PT-symmetric non-polynomial oscillators and hyperbolic potential with two known real eigenvalues in a SUSY framework

14 Pages·0.14 MB·English

by  B. Bagchi

PT-SYMMETRIC NON-POLYNOMIAL
OSCILLATORS AND HYPERBOLIC POTENTIAL
WITH TWO KNOWN REAL EIGENVALUES IN A
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SUSY FRAMEWORK
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B. BAGCHI
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J
Department of Applied Mathematics, University of Calcutta,
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92 Acharya Prafulla Chandra Road, Calcutta 700 009, India

1 E-mail: [email protected]
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C. QUESNE
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Physique Nucl´eaire Th´eorique et Physique Math´ematique,
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Universit´e Libre de Bruxelles, Campus de la Plaine CP229,
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0 Boulevard du Triomphe, B-1050 Brussels, Belgium
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E-mail: [email protected]
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Abstract
q
:
v Extending the supersymmetric method proposed by Tkachuk to the complex do-
i main, we obtain general expressions for superpotentials allowing generation of quasi-
X
exactlysolvablePT-symmetricpotentialswithtwoknownrealeigenvalues(theground
r
a state and first-excited state energies). We construct examples, namely those of com-
plexified non-polynomial oscillators and of a complexified hyperbolic potential, to
demonstrate how our scheme works in practice. For the former we provide a connec-
tion with the sl(2) method, illustrating the comparative advantages of the supersym-
metric one.
Running head: SUSY and PT-symmetric potentials
PACS: 03.65.Fd, 03.65.Ge
Keywords: supersymmetric quantum mechanics, quasi-exactly solvable potentials, PT sym-
metry, non-polynomial oscillator, hyperbolic potential
∗
Directeur de recherches FNRS
1
1 Introduction
Non-Hermitian Hamiltonians, in particular the PT-symmetric ones, are of great current
interest (see e.g. [1]–[12] and references quoted therein). The main reason for this is that
PT invariance, in a number of cases, leads to energy eigenvalues that are real. In this
regard, the early work of Bender and Boettcher [1] is noteworthy since it has sparked off
some very interesting developments subsequently.
Someyearsago,Tkachuk[13]proposedasupersymmetric (SUSY)methodforgenerating
quasi-exactly solvable (QES) potentials with two known eigenstates, which correspond to
the wavefunctions of the ground state and the first excited state. A distinctive feature of
this method is that in contrast with other ones, it does not require the knowledge of an
initial QES potential for constructing a new one. Later on, the procedure was extended to
deal with QES potentials with two arbitrary eigenstates [14] or with three eigenstates [15].
Quite recently, Brihaye et al. [16] established a connection between the Tkachuk approach
and the Turbiner one, based upon the finite-dimensional representations of sl(2) [17].
In this letter, we pursue Tkachuk’s ideas further by considering an extension of his
results to the case of PT-symmetric potentials. As a consequence, we arrive at a pair of
solutions, one of which is obtained by a straightforward and natural complexification of
Tkachuk’s results while the other is new. We demonstrate the applicability of our scheme
by focussing on two specific potentials, both of which are PT-symmetric.
2 Procedure
2.1 The Hermitian case
LetusstartwiththeHermitianSUSYcase, wherethetwo knowneigenstates aretheground
state and the first excited state. As is well known [18], the SUSY partner Hamiltonians are
given by
d2 d2
H(+) = A¯A = +V(+)(x), H(−) = AA¯ = +V(−)(x), (1)
−dx2 −dx2
2
foravanishingfactorizationenergy. HereAandA¯aretakentobefirst-derivativedifferential
operators, namely
d d
A = +W(x), A¯ = +W(x), (2)
dx −dx
where W(x) is the underlying superpotential and V(±)(x) are the usual SUSY partner
potentials
V(±)(x) = W2(x) W′(x). (3)
∓
We assume SUSY to be unbroken with the ground state of the SUSY Hamiltonian
H diag(H(+),H(−)) to belong to H(+):
s
≡
x
H(+)ψ(+)(x) = 0, ψ(+)(x) = C(+)exp W(t)dt , (4)
0 0 0 −
(cid:18) Z (cid:19)
C(+) being the normalization constant. Equation (4) implies
0
sgn(W( )) = 1. (5)
±∞ ±
In the above formulation of SUSY, the eigenvalues of H(+) and H(−) are related as
E(+) = E(−), E(+) = 0, (6)
n+1 n 0
while the corresponding eigenfunctions are intertwined according to
1 1
ψ(+)(x) = A¯ψ(−)(x), ψ(−)(x) = Aψ(+)(x). (7)
n+1 E(−) n n E(+) n+1
n n+1
q q
Following Tkachuk [13], let us consider expressing H(−) in the form
H(−) = H(+) +ǫ = A A¯ +ǫ, (8)
1 1 1
where ǫ = E(+) = E(−) corresponds to the energy of the first excited state of H(+) or of the
1 0
ground state of H(−) and the operators A and A¯ are such that relations analogous to (2)
1 1
hold in terms of a new superpotential W (x). Thus we can write
1
V(−)(x) = V(+)(x)+ǫ (9)
1
with V(+)(x) = W2(x) W′(x) from (3).
1 1 − 1
3
The ground state wave function of H(+) (or H(−)) can be read off from (4) as
1
ψ(−)(x) = C(−)exp xW (t)dt , (10)
0 0 − 1
(cid:18) Z (cid:19)
where C(−) is the normalization constant and
0
sgn(W ( )) = 1. (11)
1
±∞ ±
On the other hand, the wave function of the first excited state of H(+) is obtained in the
form
x C(−)
ψ(+)(x) = C(+)A¯exp W (t)dt , C(+) = 0 , (12)
1 1 − 1 1 √ǫ
(cid:18) Z (cid:19)
as guided by (7) and (10).
Using (3) and (9), it is clear that the superpotentials W(x) and W (x) have to satisfy
1
a constraint
W2(x)+W′(x) = W2(x) W′(x)+ǫ. (13)
1 − 1
The problem therefore amounts to finding a set of functions W(x) and W (x) satisfying
1
(13) for some ǫ > 0, along with the conditions (5) and (11). For this purpose, Tkachuk
introduced the following combinations
W±(x) = W1(x) W(x), (14)
±
which transform (13) into
′
W+(x) = W−(x)W+(x)+ǫ. (15)
An advantage with Eq. (15) is that it readily gives W−(x) in terms of W+(x):
W′ (x) ǫ
W−(x) = + − . (16)
W (x)
+
Consequently the representations
1 W′ (x) ǫ 1 W′ (x) ǫ
W(x) = W (x) + − , W (x) = W (x)+ + − (17)
2 " + − W+(x) # 1 2 " + W+(x) #
satisfy Eq. (13). In (17), W (x) is some function for which W(x) and W (x) fulfil condi-
+ 1
tions (5) and (11). This means
sgn(W ( )) = 1. (18)
+
±∞ ±
4
Restricting to continuous functions W (x), the above condition reflects that W (x)
+ +
must have at least one zero. Then from (16) and (17), W−(x), W(x), and W1(x) may have
poles. Tkachuk considered the case when W (x) has only one simple zero at x = x . In
+ 0
the neighbourhood of x , one gets
0
W′ (x) ǫ W′ (x ) ǫ+(x x )W′′(x )+
+ − + 0 − − 0 + 0 ···, (19)
W (x) ≃ (x x )W′ (x )+
+ 0 + 0
− ···
so that the superpotentials will be free of singularity if one chooses
′
ǫ = W (x ). (20)
+ 0
One is then led to
1 W′ (x) W′ (x ) 1 W′ (x) W′ (x )
W(x) = W (x) + − + 0 , W (x) = W (x)+ + − + 0 .
2 " + − W+(x) # 1 2 " + W+(x) #
(21)
To summarize, provided the continuous function W (x) with a single pole at x = x is
+ 0
such that W(x) and W (x), given by (21), satisfy conditions (5) and (11), the Hamiltonian
1
H(+) has two known eigenvalues 0 and ǫ, given by (20), with corresponding eigenfunc-
tions (4) and
x
ψ(+)(x) = C(+)W (x)exp W (t)dt . (22)
1 1 + − 1
(cid:18) Z (cid:19)
In deriving (22), use is made of (2), (12), and (14).
2.2 The non-Hermitian case
In the non-Hermitian case, we have to deal with complex potentials. Consider the decom-
position [3]
W(x) = f(x)+ig(x), V(±)(x) = V(±)(x)+iV(±)(x), (23)
R I
where f, g, V(±), V(±) R and
R I ∈
V(±) = f2 g2 f′, V(±) = 2fg g′. (24)
R − ∓ I ∓
If V(±)(x) are PT-symmetric, then f(x) and g(x) are odd and even functions, respectively.
5
It may benoted that Eqs. (4) – (13) remain truefor some realand positive ǫ. Employing
the separation
W (x) = f (x)+ig (x), V(+)(x) = V(+)(x)+iV(+)(x) (25)
1 1 1 1 1R 1I
with f , g , V(+), V(+) R, the behaviour of f (x) and g (x) also turns out to be odd and
1 1 1R 1I ∈ 1 1
even, respectively, should V(+)(x) be PT-symmetric.
In the non-Hermitian case, the conditions (5) and (11) are to be replaced by
sgn(f( )) = sgn(f ( )) = 1. (26)
1
±∞ ±∞ ±
These conditions are actually compatible with the odd character of f and f .
1
On introducing the first relations of (23) and (25) into (13) and splitting into real and
imaginary parts, we get the system of two equations
f2 g2 +f′ = f2 g2 f′ +ǫ, 2fg +g′ = 2f g g′. (27)
− 1 − 1 − 1 1 1 − 1
The problem now amounts to finding a pair of odd functions f, f and a pair of even
1
functions g, g , satisfying Eq. (27) for some ǫ > 0, as well as the conditions (26).
1
To this end, we introduce the linear combinations
f±(x) = f1(x) f(x), g±(x) = g1(x) g(x), (28)
± ±
which replace the constraints (27) by
′ ′
f+ = f−f+ −g−g+ +ǫ, g+ = f+g− +f−g+. (29)
Solving for f− and g− in terms of f+ and g+, we get
(f′ ǫ)f +g′ g (f′ ǫ)g +g′ f
f− = + − + + +, g− = − + − + + +. (30)
f2 +g2 f2 +g2
+ + + +
Hence the functions
1 (f′ ǫ)f +g′ g 1 (f′ ǫ)g g′ f
f = f + − + + + , g = g + + − + − + + ,
2 " + − f+2 +g+2 # 2 " + f+2 +g+2 #
1 (f′ ǫ)f +g′ g 1 (f′ ǫ)g g′ f
f = f + + − + + + , g = g + − + − + + (31)
1 2 " + f+2 +g+2 # 1 2 " + − f+2 +g+2 #
6
satisfy the coupled equations (27). In (31), f must be such that the conditions (26) are
+
fulfilled. These suggest
sgn(f ( )) = 1. (32)
+
±∞ ±
If we restrict ourselves to continuous functions f (x) and g (x), the condition (32)
+ +
shows that f (x) must have at least one zero. For simplicity’s sake, we assume that f (x)
+ +
has only one simple zero at x = x . This means that the parity operation is defined with
0
respect to a mirror placed at x = x . Thus in the neighbourhood of x , we get
0 0
(f′ ǫ)f +g′ g
+ − + + + (33)
f2 +g2
+ +
[f′ (x ) ǫ]f′ (x )+g (x )g′′(x )
(x x ) + 0 − + 0 + 0 + 0 + if g (x ) = 0
≃ − 0 [g (x )]2 ··· + 0 6
+ 0
1 f′ (x ) ǫ
+ 0 − + if g (x ) = 0, (34)
≃ x x f′ (x ) ··· + 0
0 + 0
−
(f′ ǫ)g g′ f
+ − + − + + (35)
f2 +g2
+ +
f′ (x ) ǫ
+ 0 − + if g (x ) = 0
≃ g (x ) ··· + 0 6
+ 0
[f′ (x )+ǫ]g′′(x )
+ 0 + 0 + if g (x ) = 0. (36)
≃ − 2[f′ (x )]2 ··· + 0
+ 0
Hence the superpotentials will be free of singularity if either g (x ) = 0 and ǫ arbitrary or
+ 0
6
g (x ) = 0 and ǫ = f′ (x ) (or ǫ = W′ (x ) since g′ (x ) = 0).
+ 0 + 0 + 0 + 0
The superpotentials may therefore be written as
1 W′ (x) ǫ 1 W′ (x) ǫ
W(x) = W (x) + − , W (x) = W (x)+ + − , (37)
2 " + − W+(x) # 1 2 " + W+(x) #
if W (x ) = ig (x ) = 0 and ǫ > 0, or
+ 0 + 0
6
1 W′ (x) W′ (x ) 1 W′ (x) W′ (x )
W(x) = W (x) + − + 0 , W (x) = W (x)+ + − + 0 ,
+ 1 +
2 " − W+(x) # 2 " W+(x) #
(38)
if W (x ) = ig (x ) = 0 and ǫ = W′ (x ).
+ 0 + 0 + 0
To summarize, in the non-Hermitian PT-symmetric case, we get two types of solutions:
the one given by (38) is obtained as a straightforward consequence of the complexification
of Tkachuk’s result, while the other given by (37) is new.
7
3 Applications
3.1 A family of complexified non-polynomial oscillators
As the first application of our scheme we consider the case of a family of complexified non-
polynomial oscillators. This corresponds to the choice f = ax, g = bx2m (a > 0, b = 0,
+ +
6
m N), which leads to
∈
W (x) = ax+ibx2m, m N. (39)
+
∈
Here x = 0 and the first type of solutions, given in (37), applies to the case m = 0, while
0
the second one, given in (38), has to be used for m N .
0
∈
The former case reduces to the well-studied exactly solvable PT-symmetric oscillator
potential [4,11, 12], thus partly accounting for the name of the family of potentials. Indeed,
to get it in the standard form we have to set a = 2, b = 2c, and ǫ = 4α. Then V(+)
−
assumes the form
α2 1
V(+)(x) = (x ic)2 +2(α 1)+ − 4 , (40)
− − (x ic)2
−
along with
α 1 α 1
W(x) = x ic+ − 2, W (x) = x ic − 2. (41)
1
− x ic − − x ic
− −
These agree with the two independent forms of the complex superpotentials proposed by
us previously [11] in connection with para-SUSY and second-derivative SUSY of (40). The
potential (40) can be looked upon as a transformed three-dimensional radial oscillator for
the complex shift x x ic, c > 0, and replacing the angular momentum parameter l by
→ −
α 1. The presence of a centrifugal-like core notwithstanding, the shift of the singularity
− 2
off the integration path makes (40) exactly solvable on the entire real line for any α > 0
like the harmonic oscillator to which (40) reduces for α = 1 and c = 0.
2
In contrast, the case m N is entirely new. For such m values, the superpotentials are
0
∈
given by
1 2mibx2m−2 1 2mibx2m−2
W(x) = ax+ibx2m , W (x) = ax+ibx2m + ,
2 " − a+ibx2m−1# 1 2 " a+ibx2m−1#
(42)
where we used ǫ = W′ (0) = a.
+
8
The corresponding potentials turn out to be
1 4m(m 1)ibx2m−3
V(+)(x) = b2x4m +2iabx2m+1 8mibx2m−1 +a2x2 2a+ −
4"− − − a+ibx2m−1
4m(m 1)iabx2m−3
+ − , m = 1,2,3,..., (43)
(a+ibx2m−1)2 #
which are QES and are seen to be PT-symmetric as well. The ground and first-excited
state wave functions corresponding to the above QES potentials are easily determined to
be
1 ib
ψ(+)(x) (a+ibx2m−1)m/(2m−1)exp ax2 x2m+1 , (44)
0 ∝ "−4 − 2(2m+1) #
1 ib
ψ(+)(x) x(a+ibx2m−1)(m−1)/(2m−1)exp ax2 x2m+1 . (45)
1 ∝ "−4 − 2(2m+1) #
It is worth noting that the first member of the set (43) obtained for m = 1,
1
V(+)(x) = b2x4 +2iabx3 +a2x2 8ibx 2a , (46)
4 − − −
(cid:16) (cid:17)
is a quartic potential differing from the known QES ones [2, 9]. All the remaining members
of the set, starting with that associated with m = 2,
1 8ibx 8iabx
V(+)(x) = b2x8 +2iabx5 16ibx3 +a2x2 2a+ + , (47)
4 "− − − a+ibx3 (a+ibx3)2#
are non-polynomial potentials. As for the PT-symmetric oscillator (40), the shift of the
singularity off the integration path makes such potentials QES.
On introducing the new variable z = x(a+ibx2m−1)−1/(2m−1), the first-excited statewave
function (45) can be rewritten in terms of the ground state one (44) as ψ(+)(z) zψ(+)(z).
1 ∝ 0
By setting in general ψ(+)(z) = ψ(+)(z)φ(+)(z), where φ(+)(z) 1 and φ(+)(z) z, the
n 0 n 0 ∝ 1 ∝
Schr¨odinger equation for the potentials (43) is transformed into the differential equation
d2 d
Tφ(+)(z) a−2/(2m−1)(1 ibz2m−1)4m/(2m−1) +az φ(+)(z) = E(+)φ(+)(z). (48)
n ≡ "− − dz2 dz# n n n
For m = 1, the coefficient of the second-order differential operator in (48) becomes a
quartic polynomial in z, thus showing that T can be expressed as a quadratic combination
of the three sl(2) generators
d d N d
J+ = z2dz −Nz, J0 = zdz − 2 , J− = dz, (49)
9
corresponding to the two-dimensional irreducible representation (i.e., N = 1 in (49)) [16,
17]. The result reads
T = a−2 −b4J+2 −4ib3J+J0 +6b2J+J− +4ibJ0J− −J−2 −2ib3J+ +6b2J0 +2ibJ− +3b2 .
(cid:16) (cid:17)
(50)
For higher m values, the differential operator T contains a non-vanishing element of
the kernel [16]. It is worth stressing that in such a case, the sl(2) method becomes quite
ineffective for constructing new QES potentials, whereas the SUSY one is not subject to
such restrictions.
3.2 A complexified hyperbolic potential
Our next example is that of a complexified hyperbolic potential induced by the represen-
tations f = Asinhαx, g = B (A, α > 0, B = 0). We then get
+ +
6
W (x) = Asinhαx+iB, (51)
+
which gives for x = 0
0
1 Aαcoshαx ǫ
W(x) = Asinhαx+iB − , (52)
2 " − Asinhαx+iB#
1 Aαcoshαx ǫ
W (x) = Asinhαx+iB + − . (53)
1
2 " Asinhαx+iB#
The resulting expression for the complexified hyperbolic potential is
1
V(+)(x) = A2sinh2αx 4Aαcoshαx+2ǫ+α2 B2 +2iABsinhαx
4" − −
ǫ2 α2(A2 B2)
+ − − . (54)
(Asinhαx+iB)2#
Clearly V(+)(x) is PT-symmetric. The two known eigenstates of (54) correspond to the
ground state and first excited state as outlined earlier. These are
ψ(+)(x) (Acoshαx ν)14(1−αǫν)(Acoshαx+ν)14(1+αǫν)
0 ∝ −
A 1 i A
exp coshαx iBx arctan sinhαx
× "−2α − 2 − 2 (cid:18)B (cid:19)
10