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Prepotential approach to quasinormal modes
Annals of Physics 326 (2011) 1394–1407
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Prepotential approach to quasinormal modes Choon-Lin Ho Department of Physics, Tamkang University, Tamsui 251, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 17 October 2010 Accepted 5 November 2010 Available online 7 January 2011 Keywords: Prepotential Quasinormal modes Exact and quasi-exact solvabilities
a b s t r a c t In this paper we demonstrate how the recently reported exactly and quasi-exactly solvable models admitting quasinormal modes can be constructed and classified very simply and directly by the newly proposed prepotential approach. These new models were previously obtained within the Lie-algebraic approach. Unlike the Lie-algebraic approach, the prepotential approach does not require any knowledge of the underlying symmetry of the system. It treats both quasi-exact and exact solvabilities on the same footing, and gives the potential as well as the eigenfunctions and eigenvalues simultaneously. We also present three new models with quasinormal modes: a new exactly solvable Morse-like model, and two new quasi-exactly solvable models of the Scarf II and generalized Pöschl–Teller types. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Quasinormal modes (QNM) has attracted great interest in recent years [1]. They arise as waves emitted by a perturbed neutron star or black hole that are outgoing to spatial infinity and the event horizon. Generally, the wave function of QNMs has discrete complex frequency, whose imaginary part leads to a damping behavior. QNM carry information of black holes and neutron stars, and thus are of importance to gravitational-wave astronomy. In fact, these oscillations, produced mainly during the formation phase of the compact stellar objects, can be strong enough to be detected by several large gravitational wave detectors under construction. As black hole potentials are generally too complicated to allow analytic treatment, so in order to understand the origin of the discrete imaginary frequencies, one can try approximating the top region of the black hole potential by some inverted potentials which are solvable. This has been done by using the inverted harmonic oscillator [2], and the Pöschl–Teller potential [3]. Recently, in [4] we have extended the number of exactly solvable models that admit QNMs. We take QNMs to include both decaying and growing modes with complex energies. Furthermore, we E-mail address: [email protected] 0003-4916/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2010.11.020
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have provided the first model with QNM that is quasi-exactly solvable (QES). A system is called QES if a part of its spectrum, but not the whole spectrum, can be determined analytically [5–12]. Our approach in [4] was to study solutions of QNM based on the sl(2)-Lie-algebraic approach to one-dimensional QES theory [5,6,8,9]. We demonstrated that, by suitably complexifying some parameters of the generators of the sl(2) algebra while keeping the Hamiltonian Hermitian, we could indeed obtain potentials admitting exact or quasi-exact QNMs. These models were later re-studied numerically by the asymptotic iteration method in [13]. In this paper we would like to show that exactly solvable and QES models with QNMs can be constructed much more simply without resorting to the machinery of Lie-algebra. This is achieved through the prepotential approach proposed recently [14–17]. This is a simple constructive approach, based on the so-called prepotential [7,18,19], which can give the potential as well as the eigenfunctions and eigenvalues simultaneously. The novel feature of the approach is that both exact and quasi-exact solvabilities can be solely classified by two integers, the degrees of two polynomials which determine the change of variables and the zero-th order prepotential. Hence this approach treats both quasi-exact and exact solvabilities on the same footing, and it provides a simple way to determine the required change of variables, say x, to a new one z = z(x). All the well-known exactly solvable models classified in supersymmetric quantum mechanics (SUSYQM) [20], the QES models discussed in [5,6,8– 10], and some new QES ones (also for non-Hermitian Hamiltonians), can be generated by appropriately choosing the two polynomials. Our approach, unlike the Lie-algebraic approach, does not require any knowledge of the underlying symmetry of the system. Furthermore, our approach can generate the Coulomb, Eckart, Rosen–Morse type I and II models [16] which are not covered by the standard Lie-algebraic program [8,9]. Compared with SUSYQM, our approach has the advantage that we do not have to assume the sufficient condition for integrability needed in SUSYQM, namely, shape invariance. In fact, shape invariance comes out automatically from this approach [17]. What is more, the transformation of the original variable z(x) is determined within the prepotential approach, whereas in SUSYQM this has to be taken as given from the known solutions of the respective models before one could solve the shape invariance condition. We shall adopt the prepotential approach here to generate and classify all one-dimensional exactly solvable and QES models with QNMs based on sinusoidal coordinates, namely, those coordinates z(x) whose derivatives (with respect to x) squared are at most quadratic in z. Our strategy is to suitably complexify some or all of the parameters in the prepotential while keeping the resulted potentials real. We find that all the systems reported in [4] can be very easily constructed. During the course of investigation, we also realize that there are three new QNM models which were missed in [4]. We thus take this opportunity to report on them. The plan of the paper is as follows. In Section 2 we briefly review the essence of the prepotential approach. Sections 3 to 5 then discuss the prepotential construction of QNM models of the Scarf II, the Morse, and the generalized Pöschl–Teller type, respectively. These three types of potentials have some interesting features, and serve as good examples to illustrate the procedure. Furthermore, there is one new QNM model in each of these three types that we would like to report. Section 6 summarizes the paper. In Appendices A–C, we list all other cases for easy reference. 2. Prepotential approach The main ideas of the prepotential approach [14,15] can be summarized as follows (we adopt the unit system in which h and the mass m of the particle are such that h = 2m = 1). Consider a wave function /N(x) (N: non-negative integer) which is defined as
/N ðxÞ eW 0 ðxÞ pN ðzÞ;
ð1Þ
with
pN ðzÞ
1; QN
N ¼ 0;
N > 0: k¼1 ðz zk Þ;
ð2Þ
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Here z = z(x) is some real function of the basic variable x, W0(x) is a regular function of z(x), and zk’s are the roots of pN(z). The variable x is defined on the full line, half-line, or finite interval, as dictated by the choice of z(x). The function pN(z) is a polynomial in an (N + 1)-dimensional Hilbert space with the basis h1, z, z2, . . . , zNi. W0(x) defines the ground state wave function. The wave function /N can be recast as
/N ¼ expðW N ðx; fzk gÞÞ;
ð3Þ
with WN given by
W N ðx; fzk gÞ ¼ W 0 ðxÞ
N X
ln jzðxÞ zk j:
ð4Þ
k¼1
Operating on /N by the operator d2/dx2 results in a Schrödinger equation HN/N = 0, where 2
HN ¼
d
þ VN; 2 dx 00 V N W 02 N WN:
ð5Þ ð6Þ
Here and below the prime represents derivative with respect to x. Since the potential VN is determined by WN, we thus call WN the Nth order prepotential. From Eq. (4), one finds that VN has the form VN = V0 + DVN: 00 V 0 ¼ W 02 0 W0; N X z00 X 1 z02 DV N ¼ 2 W 00 z0 þ : 2 k¼1 z zk ðz z k Þðz zl Þ k;l
ð7Þ
k–l
Thus the form of VN, and consequently its solvability, are determined by the choice of W0(x) and z02 02 (or equivalently by z00 ¼ ðdz =dzÞ=2Þ. Let W 00 z0 ¼ Pm ðzÞ and z02 ¼ Q n ðzÞ be two polynomials of degree m and n in z, respectively. The variables x and z are related by
xðzÞ ¼
Z
z
dz pffiffiffiffiffiffiffiffiffiffiffiffi ; Q n ðzÞ
ð8Þ
and the prepotential W0(x) is determined as
W 0 ðxÞ ¼
Z
z
dz
Pm ðzÞ : Q n ðzÞ z¼zðxÞ
ð9Þ
We assume (8) is invertible to give z = z(x). Eqs. (8) and (9) define the change of variables z(x) and the corresponding prepotential W0(x). Thus, Pm(z) and Qn(z) determine the quantum system. Of course, for bound state problems the choice of Pm and Qn must ensure normalizability of /0 = exp(W0). Now depending on the degrees of the polynomials Pm and Qn, we have the following situations [14,15]: (i) if max{m, n 1} 6 1, then in VN(x) the parameter N and the roots zk’s will only appear as an additive constant and not in any term involving powers of z. Such system is then exactly solvable; (ii) if max{m, n 1} = 2, then N may appear in the first power term in z, but zk’s only in an additive term. If N does appear before the z-term, then the system belongs to the so-called type 1 QES system defined in [6], i.e., for each N P 0, VN admits N + 1 solvable states with the eigenvalues being given by the N + 1 sets of roots zk’s. This is the main type of QES systems considered in the literature; (iii) if max{m, n 1} P 3,1 then not only N but also zk’s may appear in terms involving powers of z. If zk’s do appear before any z-dependent term, then for each N P 0, there are N + 1 different potentials VN, differing in several parameters in terms involving powers of z, have the same 1 We take this opportunity to correct a typographic error in [14,15], where the condition for m and n for this case was erroneously written as min{m, n 1} P 3.
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eigenvalue (when the additive constant, or the zero point, is appropriately adjusted). When zk’s appear only in the first power term in z, such systems are called type 2 QES systems in [6]. We see that QES models of higher types are possible. This gives a very simple algebraic classification of exact and quasi-exact solvabilities. Previously exact and QES systems were treated separately. In the rest of this paper we shall consider only cases with m, n 6 2. Coordinates with n 6 2 are called the sinusoidal coordinates. Let P2(z) = A2z2 + A1z + A0 and Q2(z) = az2 + bz + c. Hence the solvability of the system is determined solely by A2: exactly solvable if A2 = 0, or (type 1) QES otherwise. The potential VN takes the form
V N ¼ W 02 0
W 000
2
þ aN 2A1 N 2A2 Nz 2A2
N X
zk 2
k¼1
N X k¼1
( ) 1 a b X Q 2 ðzk Þ P2 ðzk Þ zk : z zk 4 l–k zk zl 2
ð10Þ Demanding the residues at zk’s vanish gives the Bethe ansatz equations satisfied by the roots zk’s:
a b X Q 2 ðzk Þ P2 ðzk Þ zk ¼ 0; 4 l–k zk zl 2
k ¼ 1; 2; . . . ; N;
ð11Þ
a b X az2k þ bzk þ c A2 z2k þ A1 zk þ A0 ¼ 0: 4 l–k 2 zk zl
ð12Þ
or
Using
P2 ðzÞ W 00 ðzÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ; Q 2 ðzÞ
0
W 000 ðzÞ ¼ z0
dW 0 Q 2 ¼ dz
dP 2 dz
2 12 P2 dQ dz ; Q2
ð13Þ
we arrive at the potential
! N 2 X P22 Q 2 dPdz2 þ 12 P2 dQ dz 2A1 N aN2 þ 2A2 Nz þ 2A2 zk Q2 k¼1 " # 2 2az þ b ðA2 z þ A1 z þ A0 Þ2 1 2 2ðN þ 1ÞA2 z þ A2 z þ A1 z þ A0 ¼ 2 az2 þ bz þ c az2 þ bz þ c z¼zðxÞ " # N X ð2N þ 1ÞA1 aN2 þ 2A2 zk ;
V N ðxÞ ¼
ð14Þ
k¼1
and the wave function
wN eW 0 pN ðzÞ R zðxÞ P2 ðzÞ dzQ ðzÞ 2 p ðzÞ: e N
ð15Þ
Eq. (14) gives the most general form of potential, based on sinusoidal coordinates, that cover both the exactly and quasi-exactly solvable systems. It turns out that there are only three inequivalent canonical forms of the sinusoidal coordinates [17], namely, (i) z02 ¼ c – 0, (ii) z02 ¼ bzðb > 0Þ, and (iii) z02 ¼ aðz2 þ dÞ (d = 0, ±1 for a > 0, and d = 1 if a < 0). Case (i) and (ii) correspond to one- and three-dimensional oscillator-type potentials, respectively. In case (iii), for a > 0, the potentials are of the Scarf II (d = 1), Morse (d = 0) and generalized Pöschl–Teller (d = 1) types, while for a < 0 and d = 1, the potentials generated belong to the Scarf I type. For clarity of presentation, in the main text we shall illustrate the prepotential construction of QNM models only for the Scarf II, Morse, and generalized Pöschl–Teller type potentials. Other cases are
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summarized in Appendices. As mentioned in Section 1, we choose to discuss these three cases because they have some interesting features that serve as good examples to illustrate the procedure, and because there are new QNM models in these types not realized in [4]. 3. Scarf II: z02 ¼ aðz2 þ 1Þ pffiffiffi Consider first the case z02 ¼ aðz2 þ 1Þ (a > 0), which is solved to give zðxÞ ¼ sinh ax . For defipffiffiffi niteness we shall take zðxÞ ¼ sinh ax . The case corresponding to the negative sign is simply the mirror image of the present case (i.e., by taking x ? x). The same applies to the other sinusoidal coordinates discussed in the rest of the paper. Putting b = 0 and c = a in (14), we get the potential (1 < x < 1)
VN ¼
" ðA0 A2 Þ2
ðz2 þ 1Þ þ
A1
A1
#
1 2A1 z A Þ þ 1 þ ðA 0 2 z2 þ 1 z2 þ 1 a a a a " # N X 2A1 A21 2A2 2 2N 1 z 2A1 N aN ðA0 A2 Þ þ 2A2 zk : þ A2
A22
þ1
a
a
a
ð16Þ
k¼1
In terms of x, it is
A22
V N ðxÞ ¼
a
cosh
2 pffiffiffi
"
ax þ
ðA0 A2 Þ2
a
2A1
A1
A1
a
þ1
# 2 pffiffiffi sech ð axÞ
pffiffiffi pffiffiffi pffiffiffi 2A1 þ 1 tanhð axÞsech ax þ A2 2N 1 sinhð axÞ þ ðA0 A2 Þ a a " # N X A2 2A2 ðA0 A2 Þ þ 2A2 zk : 2A1 N aN2 1
a
a
ð17Þ
k¼1
The prepotential W0, obtained from (9), is
A1 A0 A2 lnðz2 þ 1Þ þ tan1 z 2a a pffiffiffi A1 pffiffiffi A0 A2 pffiffiffi A2 sinh ax þ ln cosh ax þ tan1 sinh ax : ¼
W 0 ðxÞ ¼
A2
a
zþ
a
a
a
W 0
Hence the ground state wave function w0 e A 2
w0 e
a sinhð
pffiffi
axÞ
ð18Þ
is
pffiffiffi Aa1 A2 A0 tan1 sinh ðpffiffiaxÞ cosh ax e a :
ð19Þ
3.1. A2 = 0 Let us first discuss the exactly solvable case, with A2 = 0. The potential becomes
VN ¼
" A20
a
A1
A1
a
þ1
# pffiffiffi pffiffiffi 2A1 2 pffiffiffi sech ð axÞ þ A0 þ 1 tanh ax sech ax
" 2A1 N aN 2
a
A21
a
# :
ð20Þ
Suppose all Ai’s are real, the potential VN is just the Scarf II potential given in [20]. Recall that in our approach we have HN/N = 0, hence the eigenvalue of HN with potential VN is always zero. If we define the Scarf II potential V(x) by the first two N-independent terms, then VN(x) = V(x) EN, where EN ¼ 2A1 N aN 2 A21 =a are the eigenvalues of V(x). The corresponding eigenfunctions are given by (1), with /0 given by (19), and pN(z) defined by the roots zk’s of the corresponding Bethe ansatz Eq. (12), which gives the Jacobi polynomials [22].
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If we take instead
d A0 ¼ i ; 2
2A1
a
c þ 1 ¼ i ;
ð21Þ
a
then the potential is
V N ðxÞ ¼
pffiffiffi cd pffiffiffi pffiffiffi 1 2 a þ c2 d2 sech2 ax tanh ax sech ax 4a 2a " 2 # c2 1 1 : Nþ a ic N þ 2 2 4a
ð22Þ
This is the exactly solvable case 1 hyperbolic QNM system discussed in [4]. The special case where d = 0 has been employed in [3] to study black hole’s QNMs. If we take the potential to be defined by the first two terms, then the energies and ground state wave function are
EN ¼
2 c2 1 1 Nþ a ic N þ : 2 2 4a
ð23Þ
and
pffiffiffi ðicþaÞ=2a pffiffiffi w0 ðxÞ cosh ax exp id tan1 sinh ax =2a :
ð24Þ
Note that EN is independent of d, which is a general feature of the Scarf-type potentials. Also, the imaginary part is proportional to N + 1/2, which is characteristic of black hole QNMs. 3.2. A2 – 0 If A2 – 0, then reality of the first term of VN in (17) requires that A2 be real, or purely imaginary. If A2 is real, then it is clear from the first term of (19) that w0, which governs the asymptotic behaviors of wN, is not normalizable on the whole line. Hence there is no QES model with real energies in this case. Suppose A2 is purely imaginary, say A2 = ic – 0 with real constant c. Then the fourth term of VN in (17) can be real provided that
2A1
a
2N 1 ¼ id;
d : real:
ð25Þ
If d – 0, then reality of the third term of VN demands that A0 A2 = ±[2(N + 1) id]a. But then, as can be easily checked, with these values of Ai’s the second term of VN cannot be real, unless d = 0. So we are left with the choice A2 = ic – 0 and 2A1/a 2N 1 = 0. This implies that A0 A2 must be real for VN real. Let A0 A2 = aa with real a. The potential of this system assumes the form
3 2 pffiffiffi Nþ sech ax 2 a ! N X pffiffiffi pffiffiffi a zk : þ 2aaðN þ 1Þ tanh ax sech ax 2ica þ 2ic 4 k¼1
V N ðxÞ ¼
c2
2 pffiffiffi
cosh
ax þ a a2 N þ
1 2
ð26Þ
The ground state wave function is
w0 eia sinh ð c
pffiffi
axÞ
cosh
pffiffiffi ðNþ12Þ a tan1 sinh ðpffiffiaxÞ e ax :
ð27Þ
For a – 0, the potential (26) is a new QES model with QNMs which has been overlooked in our previous study based on the Lie-algebraic theory of [4]. The case a = 0 leads to a totally different model with real QES energies, which was first discussed in [21]. We briefly discuss it in the next subsection.
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3.3. Singular potential with A2 – 0 For a = 0, i.e., A2 = A0 = ic – 0, the potential (26) becomes
V N ðxÞ ¼
c2
a
cosh
2 pffiffiffi
ax a N þ
1 2
! N X 3 a 2 pffiffiffi ax þ 2ic zk : Nþ sech 2 4 k¼1
ð28Þ
This is a singular potential unbounded from below. Yet it exhibits very peculiar features, such as the existence of QES bound states with real energies, and QES total transmission modes. We refer the reader to [21] for a detailed discussion of this potential. P At first sight it may seem strange that the system has QES real energies, owing to the term 2ic k zk . P We prove below that this term is indeed real, as the sum k zk is purely imaginary. The Bethe ansatz Eq. (12) in this case are 2
iczk þ aNzk þ ic a
X z2 þ 1 k ¼ 0; z zl l–k k
k ¼ 1; 2; . . . ; N:
ð29Þ
Multiplying (29) by 1 and taking its complex conjugate, we have
icðzk Þ2 þ aNðzk Þ þ ic a
X ðzk Þ2 þ 1 ¼ 0; ðzk Þ ðzl Þ l–k
k ¼ 1; 2; . . . ; N:
ð30Þ
Here z represent the complex conjugate of z. Comparing (29) and (30), we conclude that if zk’s are the P solutions of (29), then so are their negative complex conjugates zk ’s. This implies that the sum k zk is purely imaginary, and hence the QES energies are real. 4. Morse: z02 ¼ az2 (a > 0) pffiffiffi In this case the change of variables is given by zðxÞ ¼ expð axÞ. Again we shall only consider the pffiffiffi positive case, i.e., we take zðxÞ ¼ expð axÞ. The potential is (1 < x < 1)
V N ðxÞ ¼
2A1 1 A20 1 þ 2N 1 z þ A0 þ1 z a z2 a a a ! N X A2 2A2 A0 2A1 N aN2 1 þ 2A2 zk A22
z 2 þ A2
2A1
a
a
2A1
k¼1
pffiffi pffiffi pffiffi 2A1 A2 þ A2 2N 1 e ax þ A0 þ 1 e ax þ 0 e2 ax ¼ e a a a a ! N X A2 2A2 A0 2A1 N aN2 1 þ 2A2 zk : pffiffi 2 ax
A22
a
a
ð31Þ
k¼1
The prepotential W0 is
A0 A2 z þ A1 ln z a z pffiffi A2 paffiffix A1 pffiffiffi A ¼ e þ ax 0 e ax :
W0 ¼
1
a
a
ð32Þ
a
Hence the ground state wave function is
ffi
ffiffi
ffi
p A2 pax A1 p A axþ 0 e ax
/0 e a e
a
a
ð33Þ
:
4.1. A2 = 0 For A2 = 0, the potential reads
V N ðxÞ ¼ A0
2A1
a
! pffiffi A20 2pffiffiax A21 2 ax þ1 e þ e 2A1 N aN :
a
a
ð34Þ
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To get a real potential defined by the first two terms of VN, A0 and A1 must be both real, or both complex. If A0 is real, then the first term of VN requires that A1 be real. The resulted model is the exactly solvable Morse potential. If A0 is complex, then the second term of VN demands that A0 be purely imaginary, i.e., A0 = ic with c real. The first term of VN then requires that 2A1/a + 1 = id is also purely imaginary. The potential is pffiffi ax
V N ðxÞ ¼ cde
c2
a
pffiffi 2 ax
e
" 2 # d 1 1 2 ; N N id N þ a 4 2
ð35Þ
with ground state
ffi
ffiffi
p p c ax þ1ðidþ1Þ ax 2
w0 eiae
ð36Þ
:
This gives a new exactly solvable QNM model. This system was overlooked in [4]. We note here that this system, or rather its mirror image, can be obtained by a different complexification of A0, A1 and A2. We shall discuss this in the next subsection. 4.2. A2 – 0 If A2 – 0, then A2 has to be real or purely imaginary. If A2 is real, then all Ai’s must be real. In this case, for A2 > 0 and A0 < 0, the potential VN in (31) defines a QES system. For A2 purely imaginary, A1/a (N + 1/2) must either be zero, or purely imaginary. If A1/a (N + 1/ 2) = 0, we can have
b A2 ¼ i ; 2
Nþ
A1 ¼
1 a; 2
d A0 ¼ ; 2
b; d : real:
ð37Þ
This leads to a potential
V N ðxÞ ¼
! 2 2 N pffiffi X b 2pffiffiax d 2pffiffiax a bd ax e ðN þ 1Þde þ e i ib zk : 2a 4a 4a 4 k¼1
ð38Þ
The system defined by the first three terms of VN is a QES system with complex energies given by the last term in bracket. This is indeed the very first QES QNM model presented in [4]. On the other hand, if A1/a (N + 1/2) is purely imaginary, then one can choose
d 1 ; ¼i þ Nþ 2 2 a
A1
A2 ¼ ic;
A0 ¼ 0;
ð39Þ
with c and d real constants. The resulted potential is pffiffi ax
V N ðxÞ ¼ cde
c2
a
e2
pffiffi
ax
" # 2 d 1 d c X i 2i zk : 4 2 a k
a
ð40Þ
In this case though A2 – 0, the parameter N and the roots zk’s do not appear in any x-dependent term, and hence the system is exactly solvable. This system is indeed the mirror image of the model described by (35), although the functional forms of the two potentials look very differently. To show this, we need to demonstrate that the potential, the eigenvalues and the eigenfunctions of one system are mapped into those of the other system under parity transformation x ? x. The wave function of the present system is c
p
wN eiae
ffi
N ffiffi Y ðz zk Þ;
ax 1ðidþ2Nþ1Þpax 2
k¼1
ð41Þ
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where the roots zk’s satisfy the BAE (from (12))
N X z2k d 2 iczk þ a i þ N zk a ¼ 0; 2 z zl l–k k
k ¼ 1; 2; . . . ; N:
ð42Þ
pffiffi We stress here that z is z ¼ e ax as before. Under parity transformation x ? x, we have z ! z1 ; zk ! z1 k . Eqs. (40)–(42) are mapped, respectively, into
" # 2 d 1 d c X 1 i 2i ; 4 2 a a k zk N pffi pffiffi Y 1 1 c ax 1 ; wN eiae þ2ðidþ2Nþ1Þ ax z zk k¼1
V N ðxÞ ¼ cde
pffiffi ax
c2
pffiffi ax
e2
a
ð43Þ ð44Þ
and
ic
N X 1 d 1 zl þ N þ a i a ¼ 0; 2 2 z z ðz zk Þ zk k l–k k l
k ¼ 1; 2; . . . ; N:
ð45Þ
Multiplying (45) by z2k , and using the identity N X l–k
N X zl zk ¼N1 ; zl zk z zl l–k k
ð46Þ
we can transform (45) to
N X z2k d a i þ 1 zk þ ic a ¼ 0; 2 z zl l–k k
k ¼ 1; 2; . . . ; N:
ð47Þ
This is just the BAE for the roots zk’s of the eigenfunctions of the system defined by the potential (35). Hence the BAE of this system is mapped into the BAE of the system defined by (35) under parity. Now the eigenfunctions (44) can be rewritten as N Y ffiffi pffiffi N e ax zN ðz zk Þ;
ffi
p p c ax þ1ðidþ1Þ ax 2
wN eiae
k¼1
ffi
N ffiffi Y
p p iac e ax þ12ðidþ1Þ ax
e
ð48Þ
ðz zk Þ;
k¼1
pffiffi N where we have used the fact that e ax zN ¼ 1. Eq. (48) is just the eigenfunction of the potential (35). Thus, together with the result of the last paragraph on BAE, one sees that under parity the eigenfunctions of the system given by (40) are mapped into those of given by (35). Finally we show that the last term of the potential (40) is equal to the last term of (35). This proves the equality of the eigenvalues of both systems. Dividing the BAE (47) by zk, summing over all k and using the identity N X X
zk 1 ¼ NðN 1Þ; zk zl 2
ð49Þ
N c X 1 ¼ idN N 2 N: a k¼1 zk
ð50Þ
k¼1 l–k
we get
2ic
C.-L. Ho / Annals of Physics 326 (2011) 1394–1407
1403
So the last term of the potential (40) becomes
" # " # 2 2 d 1 d c X d 1 1 2 i 2i N N id N þ a z ¼ a : 4 2 4 2 a k k
ð51Þ
This is equal to the last term of (35). Hence under parity potential (40) is mapped into (35). 5. Generalized Pöschl–Teller: z0 2 = a(z2 1) pffiffiffi In this case zðxÞ ¼ coshð axÞ. The potential is (0 6 x < 1)
A22
V N ðxÞ ¼
a
2 pffiffiffi
ax þ
sinh
" ðA0 þ A2 Þ2
a
þ A1
A1
a
# 2 pffiffiffi þ 1 cosech ax
pffiffiffi pffiffiffi pffiffiffi 2A1 2A1 þ 1 coth ax cosech ax þ A2 2N 1 cosh ax þ ðA0 þ A2 Þ a a " # N X A21 2A2 2 ðA0 þ A2 Þ þ 2A2 zk : 2A1 N aN
a
a
ð52Þ
k¼1
The ground state wave function w0 exp(W0) is pffiffi pffiffiffi A1 A2 þA0 1 sinhð axÞ a e a coth cosh ð axÞ 0
pffiffiffi A2 þA a pffiffi A2 pffiffiffi A1 a e a cosh ð axÞ sinhð axÞ a tanh : x 2 A2
w0 e a cosh ð
pffiffi
axÞ
ð53Þ
Since VN ? 1as x ? 0, we must have the boundary condition wN ? 0 as x ? 0. 5.1. A2 = 0 For A2 = 0, the potential is
VN ¼
" A20
a
þ A1
A1
a
# pffiffiffi pffiffiffi 2A1 2 pffiffiffi þ 1 cosech a x þ A0 þ 1 coth ax cosech ax
" 2A1 N aN2
a
A21
a
# :
ð54Þ
If all Ai’s are real, this gives the generalized Pöschl–Teller potential in [20]. If we take the Pöschl–Teller potential V(x) to consist of the first two N-independent terms, then the eigen-energies are EN ¼ 2A1 N aN 2 A21 =a. These eigenvalues are the same as those of the Scarf II potential. Our approach makes it very easy to see why the eigenvalues are the same for the two apparently different systems. We can also take
d A0 ¼ i ; 2
2A1
a
c þ 1 ¼ i ;
a
ð55Þ
which result in the potential
pffiffiffi cd pffiffiffi pffiffiffi 1 2 a þ c2 þ d2 cosech2 ax coth ax cosech ax 4a 2a " 2 # c2 1 1 : Nþ a ic N þ 2 2 4a
V N ðxÞ ¼
ð56Þ
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C.-L. Ho / Annals of Physics 326 (2011) 1394–1407
This is the exactly solvable case 2 hyperbolic QNM system discussed in [4]. 5.2. A2 – 0 As with the Scarf II case, if A2 – 0, then reality of the first term of VN in (52) requires that A2 be real, or purely imaginary. If A2 is real, then A0 and A1 have to be real. The boundary condition that wN ? 0 as x ? 1 is met if A2 > 0. As x ? 0, we have A2 þA1 þA0
w0 x
a
ð57Þ
:
Thus w0 ? 0 as x ? 0 is guaranteed if
A2 þ A1 þ A0 < 0:
ð58Þ
With Ai’s satisfying (58) and A2 > 0, the potential VN gives a QES system with real energies. If A2 is imaginary, say A2 = ic – 0 with real constant c, then the fourth term of VN in (52) can be real if 2A1/a 2N 1 = 0 or id – 0. As in the Scarf II case, if 2A1a 2N 1 = id – 0, then reality of the third term of VN demands that A0 + A2 = ±[2(N + 1) id]a. But then the second term of VN cannot be real, unless d = 0. So we must have 2A1/a 2N 1 = 0. In this case one has A2 + A0 = aa – 0 with real a. The potential VN is
1 3 2 pffiffiffi 2 pffiffiffi Nþ cosech sinh ð axÞ þ a a2 þ N þ ax 2 2 a pffiffiffi pffiffiffi 2aaðN þ 1Þ coth ax cosech ax ! N X a zk ; þ 2ica þ 2ic 4 k¼1
V N ðxÞ ¼
c2
ð59Þ
and the ground state wave function pffiffi axÞ
w0 eia cosh ð c
sinh
pffiffiffi a pffiffiffi ðNþ12Þ a tanh ax : x 2
ð60Þ
To satisfy the boundary condition w0 ? 0 as x ? 0, we have from (58) the condition
1 a>Nþ : 2
ð61Þ
With these Ai’s, we have a new QES system with QNMs. Unlike the Scarf II case, however, when a = 0, we do not have a QES model with real energy, since in this case w0 is not normalizable at x = 0. 6. Summary Exactly solvable models admitting QNM maybe useful in providing insights to QNMs emitted from more complicated systems such as black holes. In [4] we have found some new exactly solvable QNM models, and a new QES QNM model within the sl(2) Lie-algebraic approach to QES theory. In this paper we have demonstrated how exactly solvable and QES models admitting quasinormal modes can be constructed and classified very simply and directly by the prepotential approach. This approach, unlike the Lie-algebraic approach, does not require any knowledge of the underlying symmetry of the system. It treats both quasi-exact and exact solvabilities on the same footing, and gives the potential as well as the eigenfunctions and eigenvalues simultaneously. We also present a new exactly solvable Morse-like model with quasinormal modes, and two new quasi-exactly solvable models with quasinormal modes of the Scarf II and generalized Pöschl–Teller types.
C.-L. Ho / Annals of Physics 326 (2011) 1394–1407
1405
Acknowledgments This work is supported in part by the National Science Council (NSC) of the Republic of China under Grant NSC 96-2112-M-032-007-MY3 and NSC-99-2112-M-032-002-MY3. Appendix A. Shifted oscillator: z02 ¼ c > 0 pffiffiffi The corresponding transformation z(x) is zðxÞ ¼ cx þ constant. By an appropriate translation one pffiffiffi can always set the constant to zero. Hence we shall take zðxÞ ¼ cx without loss of generality. The potential is (1 < x < 1)
pffiffiffi pffiffiffi A1 A0 V N ¼ A22 cx4 þ 2A2 A1 cx3 þ A21 þ 2A2 A0 x2 þ 2 A2 ðN þ 1Þ cx c " # N X 1 A2 0 þ 2A2 2A1 N þ zk : 2 c k¼1
ðA1Þ
The ground state wave function w0 exp(W0) is
w0 e
A pffiffi A A 32 cx3 21 x2 p0fficx
ðA2Þ
:
We want VN to be real. If A2 – 0, then the term A2(N + 1) in the fourth term implies A2 be real. However, this implies the wave function wN, whose asymptotic behavior is governed by w0, is not normalizable on the whole line. Hence there is no QES models, with or without QNMs. For A2 = 0, the potential is
2 A0 1 : V N ¼ A1 x þ pffiffiffi 2A1 N þ 2 c
ðA3Þ
So A1 and A0 must both be real, this is just the potential of the shifted oscillator. If we take
c A1 ¼ i ; 2
A0 ¼ 0;
ðA4Þ
the potential becomes
1 1 : V N ¼ c2 x2 þ ic N þ 4 2
ðA5Þ
This is the case 5 exactly solvable QNM model discussed in [4]. Appendix B. Radial oscillator: z02 ¼ bz, (b > 0) For simplicity we take zðxÞ ¼ 4b x2 . The potential is (0 6 x < 1)
VN ¼
1 2 2 6 1 1 1 A2 b x þ A2 A1 bx4 þ A21 þ 2A2 A0 A2 bð4N þ 3Þ x2 64 8 4 2 " # N X 4A0 A0 1 1 A1 4A0 þ 2A2 þ zk : þ 4N þ 1 b b 2 x2 2 b k¼1
ðB1Þ
The ground state w0 exp(W0) is 2A0
1
w0 x b e4A1 x
2 1 A bx4 32 2
:
ðB2Þ
It can be checked that if A2 – 0, then all Ai’s have to be real. Hence there is no QES model with QNMs in this case. For A2 > 0 and A0 < 0, the wave functions wN are normalizable on the positive half-line. The system is then a QES model with real energies. Together with the discussion in Appendix A, one concludes that one-dimensional QES models start with degree six, i.e., the sextic oscillator [23]. In fact, the potential (B1) with A2 = 2a > 0, A1 = 2b, A0 = 0 and b = 4, namely,
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C.-L. Ho / Annals of Physics 326 (2011) 1394–1407
h i 4 2 V N ¼ a2 x6 þ 2abx þ b ð4N þ 3Þa x2 " # X ð4N þ 1Þb þ 4a zk ;
ðB3Þ
k
is the very first QES model discussed in the literature [5]. If A2 = 0, then (B1) becomes
VN ¼
A21 2 4A0 A0 1 1 1 2A0 A 2N þ x þ þ : 1 2 4 b b 2 x2 b
ðB4Þ
For real A1 and A0, this is just the radial oscillator. Suppose we take
b ¼ 4;
A1 ¼ 2ia;
and A0 ¼ 2c:
ðB5Þ
The potential takes the form
V N ¼ a2 x2 þ
cðc 1Þ x2
þ iað4N þ 2c þ 1Þ:
ðB6Þ
This is the Case 4 model with QNMs presented in [4]. Appendix C. Scarf I: z02 ¼ að1 z2 Þ In this case zðxÞ ¼ sin
V N ðxÞ ¼
A22
a
cos2
pffiffiffi ax . The potential is
" # pffiffiffi pffiffiffi ðA þ A2 Þ2 A1 ax þ 0 þ A1 1 sec2 ð axÞ
a
2A1
a
pffiffiffi pffiffiffi pffiffiffi 2A1 1 tanð axÞ sec ax A2 þ 2N þ 1 sin ax þ ð A0 þ A2 Þ a a " # N X A21 2A2 2 ðA0 þ A2 Þ þ 2A2 zk ; 2A1 N þ aN þ þ
a
a
ðC1Þ
k¼1
with the ground state wave function pffiffi axÞ
A2
w0 e a sinð
cos
pffiffiffi Aa1 A2 þA0 tanh1 sin ðpffiffiaxÞ ax e a :
ðC2Þ
pffiffiffi The system is defined only on a finite interval, usually taken to be ax 2 ½p=2; p=2. Hence there are no QNMs. If A2 – 0, then all Ai’s have to be real. The third term of (C2) then implies that the wave functions wN are not normalizable. So there are no QES systems with real energies. For A2 = 0, the potential becomes
VN ¼
" A20
a
þ A1
A1
a
#
1
" 2A1 N þ aN 2 þ
sec2
A21
a
pffiffiffi pffiffiffi pffiffiffi 2A1 ax þ A0 1 tan ax sec ax
a
# :
For real A1 and A0, this is Scarf I potential given in [20]. References [1] For a comprehensive review, see eg. K.D. Kokkotas, B.G. Schmidt, Living Rev. Rel. 2 (1999) 2; S. Chandrasekhar, The Mathematical Theory of Black Holes, Clarendon, Oxford, 1983. [2] S.P. Kim, J. Korean Phys. Soc. 49 (2006) 764; G. Barton, Ann. Phys. 166 (1986) 322. [3] V. Ferrari, B. Mashhoon, Phys. Rev. Lett. 52 (1984) 1361.
ðC3Þ
C.-L. Ho / Annals of Physics 326 (2011) 1394–1407 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
[16] [17] [18] [19]
[20] [21] [22] [23]
1407
H.-T. Cho, C.-L. Ho, J. Phys. A40 (2007) 1325. A. Turbiner, A.G. Ushveridze, Phys. Lett. A 126 (1987) 181. A.V. Turbiner, Commun. Math. Phys. 118 (1988) 467. A.V. Turbiner, Sov. Phys. JETP 67 (1988) 230. A. González, N. Kamran, P.J. Olver, Commun. Math. Phys. 153 (1993) 117. M.A. Shifman, Int. J. Mod. Phys. A4 (1989) 2897. A.G. Ushveridze, Sov. Phys.-Lebedev Inst. Rep. 2 (1988) 50; A.G. Ushveridze, Sov. Phys.-Lebedev Inst. Rep. 2 (1988) 54. G. Post, A. Turbiner, Russian J. Math. Phys. 3 (1995) 113. N. Kamran, R. Milson, P.J. Olver, Invariant modules and the reduction of nonlinear partial differential equations to dynamical systems, 1999. Available from:. O. Özer, P. Roy, Cent. Eur. J. Phys. 7 (2009) 747. C.-L. Ho, Ann. Phys. 323 (2008) 2241. C.-L. Ho, Prepotential approach to exact and quasi-exact solvabilities of Hermitian and non-Hermitian Hamiltonians, in: Talk presented at ‘‘Conference in Honor of CN Yang’s 85th Birthday’’, 31 October–3 November, 2007, Singapore. Available from: . C.-L. Ho, Ann. Phys. 324 (2009) 1095. C.-L. Ho, J. Math. Phys. 50 (2009) 042105. R. Sasaki, K. Takasaki, J. Phys. A34 (2001) 9533. C.-L. Ho, P. Roy, J. Phys. A36 (2003) 4617; C.-L. Ho, P. Roy, Ann. Phys. 312 (2004) 161; C.-L. Ho, Ann. Phys. 321 (2006) 2170. F. Cooper, A. Khare, U. Sukhatme, Phys. Rep. 251 (1995) 267; G. Junker, Supersymmetric Methods in Quantum and Statistical Physics, Springer-Verlag, Berlin, 1996. H.-T. Cho, C.-L. Ho, J. Phys. A41 (2008) 172002; H.-T. Cho, C.-L. Ho, J. Phys. A41 (2008) 255308. G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloquium Publications, vol. 23, Amer. Math. Soc., New York, 1939. If one allows non-Hermitian PT -symmetric Hamiltonians, then a QES polynomial potential can be quartic in its variable. See e.g. C.M. Bender, S. Boettcher, J. Phys. A31 (1998) L273. and [15].
Contents lists available at ScienceDirect
Annals of Physics journal homepage: www.elsevier.com/locate/aop
Prepotential approach to quasinormal modes Choon-Lin Ho Department of Physics, Tamkang University, Tamsui 251, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 17 October 2010 Accepted 5 November 2010 Available online 7 January 2011 Keywords: Prepotential Quasinormal modes Exact and quasi-exact solvabilities
a b s t r a c t In this paper we demonstrate how the recently reported exactly and quasi-exactly solvable models admitting quasinormal modes can be constructed and classified very simply and directly by the newly proposed prepotential approach. These new models were previously obtained within the Lie-algebraic approach. Unlike the Lie-algebraic approach, the prepotential approach does not require any knowledge of the underlying symmetry of the system. It treats both quasi-exact and exact solvabilities on the same footing, and gives the potential as well as the eigenfunctions and eigenvalues simultaneously. We also present three new models with quasinormal modes: a new exactly solvable Morse-like model, and two new quasi-exactly solvable models of the Scarf II and generalized Pöschl–Teller types. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Quasinormal modes (QNM) has attracted great interest in recent years [1]. They arise as waves emitted by a perturbed neutron star or black hole that are outgoing to spatial infinity and the event horizon. Generally, the wave function of QNMs has discrete complex frequency, whose imaginary part leads to a damping behavior. QNM carry information of black holes and neutron stars, and thus are of importance to gravitational-wave astronomy. In fact, these oscillations, produced mainly during the formation phase of the compact stellar objects, can be strong enough to be detected by several large gravitational wave detectors under construction. As black hole potentials are generally too complicated to allow analytic treatment, so in order to understand the origin of the discrete imaginary frequencies, one can try approximating the top region of the black hole potential by some inverted potentials which are solvable. This has been done by using the inverted harmonic oscillator [2], and the Pöschl–Teller potential [3]. Recently, in [4] we have extended the number of exactly solvable models that admit QNMs. We take QNMs to include both decaying and growing modes with complex energies. Furthermore, we E-mail address: [email protected] 0003-4916/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2010.11.020
C.-L. Ho / Annals of Physics 326 (2011) 1394–1407
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have provided the first model with QNM that is quasi-exactly solvable (QES). A system is called QES if a part of its spectrum, but not the whole spectrum, can be determined analytically [5–12]. Our approach in [4] was to study solutions of QNM based on the sl(2)-Lie-algebraic approach to one-dimensional QES theory [5,6,8,9]. We demonstrated that, by suitably complexifying some parameters of the generators of the sl(2) algebra while keeping the Hamiltonian Hermitian, we could indeed obtain potentials admitting exact or quasi-exact QNMs. These models were later re-studied numerically by the asymptotic iteration method in [13]. In this paper we would like to show that exactly solvable and QES models with QNMs can be constructed much more simply without resorting to the machinery of Lie-algebra. This is achieved through the prepotential approach proposed recently [14–17]. This is a simple constructive approach, based on the so-called prepotential [7,18,19], which can give the potential as well as the eigenfunctions and eigenvalues simultaneously. The novel feature of the approach is that both exact and quasi-exact solvabilities can be solely classified by two integers, the degrees of two polynomials which determine the change of variables and the zero-th order prepotential. Hence this approach treats both quasi-exact and exact solvabilities on the same footing, and it provides a simple way to determine the required change of variables, say x, to a new one z = z(x). All the well-known exactly solvable models classified in supersymmetric quantum mechanics (SUSYQM) [20], the QES models discussed in [5,6,8– 10], and some new QES ones (also for non-Hermitian Hamiltonians), can be generated by appropriately choosing the two polynomials. Our approach, unlike the Lie-algebraic approach, does not require any knowledge of the underlying symmetry of the system. Furthermore, our approach can generate the Coulomb, Eckart, Rosen–Morse type I and II models [16] which are not covered by the standard Lie-algebraic program [8,9]. Compared with SUSYQM, our approach has the advantage that we do not have to assume the sufficient condition for integrability needed in SUSYQM, namely, shape invariance. In fact, shape invariance comes out automatically from this approach [17]. What is more, the transformation of the original variable z(x) is determined within the prepotential approach, whereas in SUSYQM this has to be taken as given from the known solutions of the respective models before one could solve the shape invariance condition. We shall adopt the prepotential approach here to generate and classify all one-dimensional exactly solvable and QES models with QNMs based on sinusoidal coordinates, namely, those coordinates z(x) whose derivatives (with respect to x) squared are at most quadratic in z. Our strategy is to suitably complexify some or all of the parameters in the prepotential while keeping the resulted potentials real. We find that all the systems reported in [4] can be very easily constructed. During the course of investigation, we also realize that there are three new QNM models which were missed in [4]. We thus take this opportunity to report on them. The plan of the paper is as follows. In Section 2 we briefly review the essence of the prepotential approach. Sections 3 to 5 then discuss the prepotential construction of QNM models of the Scarf II, the Morse, and the generalized Pöschl–Teller type, respectively. These three types of potentials have some interesting features, and serve as good examples to illustrate the procedure. Furthermore, there is one new QNM model in each of these three types that we would like to report. Section 6 summarizes the paper. In Appendices A–C, we list all other cases for easy reference. 2. Prepotential approach The main ideas of the prepotential approach [14,15] can be summarized as follows (we adopt the unit system in which h and the mass m of the particle are such that h = 2m = 1). Consider a wave function /N(x) (N: non-negative integer) which is defined as
/N ðxÞ eW 0 ðxÞ pN ðzÞ;
ð1Þ
with
pN ðzÞ
1; QN
N ¼ 0;
N > 0: k¼1 ðz zk Þ;
ð2Þ
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Here z = z(x) is some real function of the basic variable x, W0(x) is a regular function of z(x), and zk’s are the roots of pN(z). The variable x is defined on the full line, half-line, or finite interval, as dictated by the choice of z(x). The function pN(z) is a polynomial in an (N + 1)-dimensional Hilbert space with the basis h1, z, z2, . . . , zNi. W0(x) defines the ground state wave function. The wave function /N can be recast as
/N ¼ expðW N ðx; fzk gÞÞ;
ð3Þ
with WN given by
W N ðx; fzk gÞ ¼ W 0 ðxÞ
N X
ln jzðxÞ zk j:
ð4Þ
k¼1
Operating on /N by the operator d2/dx2 results in a Schrödinger equation HN/N = 0, where 2
HN ¼
d
þ VN; 2 dx 00 V N W 02 N WN:
ð5Þ ð6Þ
Here and below the prime represents derivative with respect to x. Since the potential VN is determined by WN, we thus call WN the Nth order prepotential. From Eq. (4), one finds that VN has the form VN = V0 + DVN: 00 V 0 ¼ W 02 0 W0; N X z00 X 1 z02 DV N ¼ 2 W 00 z0 þ : 2 k¼1 z zk ðz z k Þðz zl Þ k;l
ð7Þ
k–l
Thus the form of VN, and consequently its solvability, are determined by the choice of W0(x) and z02 02 (or equivalently by z00 ¼ ðdz =dzÞ=2Þ. Let W 00 z0 ¼ Pm ðzÞ and z02 ¼ Q n ðzÞ be two polynomials of degree m and n in z, respectively. The variables x and z are related by
xðzÞ ¼
Z
z
dz pffiffiffiffiffiffiffiffiffiffiffiffi ; Q n ðzÞ
ð8Þ
and the prepotential W0(x) is determined as
W 0 ðxÞ ¼
Z
z
dz
Pm ðzÞ : Q n ðzÞ z¼zðxÞ
ð9Þ
We assume (8) is invertible to give z = z(x). Eqs. (8) and (9) define the change of variables z(x) and the corresponding prepotential W0(x). Thus, Pm(z) and Qn(z) determine the quantum system. Of course, for bound state problems the choice of Pm and Qn must ensure normalizability of /0 = exp(W0). Now depending on the degrees of the polynomials Pm and Qn, we have the following situations [14,15]: (i) if max{m, n 1} 6 1, then in VN(x) the parameter N and the roots zk’s will only appear as an additive constant and not in any term involving powers of z. Such system is then exactly solvable; (ii) if max{m, n 1} = 2, then N may appear in the first power term in z, but zk’s only in an additive term. If N does appear before the z-term, then the system belongs to the so-called type 1 QES system defined in [6], i.e., for each N P 0, VN admits N + 1 solvable states with the eigenvalues being given by the N + 1 sets of roots zk’s. This is the main type of QES systems considered in the literature; (iii) if max{m, n 1} P 3,1 then not only N but also zk’s may appear in terms involving powers of z. If zk’s do appear before any z-dependent term, then for each N P 0, there are N + 1 different potentials VN, differing in several parameters in terms involving powers of z, have the same 1 We take this opportunity to correct a typographic error in [14,15], where the condition for m and n for this case was erroneously written as min{m, n 1} P 3.
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eigenvalue (when the additive constant, or the zero point, is appropriately adjusted). When zk’s appear only in the first power term in z, such systems are called type 2 QES systems in [6]. We see that QES models of higher types are possible. This gives a very simple algebraic classification of exact and quasi-exact solvabilities. Previously exact and QES systems were treated separately. In the rest of this paper we shall consider only cases with m, n 6 2. Coordinates with n 6 2 are called the sinusoidal coordinates. Let P2(z) = A2z2 + A1z + A0 and Q2(z) = az2 + bz + c. Hence the solvability of the system is determined solely by A2: exactly solvable if A2 = 0, or (type 1) QES otherwise. The potential VN takes the form
V N ¼ W 02 0
W 000
2
þ aN 2A1 N 2A2 Nz 2A2
N X
zk 2
k¼1
N X k¼1
( ) 1 a b X Q 2 ðzk Þ P2 ðzk Þ zk : z zk 4 l–k zk zl 2
ð10Þ Demanding the residues at zk’s vanish gives the Bethe ansatz equations satisfied by the roots zk’s:
a b X Q 2 ðzk Þ P2 ðzk Þ zk ¼ 0; 4 l–k zk zl 2
k ¼ 1; 2; . . . ; N;
ð11Þ
a b X az2k þ bzk þ c A2 z2k þ A1 zk þ A0 ¼ 0: 4 l–k 2 zk zl
ð12Þ
or
Using
P2 ðzÞ W 00 ðzÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ; Q 2 ðzÞ
0
W 000 ðzÞ ¼ z0
dW 0 Q 2 ¼ dz
dP 2 dz
2 12 P2 dQ dz ; Q2
ð13Þ
we arrive at the potential
! N 2 X P22 Q 2 dPdz2 þ 12 P2 dQ dz 2A1 N aN2 þ 2A2 Nz þ 2A2 zk Q2 k¼1 " # 2 2az þ b ðA2 z þ A1 z þ A0 Þ2 1 2 2ðN þ 1ÞA2 z þ A2 z þ A1 z þ A0 ¼ 2 az2 þ bz þ c az2 þ bz þ c z¼zðxÞ " # N X ð2N þ 1ÞA1 aN2 þ 2A2 zk ;
V N ðxÞ ¼
ð14Þ
k¼1
and the wave function
wN eW 0 pN ðzÞ R zðxÞ P2 ðzÞ dzQ ðzÞ 2 p ðzÞ: e N
ð15Þ
Eq. (14) gives the most general form of potential, based on sinusoidal coordinates, that cover both the exactly and quasi-exactly solvable systems. It turns out that there are only three inequivalent canonical forms of the sinusoidal coordinates [17], namely, (i) z02 ¼ c – 0, (ii) z02 ¼ bzðb > 0Þ, and (iii) z02 ¼ aðz2 þ dÞ (d = 0, ±1 for a > 0, and d = 1 if a < 0). Case (i) and (ii) correspond to one- and three-dimensional oscillator-type potentials, respectively. In case (iii), for a > 0, the potentials are of the Scarf II (d = 1), Morse (d = 0) and generalized Pöschl–Teller (d = 1) types, while for a < 0 and d = 1, the potentials generated belong to the Scarf I type. For clarity of presentation, in the main text we shall illustrate the prepotential construction of QNM models only for the Scarf II, Morse, and generalized Pöschl–Teller type potentials. Other cases are
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summarized in Appendices. As mentioned in Section 1, we choose to discuss these three cases because they have some interesting features that serve as good examples to illustrate the procedure, and because there are new QNM models in these types not realized in [4]. 3. Scarf II: z02 ¼ aðz2 þ 1Þ pffiffiffi Consider first the case z02 ¼ aðz2 þ 1Þ (a > 0), which is solved to give zðxÞ ¼ sinh ax . For defipffiffiffi niteness we shall take zðxÞ ¼ sinh ax . The case corresponding to the negative sign is simply the mirror image of the present case (i.e., by taking x ? x). The same applies to the other sinusoidal coordinates discussed in the rest of the paper. Putting b = 0 and c = a in (14), we get the potential (1 < x < 1)
VN ¼
" ðA0 A2 Þ2
ðz2 þ 1Þ þ
A1
A1
#
1 2A1 z A Þ þ 1 þ ðA 0 2 z2 þ 1 z2 þ 1 a a a a " # N X 2A1 A21 2A2 2 2N 1 z 2A1 N aN ðA0 A2 Þ þ 2A2 zk : þ A2
A22
þ1
a
a
a
ð16Þ
k¼1
In terms of x, it is
A22
V N ðxÞ ¼
a
cosh
2 pffiffiffi
"
ax þ
ðA0 A2 Þ2
a
2A1
A1
A1
a
þ1
# 2 pffiffiffi sech ð axÞ
pffiffiffi pffiffiffi pffiffiffi 2A1 þ 1 tanhð axÞsech ax þ A2 2N 1 sinhð axÞ þ ðA0 A2 Þ a a " # N X A2 2A2 ðA0 A2 Þ þ 2A2 zk : 2A1 N aN2 1
a
a
ð17Þ
k¼1
The prepotential W0, obtained from (9), is
A1 A0 A2 lnðz2 þ 1Þ þ tan1 z 2a a pffiffiffi A1 pffiffiffi A0 A2 pffiffiffi A2 sinh ax þ ln cosh ax þ tan1 sinh ax : ¼
W 0 ðxÞ ¼
A2
a
zþ
a
a
a
W 0
Hence the ground state wave function w0 e A 2
w0 e
a sinhð
pffiffi
axÞ
ð18Þ
is
pffiffiffi Aa1 A2 A0 tan1 sinh ðpffiffiaxÞ cosh ax e a :
ð19Þ
3.1. A2 = 0 Let us first discuss the exactly solvable case, with A2 = 0. The potential becomes
VN ¼
" A20
a
A1
A1
a
þ1
# pffiffiffi pffiffiffi 2A1 2 pffiffiffi sech ð axÞ þ A0 þ 1 tanh ax sech ax
" 2A1 N aN 2
a
A21
a
# :
ð20Þ
Suppose all Ai’s are real, the potential VN is just the Scarf II potential given in [20]. Recall that in our approach we have HN/N = 0, hence the eigenvalue of HN with potential VN is always zero. If we define the Scarf II potential V(x) by the first two N-independent terms, then VN(x) = V(x) EN, where EN ¼ 2A1 N aN 2 A21 =a are the eigenvalues of V(x). The corresponding eigenfunctions are given by (1), with /0 given by (19), and pN(z) defined by the roots zk’s of the corresponding Bethe ansatz Eq. (12), which gives the Jacobi polynomials [22].
C.-L. Ho / Annals of Physics 326 (2011) 1394–1407
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If we take instead
d A0 ¼ i ; 2
2A1
a
c þ 1 ¼ i ;
ð21Þ
a
then the potential is
V N ðxÞ ¼
pffiffiffi cd pffiffiffi pffiffiffi 1 2 a þ c2 d2 sech2 ax tanh ax sech ax 4a 2a " 2 # c2 1 1 : Nþ a ic N þ 2 2 4a
ð22Þ
This is the exactly solvable case 1 hyperbolic QNM system discussed in [4]. The special case where d = 0 has been employed in [3] to study black hole’s QNMs. If we take the potential to be defined by the first two terms, then the energies and ground state wave function are
EN ¼
2 c2 1 1 Nþ a ic N þ : 2 2 4a
ð23Þ
and
pffiffiffi ðicþaÞ=2a pffiffiffi w0 ðxÞ cosh ax exp id tan1 sinh ax =2a :
ð24Þ
Note that EN is independent of d, which is a general feature of the Scarf-type potentials. Also, the imaginary part is proportional to N + 1/2, which is characteristic of black hole QNMs. 3.2. A2 – 0 If A2 – 0, then reality of the first term of VN in (17) requires that A2 be real, or purely imaginary. If A2 is real, then it is clear from the first term of (19) that w0, which governs the asymptotic behaviors of wN, is not normalizable on the whole line. Hence there is no QES model with real energies in this case. Suppose A2 is purely imaginary, say A2 = ic – 0 with real constant c. Then the fourth term of VN in (17) can be real provided that
2A1
a
2N 1 ¼ id;
d : real:
ð25Þ
If d – 0, then reality of the third term of VN demands that A0 A2 = ±[2(N + 1) id]a. But then, as can be easily checked, with these values of Ai’s the second term of VN cannot be real, unless d = 0. So we are left with the choice A2 = ic – 0 and 2A1/a 2N 1 = 0. This implies that A0 A2 must be real for VN real. Let A0 A2 = aa with real a. The potential of this system assumes the form
3 2 pffiffiffi Nþ sech ax 2 a ! N X pffiffiffi pffiffiffi a zk : þ 2aaðN þ 1Þ tanh ax sech ax 2ica þ 2ic 4 k¼1
V N ðxÞ ¼
c2
2 pffiffiffi
cosh
ax þ a a2 N þ
1 2
ð26Þ
The ground state wave function is
w0 eia sinh ð c
pffiffi
axÞ
cosh
pffiffiffi ðNþ12Þ a tan1 sinh ðpffiffiaxÞ e ax :
ð27Þ
For a – 0, the potential (26) is a new QES model with QNMs which has been overlooked in our previous study based on the Lie-algebraic theory of [4]. The case a = 0 leads to a totally different model with real QES energies, which was first discussed in [21]. We briefly discuss it in the next subsection.
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3.3. Singular potential with A2 – 0 For a = 0, i.e., A2 = A0 = ic – 0, the potential (26) becomes
V N ðxÞ ¼
c2
a
cosh
2 pffiffiffi
ax a N þ
1 2
! N X 3 a 2 pffiffiffi ax þ 2ic zk : Nþ sech 2 4 k¼1
ð28Þ
This is a singular potential unbounded from below. Yet it exhibits very peculiar features, such as the existence of QES bound states with real energies, and QES total transmission modes. We refer the reader to [21] for a detailed discussion of this potential. P At first sight it may seem strange that the system has QES real energies, owing to the term 2ic k zk . P We prove below that this term is indeed real, as the sum k zk is purely imaginary. The Bethe ansatz Eq. (12) in this case are 2
iczk þ aNzk þ ic a
X z2 þ 1 k ¼ 0; z zl l–k k
k ¼ 1; 2; . . . ; N:
ð29Þ
Multiplying (29) by 1 and taking its complex conjugate, we have
icðzk Þ2 þ aNðzk Þ þ ic a
X ðzk Þ2 þ 1 ¼ 0; ðzk Þ ðzl Þ l–k
k ¼ 1; 2; . . . ; N:
ð30Þ
Here z represent the complex conjugate of z. Comparing (29) and (30), we conclude that if zk’s are the P solutions of (29), then so are their negative complex conjugates zk ’s. This implies that the sum k zk is purely imaginary, and hence the QES energies are real. 4. Morse: z02 ¼ az2 (a > 0) pffiffiffi In this case the change of variables is given by zðxÞ ¼ expð axÞ. Again we shall only consider the pffiffiffi positive case, i.e., we take zðxÞ ¼ expð axÞ. The potential is (1 < x < 1)
V N ðxÞ ¼
2A1 1 A20 1 þ 2N 1 z þ A0 þ1 z a z2 a a a ! N X A2 2A2 A0 2A1 N aN2 1 þ 2A2 zk A22
z 2 þ A2
2A1
a
a
2A1
k¼1
pffiffi pffiffi pffiffi 2A1 A2 þ A2 2N 1 e ax þ A0 þ 1 e ax þ 0 e2 ax ¼ e a a a a ! N X A2 2A2 A0 2A1 N aN2 1 þ 2A2 zk : pffiffi 2 ax
A22
a
a
ð31Þ
k¼1
The prepotential W0 is
A0 A2 z þ A1 ln z a z pffiffi A2 paffiffix A1 pffiffiffi A ¼ e þ ax 0 e ax :
W0 ¼
1
a
a
ð32Þ
a
Hence the ground state wave function is
ffi
ffiffi
ffi
p A2 pax A1 p A axþ 0 e ax
/0 e a e
a
a
ð33Þ
:
4.1. A2 = 0 For A2 = 0, the potential reads
V N ðxÞ ¼ A0
2A1
a
! pffiffi A20 2pffiffiax A21 2 ax þ1 e þ e 2A1 N aN :
a
a
ð34Þ
C.-L. Ho / Annals of Physics 326 (2011) 1394–1407
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To get a real potential defined by the first two terms of VN, A0 and A1 must be both real, or both complex. If A0 is real, then the first term of VN requires that A1 be real. The resulted model is the exactly solvable Morse potential. If A0 is complex, then the second term of VN demands that A0 be purely imaginary, i.e., A0 = ic with c real. The first term of VN then requires that 2A1/a + 1 = id is also purely imaginary. The potential is pffiffi ax
V N ðxÞ ¼ cde
c2
a
pffiffi 2 ax
e
" 2 # d 1 1 2 ; N N id N þ a 4 2
ð35Þ
with ground state
ffi
ffiffi
p p c ax þ1ðidþ1Þ ax 2
w0 eiae
ð36Þ
:
This gives a new exactly solvable QNM model. This system was overlooked in [4]. We note here that this system, or rather its mirror image, can be obtained by a different complexification of A0, A1 and A2. We shall discuss this in the next subsection. 4.2. A2 – 0 If A2 – 0, then A2 has to be real or purely imaginary. If A2 is real, then all Ai’s must be real. In this case, for A2 > 0 and A0 < 0, the potential VN in (31) defines a QES system. For A2 purely imaginary, A1/a (N + 1/2) must either be zero, or purely imaginary. If A1/a (N + 1/ 2) = 0, we can have
b A2 ¼ i ; 2
Nþ
A1 ¼
1 a; 2
d A0 ¼ ; 2
b; d : real:
ð37Þ
This leads to a potential
V N ðxÞ ¼
! 2 2 N pffiffi X b 2pffiffiax d 2pffiffiax a bd ax e ðN þ 1Þde þ e i ib zk : 2a 4a 4a 4 k¼1
ð38Þ
The system defined by the first three terms of VN is a QES system with complex energies given by the last term in bracket. This is indeed the very first QES QNM model presented in [4]. On the other hand, if A1/a (N + 1/2) is purely imaginary, then one can choose
d 1 ; ¼i þ Nþ 2 2 a
A1
A2 ¼ ic;
A0 ¼ 0;
ð39Þ
with c and d real constants. The resulted potential is pffiffi ax
V N ðxÞ ¼ cde
c2
a
e2
pffiffi
ax
" # 2 d 1 d c X i 2i zk : 4 2 a k
a
ð40Þ
In this case though A2 – 0, the parameter N and the roots zk’s do not appear in any x-dependent term, and hence the system is exactly solvable. This system is indeed the mirror image of the model described by (35), although the functional forms of the two potentials look very differently. To show this, we need to demonstrate that the potential, the eigenvalues and the eigenfunctions of one system are mapped into those of the other system under parity transformation x ? x. The wave function of the present system is c
p
wN eiae
ffi
N ffiffi Y ðz zk Þ;
ax 1ðidþ2Nþ1Þpax 2
k¼1
ð41Þ
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C.-L. Ho / Annals of Physics 326 (2011) 1394–1407
where the roots zk’s satisfy the BAE (from (12))
N X z2k d 2 iczk þ a i þ N zk a ¼ 0; 2 z zl l–k k
k ¼ 1; 2; . . . ; N:
ð42Þ
pffiffi We stress here that z is z ¼ e ax as before. Under parity transformation x ? x, we have z ! z1 ; zk ! z1 k . Eqs. (40)–(42) are mapped, respectively, into
" # 2 d 1 d c X 1 i 2i ; 4 2 a a k zk N pffi pffiffi Y 1 1 c ax 1 ; wN eiae þ2ðidþ2Nþ1Þ ax z zk k¼1
V N ðxÞ ¼ cde
pffiffi ax
c2
pffiffi ax
e2
a
ð43Þ ð44Þ
and
ic
N X 1 d 1 zl þ N þ a i a ¼ 0; 2 2 z z ðz zk Þ zk k l–k k l
k ¼ 1; 2; . . . ; N:
ð45Þ
Multiplying (45) by z2k , and using the identity N X l–k
N X zl zk ¼N1 ; zl zk z zl l–k k
ð46Þ
we can transform (45) to
N X z2k d a i þ 1 zk þ ic a ¼ 0; 2 z zl l–k k
k ¼ 1; 2; . . . ; N:
ð47Þ
This is just the BAE for the roots zk’s of the eigenfunctions of the system defined by the potential (35). Hence the BAE of this system is mapped into the BAE of the system defined by (35) under parity. Now the eigenfunctions (44) can be rewritten as N Y ffiffi pffiffi N e ax zN ðz zk Þ;
ffi
p p c ax þ1ðidþ1Þ ax 2
wN eiae
k¼1
ffi
N ffiffi Y
p p iac e ax þ12ðidþ1Þ ax
e
ð48Þ
ðz zk Þ;
k¼1
pffiffi N where we have used the fact that e ax zN ¼ 1. Eq. (48) is just the eigenfunction of the potential (35). Thus, together with the result of the last paragraph on BAE, one sees that under parity the eigenfunctions of the system given by (40) are mapped into those of given by (35). Finally we show that the last term of the potential (40) is equal to the last term of (35). This proves the equality of the eigenvalues of both systems. Dividing the BAE (47) by zk, summing over all k and using the identity N X X
zk 1 ¼ NðN 1Þ; zk zl 2
ð49Þ
N c X 1 ¼ idN N 2 N: a k¼1 zk
ð50Þ
k¼1 l–k
we get
2ic
C.-L. Ho / Annals of Physics 326 (2011) 1394–1407
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So the last term of the potential (40) becomes
" # " # 2 2 d 1 d c X d 1 1 2 i 2i N N id N þ a z ¼ a : 4 2 4 2 a k k
ð51Þ
This is equal to the last term of (35). Hence under parity potential (40) is mapped into (35). 5. Generalized Pöschl–Teller: z0 2 = a(z2 1) pffiffiffi In this case zðxÞ ¼ coshð axÞ. The potential is (0 6 x < 1)
A22
V N ðxÞ ¼
a
2 pffiffiffi
ax þ
sinh
" ðA0 þ A2 Þ2
a
þ A1
A1
a
# 2 pffiffiffi þ 1 cosech ax
pffiffiffi pffiffiffi pffiffiffi 2A1 2A1 þ 1 coth ax cosech ax þ A2 2N 1 cosh ax þ ðA0 þ A2 Þ a a " # N X A21 2A2 2 ðA0 þ A2 Þ þ 2A2 zk : 2A1 N aN
a
a
ð52Þ
k¼1
The ground state wave function w0 exp(W0) is pffiffi pffiffiffi A1 A2 þA0 1 sinhð axÞ a e a coth cosh ð axÞ 0
pffiffiffi A2 þA a pffiffi A2 pffiffiffi A1 a e a cosh ð axÞ sinhð axÞ a tanh : x 2 A2
w0 e a cosh ð
pffiffi
axÞ
ð53Þ
Since VN ? 1as x ? 0, we must have the boundary condition wN ? 0 as x ? 0. 5.1. A2 = 0 For A2 = 0, the potential is
VN ¼
" A20
a
þ A1
A1
a
# pffiffiffi pffiffiffi 2A1 2 pffiffiffi þ 1 cosech a x þ A0 þ 1 coth ax cosech ax
" 2A1 N aN2
a
A21
a
# :
ð54Þ
If all Ai’s are real, this gives the generalized Pöschl–Teller potential in [20]. If we take the Pöschl–Teller potential V(x) to consist of the first two N-independent terms, then the eigen-energies are EN ¼ 2A1 N aN 2 A21 =a. These eigenvalues are the same as those of the Scarf II potential. Our approach makes it very easy to see why the eigenvalues are the same for the two apparently different systems. We can also take
d A0 ¼ i ; 2
2A1
a
c þ 1 ¼ i ;
a
ð55Þ
which result in the potential
pffiffiffi cd pffiffiffi pffiffiffi 1 2 a þ c2 þ d2 cosech2 ax coth ax cosech ax 4a 2a " 2 # c2 1 1 : Nþ a ic N þ 2 2 4a
V N ðxÞ ¼
ð56Þ
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This is the exactly solvable case 2 hyperbolic QNM system discussed in [4]. 5.2. A2 – 0 As with the Scarf II case, if A2 – 0, then reality of the first term of VN in (52) requires that A2 be real, or purely imaginary. If A2 is real, then A0 and A1 have to be real. The boundary condition that wN ? 0 as x ? 1 is met if A2 > 0. As x ? 0, we have A2 þA1 þA0
w0 x
a
ð57Þ
:
Thus w0 ? 0 as x ? 0 is guaranteed if
A2 þ A1 þ A0 < 0:
ð58Þ
With Ai’s satisfying (58) and A2 > 0, the potential VN gives a QES system with real energies. If A2 is imaginary, say A2 = ic – 0 with real constant c, then the fourth term of VN in (52) can be real if 2A1/a 2N 1 = 0 or id – 0. As in the Scarf II case, if 2A1a 2N 1 = id – 0, then reality of the third term of VN demands that A0 + A2 = ±[2(N + 1) id]a. But then the second term of VN cannot be real, unless d = 0. So we must have 2A1/a 2N 1 = 0. In this case one has A2 + A0 = aa – 0 with real a. The potential VN is
1 3 2 pffiffiffi 2 pffiffiffi Nþ cosech sinh ð axÞ þ a a2 þ N þ ax 2 2 a pffiffiffi pffiffiffi 2aaðN þ 1Þ coth ax cosech ax ! N X a zk ; þ 2ica þ 2ic 4 k¼1
V N ðxÞ ¼
c2
ð59Þ
and the ground state wave function pffiffi axÞ
w0 eia cosh ð c
sinh
pffiffiffi a pffiffiffi ðNþ12Þ a tanh ax : x 2
ð60Þ
To satisfy the boundary condition w0 ? 0 as x ? 0, we have from (58) the condition
1 a>Nþ : 2
ð61Þ
With these Ai’s, we have a new QES system with QNMs. Unlike the Scarf II case, however, when a = 0, we do not have a QES model with real energy, since in this case w0 is not normalizable at x = 0. 6. Summary Exactly solvable models admitting QNM maybe useful in providing insights to QNMs emitted from more complicated systems such as black holes. In [4] we have found some new exactly solvable QNM models, and a new QES QNM model within the sl(2) Lie-algebraic approach to QES theory. In this paper we have demonstrated how exactly solvable and QES models admitting quasinormal modes can be constructed and classified very simply and directly by the prepotential approach. This approach, unlike the Lie-algebraic approach, does not require any knowledge of the underlying symmetry of the system. It treats both quasi-exact and exact solvabilities on the same footing, and gives the potential as well as the eigenfunctions and eigenvalues simultaneously. We also present a new exactly solvable Morse-like model with quasinormal modes, and two new quasi-exactly solvable models with quasinormal modes of the Scarf II and generalized Pöschl–Teller types.
C.-L. Ho / Annals of Physics 326 (2011) 1394–1407
1405
Acknowledgments This work is supported in part by the National Science Council (NSC) of the Republic of China under Grant NSC 96-2112-M-032-007-MY3 and NSC-99-2112-M-032-002-MY3. Appendix A. Shifted oscillator: z02 ¼ c > 0 pffiffiffi The corresponding transformation z(x) is zðxÞ ¼ cx þ constant. By an appropriate translation one pffiffiffi can always set the constant to zero. Hence we shall take zðxÞ ¼ cx without loss of generality. The potential is (1 < x < 1)
pffiffiffi pffiffiffi A1 A0 V N ¼ A22 cx4 þ 2A2 A1 cx3 þ A21 þ 2A2 A0 x2 þ 2 A2 ðN þ 1Þ cx c " # N X 1 A2 0 þ 2A2 2A1 N þ zk : 2 c k¼1
ðA1Þ
The ground state wave function w0 exp(W0) is
w0 e
A pffiffi A A 32 cx3 21 x2 p0fficx
ðA2Þ
:
We want VN to be real. If A2 – 0, then the term A2(N + 1) in the fourth term implies A2 be real. However, this implies the wave function wN, whose asymptotic behavior is governed by w0, is not normalizable on the whole line. Hence there is no QES models, with or without QNMs. For A2 = 0, the potential is
2 A0 1 : V N ¼ A1 x þ pffiffiffi 2A1 N þ 2 c
ðA3Þ
So A1 and A0 must both be real, this is just the potential of the shifted oscillator. If we take
c A1 ¼ i ; 2
A0 ¼ 0;
ðA4Þ
the potential becomes
1 1 : V N ¼ c2 x2 þ ic N þ 4 2
ðA5Þ
This is the case 5 exactly solvable QNM model discussed in [4]. Appendix B. Radial oscillator: z02 ¼ bz, (b > 0) For simplicity we take zðxÞ ¼ 4b x2 . The potential is (0 6 x < 1)
VN ¼
1 2 2 6 1 1 1 A2 b x þ A2 A1 bx4 þ A21 þ 2A2 A0 A2 bð4N þ 3Þ x2 64 8 4 2 " # N X 4A0 A0 1 1 A1 4A0 þ 2A2 þ zk : þ 4N þ 1 b b 2 x2 2 b k¼1
ðB1Þ
The ground state w0 exp(W0) is 2A0
1
w0 x b e4A1 x
2 1 A bx4 32 2
:
ðB2Þ
It can be checked that if A2 – 0, then all Ai’s have to be real. Hence there is no QES model with QNMs in this case. For A2 > 0 and A0 < 0, the wave functions wN are normalizable on the positive half-line. The system is then a QES model with real energies. Together with the discussion in Appendix A, one concludes that one-dimensional QES models start with degree six, i.e., the sextic oscillator [23]. In fact, the potential (B1) with A2 = 2a > 0, A1 = 2b, A0 = 0 and b = 4, namely,
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C.-L. Ho / Annals of Physics 326 (2011) 1394–1407
h i 4 2 V N ¼ a2 x6 þ 2abx þ b ð4N þ 3Þa x2 " # X ð4N þ 1Þb þ 4a zk ;
ðB3Þ
k
is the very first QES model discussed in the literature [5]. If A2 = 0, then (B1) becomes
VN ¼
A21 2 4A0 A0 1 1 1 2A0 A 2N þ x þ þ : 1 2 4 b b 2 x2 b
ðB4Þ
For real A1 and A0, this is just the radial oscillator. Suppose we take
b ¼ 4;
A1 ¼ 2ia;
and A0 ¼ 2c:
ðB5Þ
The potential takes the form
V N ¼ a2 x2 þ
cðc 1Þ x2
þ iað4N þ 2c þ 1Þ:
ðB6Þ
This is the Case 4 model with QNMs presented in [4]. Appendix C. Scarf I: z02 ¼ að1 z2 Þ In this case zðxÞ ¼ sin
V N ðxÞ ¼
A22
a
cos2
pffiffiffi ax . The potential is
" # pffiffiffi pffiffiffi ðA þ A2 Þ2 A1 ax þ 0 þ A1 1 sec2 ð axÞ
a
2A1
a
pffiffiffi pffiffiffi pffiffiffi 2A1 1 tanð axÞ sec ax A2 þ 2N þ 1 sin ax þ ð A0 þ A2 Þ a a " # N X A21 2A2 2 ðA0 þ A2 Þ þ 2A2 zk ; 2A1 N þ aN þ þ
a
a
ðC1Þ
k¼1
with the ground state wave function pffiffi axÞ
A2
w0 e a sinð
cos
pffiffiffi Aa1 A2 þA0 tanh1 sin ðpffiffiaxÞ ax e a :
ðC2Þ
pffiffiffi The system is defined only on a finite interval, usually taken to be ax 2 ½p=2; p=2. Hence there are no QNMs. If A2 – 0, then all Ai’s have to be real. The third term of (C2) then implies that the wave functions wN are not normalizable. So there are no QES systems with real energies. For A2 = 0, the potential becomes
VN ¼
" A20
a
þ A1
A1
a
#
1
" 2A1 N þ aN 2 þ
sec2
A21
a
pffiffiffi pffiffiffi pffiffiffi 2A1 ax þ A0 1 tan ax sec ax
a
# :
For real A1 and A0, this is Scarf I potential given in [20]. References [1] For a comprehensive review, see eg. K.D. Kokkotas, B.G. Schmidt, Living Rev. Rel. 2 (1999) 2; S. Chandrasekhar, The Mathematical Theory of Black Holes, Clarendon, Oxford, 1983. [2] S.P. Kim, J. Korean Phys. Soc. 49 (2006) 764; G. Barton, Ann. Phys. 166 (1986) 322. [3] V. Ferrari, B. Mashhoon, Phys. Rev. Lett. 52 (1984) 1361.
ðC3Þ
C.-L. Ho / Annals of Physics 326 (2011) 1394–1407 [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
[16] [17] [18] [19]
[20] [21] [22] [23]
1407
H.-T. Cho, C.-L. Ho, J. Phys. A40 (2007) 1325. A. Turbiner, A.G. Ushveridze, Phys. Lett. A 126 (1987) 181. A.V. Turbiner, Commun. Math. Phys. 118 (1988) 467. A.V. Turbiner, Sov. Phys. JETP 67 (1988) 230. A. González, N. Kamran, P.J. Olver, Commun. Math. Phys. 153 (1993) 117. M.A. Shifman, Int. J. Mod. Phys. A4 (1989) 2897. A.G. Ushveridze, Sov. Phys.-Lebedev Inst. Rep. 2 (1988) 50; A.G. Ushveridze, Sov. Phys.-Lebedev Inst. Rep. 2 (1988) 54. G. Post, A. Turbiner, Russian J. Math. Phys. 3 (1995) 113. N. Kamran, R. Milson, P.J. Olver, Invariant modules and the reduction of nonlinear partial differential equations to dynamical systems, 1999. Available from: