Eigenvalue and Eigenfunction for the $PT$-symmetric Potential $V = - (ix)^N$

3.3 MB·

by  Cheng Tang

Eigenvalue and Eigenfunction for the
N
PT-symmetric Potential V = − (ix)
Cheng Tang1 and Andrei Frolov2
Department of Physics, Simon Fraser University
V5A 1S6, Burnaby, BC, Canada
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1 [email protected] [email protected]
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2 February 27, 2017

b
e
Abstract
F
If replace the Hermiticity from conventional quantum mechanics with the physi-
7
2 callytransparentconditionofparity-timereflectionsymmetry(PT-symmetry),thenon-
Hermitian Hamiltonian still guarantees that its entire energy spectrum is real if the

]
h HamiltonianhasunbrokenPT-symmetry. IfitsPT-symmetryisbroken,thentwocases
p can happen - its entire energy spectrum is complex for the first case, or a finite number
-
h of real energy levels can still be obtained for the second case. This was “officially” dis-
at coveredonapaperbyBenderandBoettchersince1998whentheenergyspectrumfrom
m the PT-symmetric Hamiltonian H = p2 − (ix)N with x ∈ C was examined within one
[ pair of Stokes wedges.

Tobetterunderstandingdifferentialequationincomplexplane,forthisHamiltonian
2
v we discuss the following three questions in this paper. First, since their paper used
0
a Runge-Kutta method to integrate along a path at the center of the Stokes wedges
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1 to calculate eigenvalues E with high accuracy, we wonder if the same eigenvalues can
7 be obtained if integrate along some other paths in different shapes. Second, what the
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corresponding eigenfunctions look like? Should the eigenfunctions be independent from
.
1 the shapes of path or not? Third, since for large N the Hamiltonian contains many
0
pairsofStokeswedgessymmetricwithrespecttotheimaginaryaxisofx,thusmultiple
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1 families of real energy spectrum can be obtained. What do they look like? Any relation
:
v among them?
i
X
r
Contents
a
1 Introduction 1
2 Local asymptotic analysis for the potential V = −(ix)N 2
3 Numerical approximation 5
3.1 Levenberg-Marquardt algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 Parametrization, eigenvalue and eigenfunction . . . . . . . . . . . . . . . . . . 7
3.2.1 When N = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2.2 When N = 3 and N = 2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1 INTRODUCTION
4 Multiple families of real energy spectrum 26
4.1 Comparison between two pairs of PT-symmetric wedges when N = 5 . . . . . 26
4.2 The leading-order WKB approximation . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Energy spectra from the first four families versus N . . . . . . . . . . . . . . 28
4.4 A comment on the WKB approximation . . . . . . . . . . . . . . . . . . . . . . 33
5 Conclusion 34
1 Introduction
In quantum mechanics, the sign of position operator xˆ and the momentum operator pˆcan
be changed by the parity reflection operator P in the following way[2]:
PxˆP = −xˆ PpˆP = −pˆ PiP = i , (1)
where, however, the sign of the complex number i is unchanged. If we apply the time
reversal operator Tˆ instead, then
TxˆT = xˆ TpˆT = −pˆ TiT = −i , (2)
where the sign of the momentum operator and of the complex number are changed. We
say a Hamiltonian H is PT-symmetric if the combined operator PT commutes with Hˆ such
that
(cid:104) (cid:105)
PT,Hˆ = Hˆ(PT)−(PT)Hˆ = 0 . (3)
For example, the Hamiltonian H = p2 −(ix)N with x ∈ C and N ∈ R is PT-symmetric.
The discovery[4, 9] that an entirely real energy spectrum can be obtained from the
non-Hermitian but PT-symmetric Hamiltonian was a surprise to scientific communities
in 1998. Since then, the developments in PT-symmetric quantum theory rapidly grew - at
least 50 experiments to observe PT-symmetric system were published during the last 10
years. Those experiments told us that it was possible to experimentally measure complex
eigenvalue, and observe broken and unbroken PT-symmetry.
It’s well-known[2] that within a specific pair of Stokes wedges the PT-symmetry of H =
p2 −(ix)N is unbroken if N ≥ 2, so that the entire energy spectrum is positive; and broken
if N < 2, so that only a finite number of positive energy levels can be found. However,
this conclusion is not satisfied for the curiosity of any enthusiastic student. As far as we
know,moststudentsfromphysicalscienceareunfamiliarwiththeconceptofStokeswedge.
Driven by curiosity, they would ask something similar to the following questions:
1. Since Bender and Boettcher[4] used a Runge-Kutta method to integrate along a path
at the center of the Stokes wedges to calculate eigenvalues E with high accuracy, we
wonderifthesameeigenvaluescanbeobtainedifintegratealongsomeotherpathsin
different shapes. In other words, are those eigenvalues independent from the shape
of path or not?
2. So far we have not seen any research about its eigenfunctions, and do not know why
the research of eigenfunction should be ignored. In this paper, can we provide a de-
tailed study to fill the gap?
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2 LOCAL ASYMPTOTIC ANALYSIS FOR THE POTENTIAL V = −(IX)N
3. DuetotheexistenceofmultiplepairsofStokeswedges,theHamiltonianH = p2−(ix)N
must contain multiple families of real energy spectrum if N is large enough. We are
really curious to see what the spectra look like.
It’sdifficulttoanswerthesethreequestionsbyusingrigorousmathematicswhichisbeyond
university-level and in fact we don’t know. To answer them pedagogically, we implement
the strategy - “seeing is believing”. A lot of figures and tables are shown in this paper to
help students visualize data, Stokes wedges and the relation among them. How we answer
the questions is based on the observation of the data, rather than rigorous mathematics.
Webelieveempiricalobservationandconjecturearethefirstandcrucialsteptodeepenour
understanding upon rather complicated concept.
Our paper is organized in the following way. In Sec.(2), the concept of Stokes wedges
is introduced. In Sec.(3), we show two ingredients needed for the numerical calculation of
the eigenvalue - algorithm and parametrization, then we provide answers for the first two
questions by discussing the results for the specific values of N. In Sec.(4), after introduced
the WKB approximation, we plot and discuss the first four families of the spectrum.
N
2 Local asymptotic analysis for the potential V = −(ix)
Consider 1D Schrodinger equation in the complex plane with N ∈ R and x ∈ C,
−ψ(cid:48)(cid:48)(x)−(ix)N ψ(x) = Eψ(x), (4)
with boundary condition ψ(x) → 0 and ψ(cid:48)(x) → 0 as |x| → ∞.
We guess that (4) has a solution with the form ψ(x) = eS(x) where S(x) = axb with
a ∈ C, a (cid:54)= 0 and b > 0. So we substitute our ansatz into (4), neglect[7] those terms whose
modulus are orders of magnitude smaller than the rest when |x| → ∞, and finally obtain
the following asymptotic relations as |x| → ∞,
(cid:34) (cid:35)
iN+1
ψ(x) ∼ C1|x|−N4 exp ±N2+1xN2+1 when N ≥ 2; (5)
2
 (cid:16) (cid:17) 
iN+1 E x−N2+1 2
ψ(x) ∼ C2|x|−N4 exp±N2+1xN2+1 ∓ (−N +2)iN+1 when 3 ≤ N < 2; (6)
2 2
 (cid:16) (cid:17) 
iN+1 E x−N2+1 E2
ψ(x) ∼ C3|x|−N4 exp±N2+1xN2+1 ∓ (−N +2)iN+1 ± 8i3N+1(cid:0)1− 3N(cid:1)x−32N+1 (7)
2 2 2 2
2
when 0 < N < ,
3
where C , C and C are some constants. Note that the asymptotic relation (5) for N ≥ 2 is
1 2 3
independent from the eigenvalue E. To satisfy the boundary condition, we expect that its
dominant contribution in the leading-order behavior vanishes such that
(cid:34) (cid:35)
iN+1
exp[S (x)] ≡ exp ± 2 xN+1 → 0 as |x| → ∞.
1,2 N +1 2
2
2
2 LOCAL ASYMPTOTIC ANALYSIS FOR THE POTENTIAL V = −(IX)N
exp[S (x)] approaches to zero in the fastest speed if the oscillatory behavior of the expo-
1,2
nential is zero, in other words[7],
(cid:104) (cid:105)
Im[S (x)−S (x)] = 0 =⇒ Im iN+1xN+1 = 0, (8)
1 2 2 2
which is our definition of “Stokes lines”. With x = reiθ, (8) becomes
(cid:18) (cid:19) (cid:18) (cid:19)
π N N
+1 +θ +1 = ±kπ for k = 0,1,2,3,···,
2 2 2
which yields
(cid:40)
θ = −π + N−4k+2π
left N+2 2 for k = 0,1,2,3,···, (9)
θ = −N−4k+2π
right N+2 2
so that
θ = −π −θ . (10)
left right
We plot all those Stokes lines on Fig.(1). When k = 1 we obtain
(cid:40)
θ = −π + N−2π
left N+22 . (11)
θ = −N−2π
right N+22
These Stokes lines are plotted on Fig.(2).
The locations of “anti-Stokes lines” are defined as
Re[S (x)−S (x)] = 0, (12)
1 2
when the exponential is purely oscillatory. Solving (12) yields
π π 2(j −1)π π π 2jπ
θ = − + θ = − + for j = 0,1,2,3,···,
1 2
N +2 2 N +2 N +2 2 N +2
which define the width of the “Stokes sector” or “Stokes wedge”
2π
(cid:52) = |θ −θ | = . (13)
1 2
N +2
The shape of a Stokes wedge is not really like a wedge or a slice of pie. They are asymptotic
concepts. The angular opening (cid:52) from (13) of the wedge only refers to the opening for |x|
at certain range of complex infinity.
3
2 LOCAL ASYMPTOTIC ANALYSIS FOR THE POTENTIAL V = −(IX)N
Figure 1: All Stokes wedges for non-negative integer N and all corresponding turning points
(yellow point with black edge). Although the locations of turning points are eigen-
value E-dependent, for the purpose of visualization here we set E as an appropriate
constant.
SSSSSSSSSSSSttttttttttttooooooooooookkkkkkkkkkkkeeeeeeeeeeeessssssssssss
IIMM IIMM IIIMMM IIIMMM
NN == 00 NN == 11 NNN === 222 NNN === 333 AAAAAAAAAAAAnnnnnnnnnnnnttttttttttttiiiiiiiiiiii------------SSSSSSSSSSSSttttttttttttooooooooooookkkkkkkkkkkkeeeeeeeeeeeessssssssssss
((hhyyppootthheettiiccaall)) ((hhyyppootthheettiiccaall))
RREE RREE RRREEE RRREEE
IIIIMMMM IIIIMMMM IIIIIMMMMM IIIIIMMMMM
NNNN ==== 4444 NNNN ==== 5555 NNNNN ===== 66666 NNNNN ===== 77777
RRRREEEE RRRREEEE RRRRREEEEE RRRRREEEEE
IIIIIIMMMMMM IIIIIIMMMMMM IIIIIIIMMMMMMM IIIIIIIMMMMMMM
NNNNNN ====== 888888 NNNNNN ====== 999999 NNNNNNN =======11111110000000 NNNNNNN ======= 11111111111111
RRRRRREEEEEE RRRRRREEEEEE RRRRRRREEEEEEE RRRRRRREEEEEEE
Except the top and bottom wedges which contain the imaginary axis, Fig.(1) shows that
the rest of all wedges form pairs symmetric with respect to the imaginary axis. Each pair
are labeled with a color - orange, green, pink, yellow, red, etc. We may sometimes call
those pairs as “PT-symmetric Stokes wedges”. Each pair contains a pair of turning points
which are also symmetric with respect to the imaginary axis. The larger the N is, the more
pairsofwedgesandofturningpointsare. Anywedgewhoseanti-Stokeslinecoincideswith
the imaginary axis only shares one singular turning point with its pair, and that singular
turning point must be located on the imaginary axis. On Fig.(1), when N = 0 and N = 1,
we label the wedges as “hypothetical wedges”, because the locations of these wedges on
the figure are actually not true for N < 2 according to (6) and (7), where these wedges are
eigenvalue E-dependent.
Sincedifferentpairofwedgeswillposedifferenteigenvalueproblem,toproceed,wenow
only focus on one pair of wedges by choosing the orange pair with the Stokes lines defined
from (11) shown on Fig.(2) to calculate the eigenvalue.
4
3 NUMERICAL APPROXIMATION
Figure 2: The chosen Stokes wedges for non-negative integer N and all corresponding turning
points (yellow point with black edge).
SSSSSSttttttooooookkkkkkeeeeeessssss
N = 0 IM N = 2 IM N = 4 IM AAAAAAnnnnnnttttttiiiiii------SSSSSSttttttooooookkkkkkeeeeeessssss
(hypothetical)
RE RE RE
N = 1 IM N = 3 IM N = 5 IM
(hypothetical)
RE RE RE
3 Numerical approximation
3.1 Levenberg-Marquardt algorithm
To solve the eigenvalue problem (4), by Levenberg-Marquardt algorithm (LMA) we choose
to minimize a square-function F (x ,E) of the following complex modulus with respect to
∞
the energy E,
F (x ,E) ≡ |f (x ,E)−ψ(x ,E)|2
∞ ∞ ∞
wherex istherightboundarypointlocatedwithinarightStokeswedgeandf (x ,E)can
∞ ∞
be calculated by Gauss-Legendre integration method[10] (GLI), an implicit Runge-Kutta
method. Due to the boundary condition lim ψ(x) = 0,
|x|→∞
F (x ,E) = |f (x ,E)|2. (14)
∞ ∞
LMA is an iterative procedure, where the previous estimate E is replaced by a new esti-
mate, E +δE, for each iterative step. We can approximate f (x ,E +δE) by
∞
f (x ,E +δE) ≈ f (x ,E)+J δE, (15)
∞ ∞
where J is the gradient of f (x ,E) with respect to E,
∞
∂f (x ,E)
∞
J = .
∂E
5
3 NUMERICAL APPROXIMATION
In our case, f (x ,E) is complex and so is E. Let u, v, a and b be real such that
∞
f (x ,E) = u(a,b)+iv(a,b) E = a+ib,
∞
then we have the following Jacobian matrix J
(cid:20) (cid:21)
∂u ∂u
J = ∂a ∂b , (16)
∂v ∂v
∂a ∂b
and (15) in vector notation is
f (x ,E+δE) ≈ f (x ,E)+JδE, (17)
∞ ∞
where
(cid:20) (cid:21) (cid:20) (cid:21)
u δa
f (x ,E) = δE = .
∞ v δb
By (14) and (17), we obtain
F (x ,E+δE) = [f (x ,E)+JδE]2 = [f (x ,E)+JδE]T [f (x ,E)+JδE]. (18)
∞ ∞ ∞ ∞
To find the minimum, we set
∂F (x ,E+δE)
∞
= 0.
∂(δE)
Hence,
δE = −(cid:0)JTJ(cid:1)−1JTf (x ,E).
∞
Due to Levenberg’s and Marquardt’s modification on the last equation, we have a damped
factor λ, which is a positive parameter, such that
δE=−(cid:2)JTJ+λdiag(cid:0)JTJ(cid:1)(cid:3)−1JTf (x ,E), (19)
∞
(cid:0) (cid:1)
where diag JTJ means a diagonal matrix with entries on the diagonal from the matrix
JTJ. If the function F (x ,E ) ≤ F (x ,E ) after a single iterative step, we update E
∞ new ∞ old old
by
E = E +δE = E −(cid:2)JTJ+λdiag(cid:0)JTJ(cid:1)(cid:3)−1JTf (x ,E), (20)
new old old ∞
√
and meanwhile decrease the value λold by a factor, for example, λnew = λold/ 2. If after a
single iterative step F (x ,E ) > F (x ,E ), this means our λ value is too small and we
∞ new ∞ old
then increase λ by a factor, for example, λ = 10λ . And the eigenvalue will not be
old new old
updated so that we still have
E = E .
new old
How we adjust the value of λ becomes important to efficiently find the eigenvalue, yet so
far there is no absolutely best way to optimize the value of λ.
6
3 NUMERICAL APPROXIMATION
3.2 Parametrization, eigenvalue and eigenfunction
We set up the following initial condition at the numerical infinity x ≡ ∞ within the left
0 left
Stokes wedge on Fig.(2):
dψ(x )
x = r exp(iθ ) ψ(x ) = 0 0 = 10−7, (21)
0 0 left 0
dx
where θ is defined by (11) and r is the complex modulus of the numerical infinity x . In
left 0 0
other words, r is the distance between the origin and the point where the wave function
0
and the derivative of the wave function almost vanish. By observation, we set r = 4.
0
To have the fastest convergence to the eigenvalue, we’re tempted to use GLI to integrate
along the two Stokes lines given by (11). However, they are connected at the origin where
is non-differentiable for N (cid:54)= 2. Since this causes non-smoothness (See Fig.(3)) on the
eigenfunction at the origin, one way to have smooth-looking eigenfunction is to integrate
along some new paths, which should satisfy the following four conditions:
1. The potential −(ix)N has a numerical cut on the positive-imaginary axis. So the new
path must not cross it; otherwise we must have a different eigenvalue problem.
2. The new path must go from one complex infinity within one Stokes wedge and back
to the other complex infinity in the other Stokes wedge. These two complex infinities
are symmetric with respect to the imaginary axis of x.
3. Thenewpathissmootheverywhereandcanbeparametrizedbyadifferentiablefunc-
tion.
4. Since GLI converges fastest if integrate along the two Stokes lines, it would be more
efficient if the path or the differentiable function has two asymptotic lines coincident
with the locations of the two Stokes lines.
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3 NUMERICAL APPROXIMATION
Figure 3: The eigenfunction of the ground state for N = 4 along the Stokes lines (non-
differentiable at origin) with r = 4. The vertical-blue dotlines represent two numer-
0
icalinfinities±Re(x ) = ±r cos30◦. AlongtheStokeslines,thenumericalresultfor
0 0
the eigenvalue E does not change even though the shape of the eigenfunction is not
smooth,incomparisonwiththesmootheigenfunctionsassociatedwiththehyperbolic
paths.
NN == 44 IIMM RE[Φ(x)]
IM[Φ(x)]
RREE
Not smooth! :(
Φ(x)
0
E =1.4771498
-4 -3 -2 -1 0 1 2 3 4
RE(x)
Figure 4: The eigenfunctions of the ground state for N = 4 along the differentiable (hy-
perbolic) paths with three different values of a defined by the hyperbolic equation
(cid:113)
Y = −a 1+ X2. All three paths have the same eigenvalue. The vertical-blue dot-
− b2
lines represent two numerical infinities ±Re(x ) = ±r cos30◦ with r = 4.
0 0 0
N = 4 E =1.4771498 Level 0
RE[ψ(x)]
ψ(x) IM[ψ(x)] (α = 0.2)
0
ψ(x) IIMM (α = 0.6 )
RREE
0

(α = 0.9)
ψ(x)
0
-4 -3 -2 -1 0 1 2 3 4
RE(x)
8
3 NUMERICAL APPROXIMATION
To satisfy all four conditions and since Fig.(2) shows that the two Stokes lines move
below the real axis when N > 2, the best differentiable function used for the parametriza-
tion when N > 2 must be hyperbola shown on the miniplot on Fig.(4). In this hyperbolic
parametrization, we treat Re(x) as the parameter so that
(cid:113)
x = Re(x)+iIm(x) = Re(x)−i a2 +[Re(x)]2(tanθ)2, (22)
where the angle θ between one of the asymptotic lines and the horizontal axis is θ =
(cid:0) (cid:1)
arctan a . Also, θ = θ from (11). Fig.(4) shows that the shapes of the corresponding
b right
eigenfunctions are smooth and different-looking since the values of a defined by the hyper-
(cid:113)
bolicequationY = −a 1+ X2 isdifferent. Fortheupcomingwork,wechoosea = 0.2since
− b2
this hyperbola is quite close to the location of the two Stokes lines and meanwhile keeps
the shape of eigenfunction smooth.
For 0 < N < 2, Fig.(2) shows that the two Stokes lines move above the real axis. Does
the function satisfy the four conditions exist? Yes. As X → ±∞ the following function with
k > 0, real parameters c and t
X2 −c
f (X) = √ (23)
kX2 +t
has two asymptotic lines:
1 1
Y = √ X as X → +∞, Y = −√ X as X → −∞.
k k
The angle θ between the asymptotic line associated with X → +∞ and the horizontal axis
satisfies
(cid:18) 1 (cid:19)2
k = .
tanθ
Fig.(5) shows a good news that the function satisfies all four conditions.
Figure 5: The differentiable path for N < 2 by choosing c = 1 and t = 8.
10
0.8
0.7
0.6
Y (X2−1 ) k .
= 10 with = 39 86
0.5 √kX2+8
0.4
Y
0.3
0.2
0.1
0
−0.1
−5 −4 −3 −2 −1 0 1 2 3 4 5
X
9