (3) (χ)
,
= V (0) (χ → −χ).
From these expressions, one can conclude that in terms of the spherical coordinates in 3dimensional space the matrix potentials V (1) , V (2) are also diagonal for spherically symmetrical case χ = χ(r). Thus, the scalar problem amenable to separation of variables produces the matrix problems which allow the separation of variables as well. 5
From this point of view, the problem of the Coulomb potential formation, considered earlier in [9], looks interesting: α α V (0) = − ; V (3) = + ; r α r α +r 0 −r 0 0 (2) k (1) k 0 0 0 ; Vi 0 0 = Vi = 0 0 0 0 0
and its Darboux trans-
0 0 . 0
(10)
Thus, the Factorization Method developed above in curvilinear coordinates allows to un(1) ravel the structure of the matrix Coulomb potential. When interpreting Ψi as a vector particle wave function, then it follows from (10) that bound states in matrix Coulomb potential exist for one polarization (along r) only, and the motion with two other polarizations is free.
3
Matrix potentials with pairing of levels.
The interrelation of spectra of the Hamiltonians H (n) for arbitrary space dimension d was studied in [4] - [6], and it was proven there that the spectrum of the joint Hamiltonian (0) H 0 0 0 H (1) 0 b = 0 (11) H 0 0 ... 0 0 0 0 H (d) is twofold degenerate (may be, excluding the ground state). This degeneracy has [7], [8] a supersymmetric interpretation: the spectra of Hamiltonians coincide for even and odd number of fermionic excitations: (0) (1) H 0 0 H 0 0 HB = 0 H (2) 0 ; HF = 0 H (3) 0 . 0 0 ... 0 0 ...
At the same time, a class of potentials with higher degree of degeneracy exists. In order to construct the potentials with additional degeneracy in 2-dimensional space, we examine the case with identical potentials V (0) and V (2) : δV ≡ V (2) − V (0) = △χ = 0. The real solutions of the Laplace equation are simply expressed in Cartesian coordinates in terms of arbitrary analytical functions of complex variable z ≡ x + iy : χ = F (z) + F (z) = 2ReF (z), z¯ ≡ x − iy. However, polar coordinates are more convenient to study the physical properties of potentials V (n) . The general solution of the Laplace equation, nonsingular for finite ρ, is: χ=
∞ X
αm χm (ρ, ϕ),
χm = ρm sin(mϕ + δm ),
m=0
6
(12)
where αm , δm are arbitrary real numbers. Substituting these solutions into (7) - (9), one obtains a class of nonsingular potentials: V
(0)
(1) k
Vi
=V
=
(2)
∞ ∞ X 1 X 2 2 2m−2 mα ρ + mnαm αn ρm+n−2 cos[(m − n)ϕ + δm − δn ], = 2 m=1 n
