Prepotential approach to exact and quasi-exact solvabilities of Hermitian and non-Hermitian Hamiltonians

0.16 MB·English

by  Choon-Lin Ho

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8 Prepotential approach to exact and quasi-exact solvabilities of
0
Hermitian and non-Hermitian Hamiltonians
0
2

n
a Choon-LinHo
J
Department of Physics, Tamkang University, Tamsui 251, Taiwan, Republic of China
7

]
h In this talk I present a simple and unified approach to both exact and quasi-exact
t solvabilitiesoftheone-dimensionalSchro¨dingerequation.Itisbasedontheprepotential
- together with Bethe ansatz equations. This approach gives the potential as well as the
p
eigenfunctionsandeigenvaluessimultaneously.Inthisapproachthesystemiscompletely
e
defined by the choice of the change of variables, and the so-called zero-th order prepo-
h
tential.WeillustratetheapproachbyseveralexamplesofHermitianandnon-Hermitian
[
Hamiltonianswithrealenergies.Themethodcanbeeasilyextendedtotheconstructions

1 ofexactlyandquasi-exactlysolvableDirac,Pauli,andFokker-Planckequations, andto
v quasinormalmodes.
4
4 Keywords: Prepotential, exact/quasi-exact solvability, Bethe-ansatz equations, non-
9 HermitianHamiltonians
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1. 1. Introduction
0
I am greatly honored to be able to participate in this conference celebrating Prof.
8
0 Yang’s 85th birthday. We wish Prof. Yang a very happy birthday, and many more
: to come.
v
i In this talk I would like to present a constructive approach to both exact and
X
quasi-exact solvable one-dimensional Schr¨odinger equations. Everyone knows what
r
a exact solvability means. It is fair to say that all of us learned the principles of
quantum mechanicsthroughseveralwell-knownexactly solvable (ES) models,such
as the infinite square well, the harmonic oscillator, the hydrogen atom, etc. But
in actual life, exactly solvable systems are hard to come by. Most systems one
encounters are non-solvable, and one has to adopt various approximation schemes
to solve them.
Twodecades ago,a new class ofpotentials which areintermediate to ESpoten-
tials and non solvable ones have been found for the Schr¨odinger equation. These
arecalledquasi-exactlysolvable(QES)models for whichitis possible to determine
algebraicallyapartofthe spectrumbutnotthewholespectrum1,2,3,4,5,6.Thesim-
plestQESmodelfirstdiscoveredisthesexticoscillator1.Thediscoveryofthisclass
of spectralproblems has greatlyenlargedthe number of physicalsystems whichwe
can study analytically.
Here I would like to discuss a novel approach that treats both ES and QES
systems on the same footing 7. It is based on the so-called prepotential together
with the Bethe ansatz equations.
1
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2
Since QES systems are less familiar to most people, I shall give a very brief
introduction to QES theory before the prepotential approach is discussed.
2. A brief introduction to QES theory
The essence of quasi-exact solvability is most easily explained as follows 4. It is
well known that any Hamiltonian H can be represented as an infinite-dimensional
Hermitian matrix
H H H
00 01 0N
··· ···
H H H
10 11 1N
 ··· ···
H = , (1)
··· ··· ··· ··· ···
H H H 
 N0 N1 ··· NN ···
 
 ··· ··· ··· ··· ···
 
where the elements H = ψ H ψ depend on the choice of a complete set of
nm n m
h | | i
orthonormalfunctions ψ forming a basis of the Hilbert space. The solution of the
n
spectralproblemthenreducestoadiagonalizationofmatrix H .Unfortunately,
nm
{ }
contrarytothecaseoffinitematrices,thereisnogeneralalgebraicrulesthatwould
allow one to diagonalize the infinite-dimensional H.
If the matrix H is very specific so that it can be reduced to the diagonal
nm
{ }
formwiththeaidofanalgebraicprocess,thenthequantalsystemisexactlysolvable.
H 0 0 0 0
00
···
0 H 0 0 0
11
 ···
H = 0 0 H 0 0 . (2)
22
···
 0 0 0 H 0 
 33 ···
 
 ··· ··· ··· ··· ··· ···
 
The most well-known example is the harmonic oscillator.
Suppose a Hamiltonian H is reducible to a block form
H H H 0 0 0
00 01 0N
··· ···
H H H 0 0 ... 0
10 11 1N
 ··· 
··· ··· ··· ··· ··· ··· ··· ···
H H H 0 0 ... 0 
 N0 N1 ··· NN 
H = 0 0 0 , (3)
 
 ··· ∗ ∗ ··· ···
 0 0 0 
 ··· ∗ ∗ ··· ···
 
 ··· ··· ··· ··· ∗ ∗ ··· ···
 0 0 0 
 ··· ∗ ∗ ··· ···
 
 ··· ··· ··· ··· ∗ ∗ ··· ···
 
where the block in the upper left corner is an (N +1) (N +1) Hermitian matrix,
×
and the block with asterisksis an infinite matrix with non-vanishingelements. The
upper left block can be diagonalizedwithout touching the infinite one. This means
that one can determine only a part of the spectrum of H with N +1 eigenvalues
and eigenfunctions. This system is called QES.
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3
QES models are not simply mathematical constructs. Although the first and
simplest example of QES model is the sextic oscillator,physical QES systems have
beenfound.Forexamples,thesystemoftwoelectronsinathree-dimensionaloscilla-
torpotential8,two-dimensionalchargedparticleinCoulombanduniformmagnetic
fields (the Schr¨odinger, Klein-Gordon, and Dirac cases are discussed in Refs. 9, 10
and 11, respectively), and electron on a lattice in magnetic field (Azbel-Hofstadter
problem) 12.
Usually a QES problem admits a certain underlying Lie algebraic symmetry
which is responsible for the quasi-exact solutions. Such underlying symmetry is
most easily studied in the Lie-algebraic approach 3,5. The essence of this approach
is as follows. Consider a Schr¨odinger equation Hψ = Eψ with Hamiltonian H =
d2/dx2 + V(x) and wave function φ(x). Here x belongs either to the interval
−
( , )or[0, ).Nowsupposewemakean“imaginarygaugetransformation”on
−∞ ∞ ∞
the function φ: φ(x) = φ˜(x)e−W0(x). For physical systems which we are interested
in, the phase factor exp( W (x)) is responsible for the asymptotic behaviors of
0
−
the wave function so as to ensure normalizability. The function φ˜(x) satisfies a
Schr¨odinger equation with a gauge transformed Hamiltonian H = eW0He−W0.
W
Suppose H can be written as a quadratic combination of the generators Ja of
W
some Lie algebra with a finite dimensional representation, i.e.,
H = c JaJb+ c Ja+real constant, (4)
W ab a
where the constants c Xand c are uXsually taken to be real 3,5,6. Then within
ab a
this finite dimensionalHilbert spacethe HamiltonianH canbe diagonalized,and
W
therefore a finite number of eigenstates are solvable. Hence the system described
by H is QES. For one-dimensional QES systems the most general Lie algebra is
sl(2). The generators Ja of the sl(2) Lie algebra take the differential forms: J+ =
z2d nz , J0 = zd n/2 , J− = d (n = 0,1,2,...). The variables x and z
z z z
− −
are related by some function. n is the degree of the eigenfunctions φ˜, which are
polynomials in a (n+1)-dimensionalHilbert space with the basis 1,z,z2,...,zn .
h i
The Lie-algebraic approach to QES models excels in revealing the underlying
symmetry of a QES system explicitly. However, solutions of QES states are more
directly found in the analytic approach based on the Bethe ansatz equations 4. In
this analytic approach the form of the wave functions containing some parameters
are assumed from the very beginning, and these parameters are fitted to make the
ansatz compatible with the potential under consideration .
Recently,adifferentapproachtoQESmodelshasemerged13,14.Inthisapproach
the emphasis shifts fromthe potential to the so-calledprepotential(or superpoten-
tial), a concept which plays a fundamental role in supersymmetric quantum me-
chanics (we note here that the prepotential considered here is the integral of the
superpotetial in Refs. 13 and 15). Prepotential has been extensively employed to
study classicaland quantum integrability in Calogero-Mosersystems 16. The merit
of this approach is that the form of the potential of the system concerned needs
not be assumed from the beginning. All information about the system is contained
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in the prepotential and the solutions, or roots, of the Bethe ansatz equations (in
termsofcertaintransformedcoordinates).Theprepotentialandtherootsdetermine
the potential as well as the eigenfunctions and eigenvalues simultaneously. Also, in
this approach exact and quasi-exact solvabilities can be treated on the same foot-
ing. Furthermore, such approach facilitates extension of the QES theory from the
Schr¨odinger equation to equations for multi-component wave functions 13,23. More
recently, QES theory was extended to the Fokker-Planck equations also via the
prepotential approach 18.
The emphasis of the works just mentioned was placed on the feasibility and
elegance of the prepotential approach. But in these works the forms of the prepo-
tentialandtherequiredchangeofcoordinateswereeitherdirectlyadaptedfromthe
known ES and QES models, or given as known for the new QES systems. To make
theprepotentialapproachasatisfactoryapproachonemustbeabletodeterminethe
choice of coordinate transformation and the prepotential from the very beginning.
This was recently achieved for certain classes of coordinates and prepotentials 7.
In what follows we shall present the main ideas of this approach, and demonstrate
how the simplest ES and QES models could be easily derived.
3. Prepotential approach
Supposeφ (x)( <x< ))isthegroundstate,withzeroenergy,ofaHamilto-
0
−∞ ∞
nian H : H φ =0. By the wellknown oscillationtheoremφ is nodeless,and thus
0 0 0 0
can be written as φ e−W0(x), where W (x) is a regular function of x. For the
0 0
≡
square-integrable φ , this is the simplest example of quasi-exact solvability. This
0
impliesthatthepotentialV iscompletelydeterminedbyW :V =W′2 W′′,and
0 0 0 0 − 0
consequently,the Hamiltonianis factorizable (we adoptthe unit system in which ~
and the mass m of the particle are such that ~=2m=1):
d d
H = +W′ +W′ . (5)
0 −dx 0 dx 0
(cid:18) (cid:19)(cid:18) (cid:19)
We shall call W (x) the zero-th order prepotential.
0
Consider now a wave function φ (N 0) which is related to φ of H by
N 0 0
≥
φ =φ φ˜ , where
N 0 N
φ˜ =(z z )(z z ) (z z ), φ˜ 1. (6)
N 1 2 N 0
− − ··· − ≡
Here z = z(x) is some function of x. In taking the form of φ˜ in Eq. (6) I have
N
assumed that the only singularities of the system are z = . Other situations
±∞
are discussedin Ref. 7. The function φ˜ is a polynomialin an(N+1)-dimensional
N
Hilbert space with the basis 1,z,z2,...,zN . One can rewrite φ as
N
h i
φ =exp( W (x, z )), (7)
N N k
− { }
with the N-th order prepotential W being defined by
N
N
W (x, z )=W (x) ln z(x) z . (8)
N k 0 k
{ } − | − |
k=1
X
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Operating on φ by the operator d2/dx2 results in a Schr¨odinger equation
N
−
H φ =0, where
N N
d2
H = +V , (9)
N −dx2 N
V W′2 W′′. (10)
N ≡ N − N
From Eq. (8), V has the form V =V +∆V , where V =W′2 W′′, and
N N 0 N 0 0 − 0
z′′ N 1 z′2
∆V 2 W′z′ + . (11)
N ≡− 0 − 2 z z (z z )(z z )
(cid:18) (cid:19)k=1 − k k,l − k − l
X Xk6=l
Here the prime denotes derivative w.r.t. the variable x.
∆V is generally a meromorphic function of z with at most simple poles. Let
N
us demandthatthe residuesofthe simple poles,z ,k =1,...,N shouldallvanish.
k
This will result in a set of algebraic equations which the parameters z must
k
{ }
satisfy. These equations are called the Bethe ansatz equations for z . With z
k k
{ } { }
satisfying the Bethe ansatz equations, ∆V will have no simple poles at z but
N k
{ }
it still generallydepends on z .Thus the form of V is determined by the choice
k N
{ }
of z′2 and W (x).
0
In what follows we would like to demonstrate that the choice of z′2 and W′z′
0
determine the nature of solvability of the quantal system. We shall restrict our
considerationonlytothosecaseswhereW′z′ =P (z),z′2 =Q (z)andz′′ =R(z)
0 m n l
are polynomials in z of degree m, n and l, respectively.Of course,Q (z) and R(z)
n l
are not independent. In fact, from 2z′′ =dz′2/dz we have l=n 1 and
−
1dQ (z)
z′′ =R (z)= n . (12)
n−1
2 dz
Consequently,the variablesx andz are relatedby (we assumez(x) is invertible for
practical purposes)
z dz
x(z)= , (13)
± Q (z)
Z n
and the prepotential W (x) is determined aps
0
x P (z)
m
W (x)= dx . (14)
0
Z Qn(z)!z=z(x)
p
Eqs. (13) and (14) define the transformation z(x) and the corresponding prepo-
tential W (x). Thus, P (z) and Q (z) determine the quantum system. Of course,
0 m n
the choice of P and Q must be such that W derived from (14) must ensure
m n 0
normalizability of φ =exp( W ).
0 0
−
Now depending on the degrees of the polynomials P and Q , we have the
m n
following situations:
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6
(i) if max m,n 1 1, then in V (x) the parameter N and the roots z ’s will
N k
{ − } ≤
only appear as anadditive constantandnot in any terminvolving powersof z.
Such system is then exactly solvable;
(ii) if max m,n 1 =2, then N will appear in the first power term in z, but z ’s
k
{ − }
onlyinanadditiveterm.This systemthen belongsto the so-calledtype 1 QES
system defined in Ref. 3, i.e., for each N 0, V admits N +1 solvable states
N
≥
with the eigenvalues being given by the N +1 sets of roots z ’s. This is the
k
main type of QES systems considered in the literature;
(iii) if min m,n 1 3, then not only N but also z ’s will appear in terms
k
{ − } ≥
involvingpowersofz.ThismeansthatforeachN 0,thereareN+1different
≥
potentials V , differing in several parameters in terms involving powers of z,
N
have the same eigenvalue (when the additive constant, or the zero point, is
appropriately adjusted). When z ’s appear only in the first power term in z,
k
suchsystemsarecalledtype 2 QESsystemsinRef.3.We seethatQESmodels
of higher types are possible.
Thesegeneralsituationsareconsideredforcaseswherez′2 =Q (z)arequadratic
2
in z (this includes Q and Q as special cases if some of the coefficients vanish) in
1 0
Ref. 7. This choice of z′2 covers most of the known ES shape-invariant potentials
in Ref. 15 and the sl(2)-based QES systems in Ref. 3, and a new one discussed in
Refs.14and18.Suchcoordinatesarecalled“sinusoidalcoordinates”,whichinclude
quadratic polynomials, trigonometric, hyperbolic, and exponential types. The con-
nection of sinusoidal coordinates with ES theory has been extensively discussed in
Ref. 19. For simplicity, in this talk I shall only discuss cases in which z′2 is linear
in z, i.e., z′2 =Q (z)=q z+q , where q and q are real constants.
1 1 0 1 0
With this choice of z′2, we have
N
1 q Q (z )
V =W′2 W′′ 2 P (z) 1 1 k . (15)
N 0 − 0 − z z  m − 4 − z z 
k k l
Xk=1 −  Xl6=k − 
In deriving Eq. (15) use has been made of the following identities:
 
N N
1 1 1
=2 , (16)
(z z )(z z ) z z z z
k,l=1 − k − l k,l=1 − k (cid:18) k− l(cid:19)
Xk6=l Xk6=l
N N
z 1 z
k
=2 . (17)
(z z )(z z ) z z z z
k,l=1 − k − l k,l=1 − k (cid:18) k− l(cid:19)
Xk6=l Xk6=l
Demanding the residues at z ’s vanish gives the set of Bethe ansatz equations
k
q Q (z )
1 1 k
P (z ) =0, k =1,2,...,N. (18)
m k
− 4 − z z
k l
l6=k −
X
Putting back the set ofrootsz into Eq.(15),we obtaina potentialV (x) without
k N
simple poles. The degree of P (z) determines the nature of the solvability of the
m
February 2, 2008 16:38 WSPC/INSTRUCTION FILE HoCL-CNYang85
7
system, namely, for m = 1,2,3,..., the system is, respectively, ES, type 1 QES,
type 2 QES, and higher types QES, as discussed generally in the last section.
To proceed further, we must specify P (z). Below I shall discuss cases with
m
m 2, i.e., only ES and type 1 QES models.
≤
4. Examples
Let us take Q (z) = 4Az +q (q = 4A), where A, q are real constants. This
1 0 1 0
impliesz′′(x)=R (z)=2A.Hence,thegeneralsolutionofz(x)isaquadraticform
0
of x:
z(x)=Ax2+Bx+C, A, B, C:real constants. (19)
Itiseasilycheckedthatq isrelatedtoA, B andC throughz′2 =4Az+B2 4AC.
0
−
We now show how some ES and QES models can be constructed in the prepo-
tential approachby taking different values of m.
4.1. Exactly solvable cases: m = 1
Suppose m = 1 so that P (z) = A z+A (A , A real). By writing P = A (z
1 1 0 1 0 1 1
−
z )+A z +A , one obtains from Eq. (15) that
k 1 k 0
N N
1 Q (z )
1 k
∆V = 2A 1 2 P (z ) A
N − 1 − z z  1 k − − z z 
k k l
Xk=1 Xk=1 −  Xl6=k − 
= 2A N. (20)
− 1  
The last term in braces in Eq. (20) vanishes when z ’s satisfy the Bethe ansatz
k
equations (18). Now N only appears as a parameter in an additive term, and not
in terms involving powers of z in ∆V . The roots z ’s do not appear at all. The
N k
additive term can be treated as the eigenvalue. The Schr¨odinger equation reads
d2
+W′2 W′′ e−WN =2A Ne−WN. (21)
−dx2 0 − 0 1
(cid:18) (cid:19)
We see that the potential W′2 W′′ is ES: by varying N, one obtains all the
0 − 0
eigenvalues 2A N and the eigenfunctions φ =exp( W ).
1 N N
−
As an example, let us take Q (z) = 1 and P (z) = bz. A particular solution of
1 1
z is z(x)=x. From Eq. (14) one gets
x bx bx2
W (x)= dx= +const. (22)
0
1 2
Z
Theintegrationconstantistobedeterminedbynormalizationofthewavefunction.
Forφ =exp( W )tobe square-integrable,onemustassumeb>0.The BAE(18)
0 0
−
are:
1
bx =0, k =1,...,N, (23)
k
− x x
k l
l6=k −
X
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and the potential is
V =b2x2 b, ∆V = 2Nb. (24)
0 N
− −
This leads to the Schr¨odinger equation:
d2
+b2x2 e−WN =b(2N +1)e−WN. (25)
−dx2
(cid:18) (cid:19)
This system is just the well-known simple harmonic oscillator.
We note here that by rescaling √bx x , Eq. (23) will have b = 1. The
k k
→
resulted equations are the equations that determine the zeros of the Hermite poly-
nomials H (x) as found by Stieltjes 20,21,16. Hence we have reproduced the well
N
known wave functions for the harmonic oscillator, namely, φ = exp( W )
N N
− ∼
exp( bx2/2)H (√bx).
N
−
4.2. Type 1 quasi-exactly solvable cases: m = 2
Next we consider P (z)=A z2+A z+A . By a similar argument we obtain
2 2 1 0
N
∆V = 2A Nz 2A z 2A N. (26)
N 2 2 k 1
− − −
k=1
X
The Schr¨odinger can be written as
d2 N
+W′2 W′′ 2A Nz e−WN =2 A z +A N e−WN. (27)
(cid:18)−dx2 0 − 0 − 2 (cid:19) 2k=1 k 1 !
X
Unlike the previouscase,nowN notonly appearsinanadditive constanttermbut
also in the term with z, and the set of roots z ’s appear in the additive term. This
k
system is the so-called type 1 QES models. Type 1 QES models classified as class
VI in Ref. 3 belong to this category.
A well-known example is the sextic oscillator, the simplest QES model of this
type 1, which is termed as class VI in Ref. 3. In our prepotential approach, this
system is defined by z(x)=x2 and P (z)=2(az2+bz). Then Q (z)=4z, and
2 1
x ax4+bx2 1 1
W (x)= dx= ax4+ bx2+const. (28)
0
√x2 4 2
Z
Here a>0 to ensure square-integrability of the wave function. The BAE (18) are:
z
2az2+2bz 1 4 k =0, k=1,...,N, (29)
k k− − z z
k l
l6=k −
X
and the potential is
V =a2x6+2abx4+ b2 (4N +3)a x2 4a z (4N +1)b. (30)
N k
− − −
k
(cid:2) (cid:3) X
This is just the well-known QES sextic oscillator first discovered in Ref. 1.
These two examples show how simply and directly the two most classic repre-
sentatives of ES and QES models are derived in the prepotential approach. Other
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9
ES and QES models can be constructed in a similar way. Particularly,we can con-
sider cases where z′2(x) = Q (z) with q = 0. Here z(x) is again some sinusoidal
2 2
6
coordinates,whichincludetheexponential,hyperbolicandtrigonometricfunctions.
By taking appropriate P (z), one can reconstruct class I, II and X QES models in
m
Ref. 3, and some of the ES models listed in Ref. 15, namely, the Morse potential,
the Scarf I and II potentials, and the Po¨schl-Teller potential 7.
5. Non-Hermitian potentials with real spectra
Now we turn to a different kind of spectral problem. About ten years ago, it was
noted that with properly defined boundary conditions the spectrum of the non-
Hermitian Hamiltonian H = p2 + x2(ix)ν (ν 0) is real and positive 22. It is
≥
believed that the reality of the spectrum of this Hamiltonian is a consequence of
its unbroken symmetry. This discovery immediately sparked great interest in
PT
searching for new systems with such properties, and in understanding the reasons
behind the reality of their eigenvalues (for review, see e.g. Ref. 23).
Soon after the appearance of Ref. 22, it was found that, by allowing non-
Hermitian -symmetricHamiltonians,aQESpolynomialpotentialcanbequartic
PT
initsvariable24.Previously,itwasbelievedthelowest-degreeone-dimensionalQES
polynomialpotentialissextic,namely,thatpresentedinSect.4.Nowletusdemon-
strate how simply the Hamiltoniandiscoveredin Ref. 24 is derivedin our approach
if we allow P (z), or equivalently W (x), to be complex.
m 0
5.1. PT-symmetric quartic case
We take z = x (i.e. Q = 1), and P (x) = ax2 +bx+c for type 1 QES system.
0 2
Eq. (14) then gives W (x) = ax3/3+bx2/2+cx. It is obvious that, with a,b and
0
c real, the wave functions are not normalizable as the factor exp( W ) does not
0
−
converge on the whole line. Hence a QES Hamiltonian with quartic polynomial
potential is impossible with real coefficients in W .
0
What if we allow P (x) to be complex? Suppose we take a = iα, b = β and
2
c=iγ,whereα, β >0andγarereal.Obviouslyexp( W )isnowsquare-integrable
0
−
on the whole line for b > 0. From Eqs. (15) and (18) we obtain straightforwardly
the type 1 QES potential
V (x) = α2x4+2iαβx3+ β2 2αγ x2 2i[(N +1)α βγ]x
N
− − − −
(cid:0) (cid:1)
γ2+β+2iα x , (31)
k
−" #
k
X
with the BAE
1
iαx2 +βx +iγ =0, k =1,2,...,N. (32)
k k − x x
k l
l6=k −
X
V givenbyEq.(31)ispreciselythe -symmetricQESquarticpotentialobtained
N
PT
in Ref. 24 (in the notation of Ref. 24, the potential has parameters α = 1, β =
February 2, 2008 16:38 WSPC/INSTRUCTION FILE HoCL-CNYang85
10
a, γ = b, and N + 1 = J). In Ref. 24 it was shown that this -symmetric
PT
Hamiltonian is also related to the Lie-algebra sl(2,C). It is nice to see that in our
prepotentialapproachthis potentialcanbe so simply anddirectlyderived,without
any knowledge of its underlying symmetries.
The properties of this system have been fully described in Ref. 24, so we will
not attempt a full discussion of it. Let us just consider the two simplest cases with
N =0 and N =1.
For N = 0, the potential is just V with only one solvable state corresponding
0
to energy E = 0. For N = 1, there are two solvable states. From Eq. (31) we can
take the potential to be
V = α2x4+2iαβx3+ β2 2αγ x2 2i(2α βγ)x (33)
1
− − − −
with eigenvalues γ2+β+2iαx . The r(cid:0)oots x sa(cid:1)tisfy the BAE (32): iαx2+βx +
1 1 1 1
iγ = 0. The solutions are: x = i(β β2+4αγ)/2α. Hence we see that for
1
±
β2+4αγ >0 the non-Hermitianpotential V has two normalizablestates with real
p1
energies γ2 β2+4αγ. But if β2 +4αγ < 0, the energies are complex. This is
±
consistentwiththe numericalresultsinRef.24.ForhighervaluesofN theanalysis
p
is more complicated and numerics are needed 24.
5.2. Sextic case without PT symmetry
We now consider a second non-Hermitian QES system. Suppose we assume the
parameter b in the sextic oscillator (class VI in Ref. 3) in Sect. 4.2 to be purely
imaginary: b = iβ, where β and a > 0 are real. The factor exp( W ) is square-
0
−
integrable. Now the potential and BAE are:
V = a2x6+2iaβx4 β2+(4N +3)a x2 4a z i(4N +1)β, (34)
N k
− − −
k
(cid:2) z (cid:3) X
2az2+2iβz 1 4 k =0, k =1,...,N. (35)
k k− − z z
k l
l6=k −
X
Again, this non-Hermitian system could have real eigenvalues. Trivially for N = 0
the potential V admits one normalizable eigenstate with real energy E = 0. For
0
N = 2, the solutions of Eq. (35) are: z = ( iβ 2a β2)/2a. Then Eq. (34)
1
− ± −
becomes
p
V =a2x6+2iaβx4 (β2+7a)x2 3iβ 2 2a β2. (36)
1
− − ± −
Hence the non-Hermitian potential defined by the first fopur terms in Eq. (36) has
two real eigenvalues E = 2 2a β2 if β2 <2a. The energies become complex if
± −
β2 >2a. Like the quartic case just discussed, for higher values of N the analysis of
p
its spectrumrequiresthe aidofnumerics.Letusemphasizethatthis systemsisnot
-symmetric.
PT
We note here that the same analysis can be applied to class I and VII QES
systems in Ref. 3 to get other non-Hermitian QES systems with real energies.