Application of Pseudo-Hermitian Quantum Mechanics to a PT-Symmetric Hamiltonian with a Continuum of Scattering States

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with a continuum of scattering states

Ali Mostafazadeh

Citation: Journal of Mathematical Physics 46, 102108 (2005); doi: 10.1063/1.2063168 View online: http://dx.doi.org/10.1063/1.2063168

View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/46/10?ver=pdfcov Published by the AIP Publishing

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Application of pseudo-Hermitian quantum mechanics

to a

PT-symmetric Hamiltonian with a continuum

of scattering states

Ali Mostafazadeha兲

Department of Mathematics, Koç University, 34450 Sariyer, Istanbul, Turkey

共Received 20 May 2005; accepted 23 August 2005; published online 21 October 2005兲 We extend the application of the techniques developed within the framework of the pseudo-Hermitian quantum mechanics to study a unitary quantum system described by an imaginaryPT-symmetric potential v共x兲 having a continuous real spectrum. For this potential that has recently been used, in the context of optical potentials, for modeling the propagation of electromagnetic waves traveling in a waveguide half and half filled with gain and absorbing media, we give a perturbative construc-tion of the physical Hilbert space, observables, localized states, and the equivalent Hermitian Hamiltonian. Ignoring terms of order three or higher in the non-Hermiticity parameter ␨, we show that the equivalent Hermitian Hamiltonian has the form p2_{/ 2m +}_共_␨2_{/ 2兲兺}

n=0 ⬁ _兵_␣

n共x兲,p2n其 with ␣n共x兲 vanishing outside an interval

that is three times larger than the support of v共x兲, i.e., in 2/3 of the physical

interaction region the potential v共x兲 vanishes identically. We provide a physical

I. INTRODUCTION

During the past seven years there have appeared over 200 research papers onPT-symmetric quantum systems. This was initially triggered by the surprising observation of Bessis and Zinn-Justin and its subsequent numerical verification by Bender and his co-workers1 that certain non-Hermitian butPT-symmetric Hamiltonians, such as

H = p2_{+ x}2_{+ i}_⑀_x3_with_⑀_{苸 R}+_, _共1兲

have a purely real spectrum. This observation suggested the possibility to use these Hamiltonians in the description of certain quantum systems. Since thePT-symmetry of a non-Hermitian Hamil-tonian H, i.e., the condition关H,PT 兴=0, did not ensure the reality of its spectrum, a crucial task was to seek the necessary and sufficient conditions for the reality of the spectrum of a given non-Hermitian Hamiltonian H. This was achieved in Ref. 2 where it was shown, under the assumptions of the diagonalizability of H and discreteness of its spectrum, that the reality of the spectrum was equivalent to the existence of a positive-definite inner product具·, ·典+that rendered

the Hamiltonian self-adjoint, i.e., for any pair共␺,␾兲 of state vectors 具␺, H␾典+=具H␺,␾典+.

Another condition that is equivalent to the reality of the spectrum of H is that it can be mapped to a Hermitian Hamiltonian h via a similarity transformation;2,3 there is an invertible Hermitian operator␳ such that

H =␳−1h␳. 共2兲

The positive-definite inner product 具·, ·典₊ and the operator ␳ entering 共2兲 are determined by a positive-definite operator␩+ according to2,3

a兲_{Electronic mail: amostafazadeh@ku.edu.tr}

46, 102108-1

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具·, · 典+ª 具·兩␩+·典, 共3兲

␳=

冑

␩+, 共4兲

and the Hamiltonian satisfies the␩+-pseudo-Hermiticity condition:4

H†=␩+H␩+−1. 共5兲

Here具·兩·典 stands for the standard 共L2_{兲 inner product that determines the 共reference兲 Hilbert space H}

as well as the adjoint H†_{of H.}5_{共The adjoint A}†_{of an operator A is the unique operator satisfying,}

for all␺,␾苸H, 具␺兩A†_␾_典=具A_␺_兩_␾_{典. A is called Hermitian if A}†_{= A.}_兲

It is this, so-called metric operator, ␩+ that determines the kinematic structure共the physical

Hilbert space and the observables兲 of the desired quantum system. Note however that␩+ is not

unique6–8共it is only unique up to symmetries of the Hamiltonian7兲. In Ref. 2 we have not only established the existence of a positive definite metric operator␩+and the corresponding

positive-definite inner product具·, ·典+for a diagonalizable Hamiltonian with a discrete real spectrum, but we

have also explained the role of antilinear symmetries such as PT and offered a method for computing the most general␩+.关For a treatment of nondiagonalizable pseudo-Hermitian

Hamil-tonians see Refs. 9–11. Note that diagonalizability of the Hamiltonian is a necessary condition for applicability of the standard quantum measurement theory.5It is also necessary for the unitarity of the time-evolution, for a nondiagonalizable Hamiltonian is never Hermitian共its evolution operator is never unitary11兲 with respect to a positive-definite inner product.9,10兴 An alternative approach that yields a positive-definite inner product for a class ofPT-symmetric models is that of Ref. 12. As shown in Refs. 7 and 13, theCPT-inner product proposed in Ref. 12 is identical to the inner product具·, ·典₊=具·兩␩₊·典 for a particular choice of ␩₊.

Under the above-mentioned conditions every Hamiltonian having a real spectrum determines a setUH+of positive-definite metric operators. To formulate a consistent unitary quantum theory

having H as its Hamiltonian, one needs to choose an element␩+ofUH+.共Alternatively one may

choose sufficiently many operators with real spectrum to construct a so-called irreducible set of observables which subsequently fixes a metric operator␩+.14兲 Each choice fixes a positive-definite

inner product具·, ·典+and defines the physical Hilbert space Hphysand the observables. The latter

are by definition15the operators O that are self-adjoint with respect to具·, ·典+, alternatively they are

␩+-pseudo-Hermitian. These can be constructed from Hermitian operators o acting inH according

to5

O =␳−1o␳. 共6兲

In particular, one can define ␩₊-pseudo-Hermitian position X and momentum P operators,5,15 express H as a function of X and P, and determine the underlying classical Hamiltonian for the system by lettingប→0 in the latter expression.5,16Alternatively, one may calculate the equivalent Hermitian Hamiltonian h and obtain its classical limit共again by letting ប→0兲.

Another application of the␩₊-pseudo-Hermitian position operator X is in the construction of the physical localized states:

兩␰共x兲_{典 ª}_␳−1_兩x典. _共7兲

These in turn define the physical position wave function, ⌿共x兲ª具␰共x兲,␺典₊=具x兩␳兩␺典, and the in-variant probability density,

共x兲 ª 兩⌿共x兲兩 2

冕

−⬁ ⬁ 兩⌿共x兲兩2_dx =兩具x兩␳兩␺典兩 2 具␺,␺典+ , 共8兲

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The above-mentioned prescription for treating PT-symmetric and more generally pseudo-Hermitian Hamiltonians with a real spectrum has been successfully applied in the study of the

PT-symmetric square well in Ref. 5 and the cubic anharmonic oscillator 共1兲 in Ref. 16—See also

Ref. 17. Both these systems have a discrete nondegenerate energy spectrum, and the results of Refs. 4 and 2 are known to apply to them. The aim of the present paper is to seek whether these results 共in particular the construction method for␩+兲 may be used for treating a system with a

continuous spectrum.共The question whether the theory of pseudo-Hermitian operators as outlined in Refs. 4 and 2 is capable of treating a system having scattering states was posed to the author by Zafar Ahmed during the second International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics, held in Prague, 14–16 June, 2004.兲 This question is motivated by the desire to understand field-theoretical analogues ofPT-symmetric systems which should admit an S-matrix formulation. Furthermore, there are some basic questions related to the nonlocal nature of the Hermitian Hamiltonian h and the pseudo-Hermitian observables such as X and P especially for

PT-symmetric potentials with a compact support 共i.e., potentials vanishing outside a compact

region兲.

To achieve this aim we will focus our attention on a simple toy model recently considered as an effective model arising in the treatment of the electromagnetic waves traveling in a planar slab waveguide that is half and half filled with gain and absorbing media.18This model has a standard Hamiltonian,

H = p 2

2m+v共x兲, 共9兲

and aPT-symmetric imaginary potential,

共11兲

The Hamiltonian 共9兲 differs from a free particle Hamiltonian only within 共−L/2,L/2兲 where it coincides with the Hamiltonian for thePT-symmetric square well.5,19

It is important to note that unlike in Ref. 18 we will consider the potential共10兲 as defining a fundamental 共noneffective兲 quantum system having a unitary time-evolution 共and S-matrix兲. Therefore our approach will be completely different from that pursued in Ref. 18 and the earlier studies of effective共optical兲 non-Hermitian Hamiltonians.20

Among the main reasons for our consideration of the potential 共10兲 is that its eigenvalue problem can be solved exactly and analytically. However, the computation of the metric operator and consequently that of physical observables, localized states, associated Hermitian Hamiltonian, etc., are extremely involved, and we could only carry them out using first-order perturbation theory.

To the best of the author’s knowledge, the only other non-Hermitian Hamiltonian with a continuous共and doubly degenerate兲 spectrum that is shown to admit a similar treatment is the one

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arising in the two-component formulation of the free Klein-Gordon equation.21,22Compared to共9兲, this Hamiltonian defines a technically much simpler system to handle, because it is essentially a tensor product of an ordinary Hermitian Hamiltonian and a 2⫻2 matrix pseudo-Hermitian Hamil-tonian.

II. METRIC OPERATOR

The essential ingredient of our approach is the metric operator ␩+. For a diagonalizable

Hamiltonian with a discrete spectrum it can be expressed as

␩+=

兺

n

兺

a=1 d_a

兩␾n,a典具␾n,a兩, 共12兲

where n, a, and dn are a spectral label, a degeneracy label, and the multiplicity 共degree of

degeneracy兲 for the eigenvalue Enof H, respectively, and兵兩␾n, a典其 is a complete set of

eigenvec-tors of H†that together with the eigenvectors兩␺n, a典 of H form a biorthonormal system.4,2

Now, consider a diagonalizable Hamiltonian with a purely continuous doubly degenerate real spectrum 兵Ek其, where k苸共0,⬁兲. We will extend the application of 共12兲 to this Hamiltonian by

changing兺n¯ to 兰dk¯. This yields

␩+=

冕

0

⬁

dk共兩␾k, +典具␾k, +兩 + 兩␾k,−典具␾k,−兩兲, 共13兲

where we have used ⫾ as the values of the degeneracy label a.22 The biorthonormal system 兵兩␺k, a典,兩␾k, a典其 satisfies

H兩␺k,a典 = Ek兩␺k,a典, H†兩␾k,a典 = Ek兩␾k,a典, 共14兲

具␾k,a兩␺_ᐉ,b典 =␦ab␦共k − ᐉ兲,

冕

0

⬁

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册

, 共28兲

and u ,v苸R are arbitrary constants 共possibly depending on k and/or Z and not both vanishing兲.

The presence of the free parameters u andv is an indication of a double degeneracy of the

eigenvalues Ek= k2. We will select u andv in such a way as to ensure that in the limit Z→0 we

recover the plane-wave solutions of the free particle Hamiltonian, i.e., we demand limZ_→0␺共x兲

= e±ikx/

冑

2␲. This condition is satisfied if we set

u = 1, v = ± 1. 共29兲

In the following we use the superscript⫾ to identify the value of a quantity obtained by setting

u = 1 andv = ± 1. In this way we introduce A₁±, B₁±, A₂±, B₂±, A_±±, B_±±, and␺±. The latter define the basis共generalized23兲 eigenvectors 兩␺k, ±典 by 具x兩␺k, ±典ª␺±共x兲.

The next step is to obtain兩␾k, ±典. In view of the identity H†=兩H兩Z→−Z, we can easily obtain

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r −

冉

k+ k₋

冊

±1/2 s

册

, 共34兲

and r , s苸R are 共possibly k- and/or Z-dependent兲 parameters that are to be fixed by imposing the biorthonormality condition 共15兲. The latter is equivalent to a set of four 共complex兲 equations 关corresponding to the four possible choices for the pair of indices 共a,b兲 in the first equation in 共15兲兴 which are to be solved for the two real unknowns r and s. This together with the presence of the delta function in two of these equations make the existence of a solution quite nontrivial.

We checked these equations by expanding all the quantities in powers of the non-Hermiticity parameter Z up to 共but not including兲 terms of order two and found after a long and tedious calculation共partly done usingMATHEMATICA兲 that indeed all four of these equations are satisfied,

if we set r = u = 1 and s =v = ± 1. Again we will refer to this choice using superscript ⫾. In

par-ticular, we have␾±=兩␺±兩Z→−Zand具x兩␾k, ±典ª␾±共x兲.

Having obtained兩␾k, ±典 we are in a position to calculate the metric operator 共13兲. We carried

out this calculation using first-order perturbation theory in Z. It involved expanding the ␾₁±共x兲, ␺2

±_{共x兲, and}_␺ ±

±_{共x兲 in powers of Z, substituting the result in}

具x兩␩+兩y典 =

冕

0 ⬁

关␾+_共y兲*_␾+_{共x兲 +}_␾−_共y兲*_␾−_共x兲兴dk _共35兲

which follows from共13兲, and using the identities:

冕

−⬁ ⬁ eiak_{dk = 2}_␲␦_共a兲,

冕

−⬁ ⬁ _eiak k dk = i␲sign共a兲,

冕

_−⬁ ⬁ _eiak − eibk k2 dk =␲共兩b兩 − 兩a兩兲 共36兲

关where a,b苸R and sign共a兲ª␪共a兲−␪共−a兲兴 to perform the integral over k for all 16 possibilities for the range of values of the pair of independent variables共x,y兲 in 共35兲. This is an extremely lengthy calculation whose detail we will not include here. It is absolutely remarkable that the expressions for具x兩␩+兩y典 that we obtain for these 16 possibilities may be combined to yield a single

formula that is valid for all x , y苸R, namely 具x兩␩+兩y典 =␦共x − y兲 +

i

8共4 + 2兩x + y兩 − 兩x + y + 2兩 − 兩x + y − 2兩兲sign共x − y兲Z + O共Z

2_兲, _共37兲

where O共Z2兲 stands for terms of order two and higher in powers of Z. Note that 具x兩␩+兩y典*

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III. PHYSICAL OBSERVABLES AND LOCALIZED STATES

The physical observables of the system described by the Hamiltonian共9兲 are obtained from the Hermitian operators acting in H=L2_{共R兲 by the similarity transformation 共6兲. This equation}

involves the positive square root␳ of ␩+which takes the form16

␳±1_{= e}⫿Q/2_, _共38兲

if we express␩ in the exponential form

␩+= e−Q. 共39兲

In view of共38兲 and the Backer-Campbell-Hausdorff identity,

e−ABeA= B +关B,A兴 + 1

2!关关B,A兴,A兴 + ... 共40兲

共where A and B are linear operators兲, physical observables 共6兲 satisfy16

O = o −1₂关o,Q兴 +1₈关关o,Q兴,Q兴 + ... . 共41兲

If we expand␩+and Q in powers of Z,

␩+= 1 +

兺

ᐉ=1 ⬁ ␩+_ᐉZᐉ, Q =

兺

ᐉ=1 ⬁ Q_ᐉZᐉ, 共42兲

where␩+_ᐉ and Qᐉ are Z-independent Hermitian operators, we find using共39兲 that Q1= −␩+₁, Q2= −␩+₂+

1 2␩+1

2

. 共43兲

Combining this relation with共41兲, we have

O = o −1₂关o,Q₁兴Z +1₈共− 4关o,Q₂兴 + 关关o,Q₁兴,Q₁兴兲Z2+O共Z3兲. 共44兲 In the following we calculate the ␩+-pseudo-Hermitian position 共X兲 and momentum 共P兲

operators,16 up to 共but not including兲 terms of order Z2_{. This is because so far we have only}

calculated␩+₁ which in view of共37兲 satisfies

具x兩␩+1兩y典 =

i

8共4 + 2兩x + y兩 − 兩x + y + 2兩 − 兩x + y − 2兩兲sign共x − y兲, ∀ x,y 苸 R. 共45兲

Substituting the scaled position 共x兲 and momentum 共p兲 operator for o in 共44兲, using 共45兲, and doing the necessary algebra, we find

具x兩X兩y典 = x␦共x − y兲 + i

16共4 + 2兩x + y兩 − 兩x + y + 2兩 − 兩x + y − 2兩兲兩x − y兩Z + O共Z

2_兲, _共46兲

具x兩P兩y典 = − i⳵x␦共x − y兲 + 1

8关2 sign共x + y兲 − sign共x + y + 2兲 − sign共x + y − 2兲兴sign共x − x兲Z + O共Z 2_兲,

共47兲 where Xª2X/L and PªLP/共2ប兲 are dimensionless␩+-pseudo-Hermitian position and

momen-tum operators, respectively.

As seen from 共46兲, both X and P are manifestly nonlocal and non-Hermitian 共but pseudo-Hermitian兲 operators. If we scale back the relevant quantities in 共46兲 and 共47兲 according to 共16兲, we find

具x兩X兩y典 = x␦共x − y兲 + im

4ប2共2L + 2兩x + y兩 − 兩x + y + L兩 − 兩x + y − L兩兲兩x − y兩␨+O共␨

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具x兩P兩y典 = − iប⳵x␦共x − y兲 + m

4ប关2 sign共x + y兲 − sign共x + y + L兲 − sign共x + y − L兲兴sign共x − y兲␨+O共␨2兲. 共49兲 Note that the contributions of order ␨ to P vanish, if both x and y take values outside 关−L/2,L/2兴.

Next, we compute the localized states␰共x兲of the system. The corresponding state vectors are defined by共7兲. Using this equation as well as 共38兲, 共42兲, 共43兲, 共45兲, and 共16兲 we have the following expression for the x-representation of a localized state␰共y兲centered at y苸R:

具x兩␰共y兲_{典 =}_␦_{共x − y兲 −}im␨

8ប2共2L + 2兩x + y兩− 兩x + y + L兩− 兩x + y − 1兩兲sign共x − y兲 + O共␨

2_兲. _共50兲

Because the linear term in␨is imaginary, the presence of a weak non-Hermiticity only modifies the usual共Hermitian兲 localized states by making them complex 共nonreal兲 while keeping their real part intact. Note however that for a fixed y the imaginary part of具x兩␰共y兲典 does not tend to zero as 兩x−y兩→⬁. This observation which seems to be in conflict with the usual notion of localizability has a simple explanation. Because the usual x operator is no longer an observable, it does not describe the position of the particle. This is done by the pseudo-Hermitian position operator X; it is the physical position wave function ⌿共x兲ª具␰共x兲,␺典₊ that defines the probability density of localization in space共8兲. The physical position wave function for the localized state␰共y兲is given by具␰共x兲,␰共y兲典+=具x兩y典=␦共x−y兲 which is the expected result.

IV. EQUIVALENT HERMITIAN HAMILTONIAN AND CLASSICAL LIMIT

The calculation of the equivalent Hermitian Hamiltonian h for the Hamiltonian共9兲 is similar to that of the physical observables. In view of共2兲, 共38兲, 共40兲, and 共42兲, and the last equation in 共16兲 which we express as H = p2+ i␯共x兲Z with␯共x兲 ª␪共x + 1兲 +␪共x − 1兲 − 2␪共x兲, 共51兲 we have h = p2+ h1Z + h2Z2+O共Z3兲, 共52兲 h₁ª i␯共x兲 +12关p 2_,Q 1兴, 共53兲 h2ª 1 8兵4关p 2_,Q 2兴 + 4i关␯共x兲,Q1兴 + 关关p2,Q1兴,Q1兴其. 共54兲 where hª␳H␳−1= mL2h/共2ប2兲共55兲

is the dimensionless Hermitian Hamiltonian associated with H. Next, we substitute共43兲 and 共45兲 in the identity 具x兩关p2_{, Q}

1兴兩v典=共⳵y2−⳵x2兲具x兩Q1兩y典, and perform

the necessary algebra. We then find具x兩关p2_{, Q}

1兴兩v典=−2i␯共x兲␦共x−y兲. Therefore,

关p2_,Q

1兴 = − 2i␯共x兲, 共56兲

and in view of共53兲

h1= 0. 共57兲

This was actually to be expected, for both the operators appearing on the right-hand side of共53兲 are anti-Hermitian, while its left-hand side is Hermitian. The fact that an explicit calculation of the right-hand side of共53兲 yields the desired result, namely 共57兲, is an important check on the validity

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of our calculation of␩+₁. It may also be viewed as an indication of the consistency and general

applicability of our method, that was initially formulated for systems with a discrete spectrum.5,16 According to共57兲,

h = p2_{+ h}

2Z2+O共Z3兲. 共58兲

Hence, in order to obtain a better understanding of the nature of the system described by the Hamiltonian H, we need to calculate h2. As we will next show, the knowledge of具x兩␩+₁兩y典 turns

out to be sufficient for the calculation of h2. To see this we first employ共56兲 to express h2in the

form h2= 1 4共2关p 2_,Q 2兴 + i关␯共x兲,Q1兴兲. 共59兲

Now, we recall that p2_{, Q}

2, ␯共x兲, and Q1 are all Hermitian operators. Therefore 关p2, Q2兴 and i关␯共x兲,Q1兴 are, respectively, anti-Hermitian and Hermitian. In view of 共59兲 and the Hermiticity of

h₂, this implies that

关p2_,Q 2兴 = 0. 共60兲 Hence, h2= i 4关␯共x兲,Q1兴 = i 4关␩+1,␯共x兲兴, 共61兲

where we have also made use of the first equation in 共43兲. We should also mention that the identities 共56兲 and 共60兲 can be directly obtained from the pseudo-Hermiticity condition 共5兲 by substituting共39兲 in 共5兲 and using 共40兲 and 共42兲.

We can easily use共45兲 and 共61兲 to yield the expression for the integral kernel of h2, namely

具x兩h2兩y典 = 1

32共4 + 2兩x + y兩− 兩x + y + 2兩− 兩x + y − 2兩兲sign共x − y兲关␯共x兲 −␯共y兲兴, ∀ x,y 苸 R.

共62兲 As seen from this equation,具x兩h2兩y典=0, if x苸关−1,1兴 and y苸关−1,1兴.

We can express h2as a function of x and p by performing a Fourier transformation on the y

variable appearing in共62兲, i.e., computing

具x兩h2兩p典 ª 共2␲兲−1/2

冕

−⬁

⬁

具x兩h2兩y典eipydy. 共63兲

This yields h2as a function of x and p, if we order the factors by placing x’s to the left of p’s. We

can easily do this by expanding具x兩h2兩p典 in powers of p. Denoting the x-dependent coefficients by

␻n, we then have

h2=

兺

n=0 ⬁

␻n共x兲pn, 共64兲

where we have made the implicit assumption that具x兩h2兩p典 is a real-analytic function of p.

The Fourier transform of具x兩h2兩y典 can be performed explicitly. 关One way of doing this is to use

the integral representations of the absolute value and sign function, as given in共36兲, to perform the y-integrations in共63兲 and use the identities

冕

−⬁ ⬁ _eiau du u共u − k兲= i␲ k共e iak_{− 1兲sign共a兲,}

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冕

−⬁ ⬁ _eiau du u共u − k兲2= i␲ k2关1 + 共iak − 1兲e

iak_{兴sign共a兲,} _{∀ a,k 苸 R,}

to evaluate the remaining two integrals. The resulting expression is too lengthy and complicated to be presented here.兴 We have instead used MATHEMATICA to calculate 具x兩h2兩p典 and found the

coefficients␻nfor n艋5. It turns out that indeed 具x兩h2兩p典 does not have a singularity at p=0, and

that␻0,␻2,␻4are real and vanish outside共−3,3兲 while␻1,␻3,␻5are imaginary and proportional

to␪共x兲−1/2 outside 共−3,3兲. As we will explain momentarily these properties are necessary to ensure the Hermiticity of h.

Figures 1, 2, and 3 show the plots of real part of␻nfor n = 0 , 2 , 4 and the imaginary part of␻n FIG. 1. Graph of the real part of␻0共dashed curve兲 and␻2共full curve兲.

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for n = 1 , 3 , 5. As seen from these figures 共the absolute value of兲 ␻n sharply decreases with n,

which suggests that a truncation of共64兲 yields a good approximation for the action of h2on the

wave functions with bounded and sufficiently small x-derivatives.

If we use具p兩h2兩x典=具x兩h2兩p典*to determine the form of h2and suppose that␻2n共x兲 are real and

␻2n+1共x兲 are imaginary for all n=0,1,2,3,..., we find

h2=

兺

n=0 ⬁ pn␻n共x兲*=

兺

n=0 ⬁ 关p2n_␻ 2n共x兲 − p2n+1␻2n+1共x兲兴.

Adding both sides of this relation to those of共64兲 and diving by two, we obtain

h2=

1 2

兺

_n=0

⬁

兵an共x兲,p2n其, an共x兲 ª␻2n共x兲 + i␻2n+1

⬘

共x兲, 共65兲

where兵·,·其 stands for the anticommutator, a prime denotes a derivative, and we have made use of the identity: 关f共x兲,pm_兴=兵if

_⬘

_共x兲,pm−1_{其. It is important to note that because} _␻

2n共x兲 are real and

␻2n+1共x兲 are imaginary, an共x兲 are real. Moreover, outside 共−3,3兲,␻2n共x兲, ␻2n+1

⬘

共x兲, and

conse-quently anvanish. Therefore, we can express h2in the manifestly Hermitian form共65兲 with all the

x-dependent coefficient functions vanishing outside 共−3,3兲. Figure 4 shows the plots of an for n = 0 , 1 , 2. They are all even functions of x with an amplitude of variations that decreases rapidly

as n increases.

Next, we scale back the relevant quantities and use共16兲, 共55兲, 共58兲, and 共65兲 to obtain

h = p 2 2m+ ␨2 2

兺

_n=0 ⬁ 兵␣n共x兲,p2n其 + O共␨3兲, ␣n共x兲 ª 2m

冉

L 2ប

冊

FIG. 5. Graph of the effective mass m_eff共full curve兲 for m=1₂,ប=1, L=2, and␨=1₃. The dashed curve represents m =1₂.

FIG. 6. Phase space trajectories of the Hamiltonian H˜c共xc, pc兲 for m=

1

2,ប=1, L=2, and␨= 1

3. The horizontal and vertical

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generated. These describe a particle that is trapped inside the physical interaction region. This is consistent with the fact that for small pc, H˜c is dominated by the potential term w共xc兲 which in

view of its relation to a₀共x兲 and Fig. 4 can trap the particle.

We wish to emphasize that because we have not yet taken theប→0 limit of H˜c, we cannot

identify it with the the true classical Hamiltonian Hcfor the quantum Hamiltonian h and

conse-quently H. Given the limitations of our perturbative calculation of H˜c, we are unable to determine

this limit. 共This is in contrast with both the PT-symmetric square well and the PT-symmetric cubic anharmonic oscillator studied in Refs. 5 and 16, respectively. In the former system the presence of an exceptional spectral point imposes the condition that ␨ must be of order ប2 or higher and consequently the classical system is the same as that of the Hermitian infinite square well.5 In the latter system, theប→0 limit of the associated Hermitian Hamiltonian can be easily evaluated and classical Hamiltonian obtained.16兲 Therefore, we cannot view the presence of closed phase space trajectories for H˜cas evidence for the existence of bound states of h and H. This is

especially because these trajectories are associated with very low momentum values where the quantum effects are expected to be dominant.

V. CONCLUSION

In this paper we explored for the first time the utility of the methods of pseudo-Hermitian quantum mechanics in dealing with a non-Hermitian PT-symmetric potential v共x兲 that has a continuous spectrum. We were able to solve the eigenvalue problem for this potential exactly and obtain the explicit form of the metric operator, the pseudo-Hermitian position and momentum operators, the localized states, and the equivalent Hermitian Hamiltonian perturbatively.

Our analysis revealed the surprising fact that the physical interaction region for this model is three times larger than the support of the potential, i.e., there is a region of the configuration space in whichv共x兲 vanishes but the interaction does not seize.

A simple interpretation for this peculiar property is that the argument x of the potentialv共x兲 is

not a physical observable and the support共−L/2,L/2兲 of v共x兲 being a range of eigenvalues of x does not have a direct physical meaning. This observation underlines the importance of the Hermitian representation of non-Hermitian 共inparticular PT-symmetric兲 Hamiltonians having a real spectrum.

The Hermitian representation involves a nonlocal Hamiltonian that is not suitable for the computation of the energy spectrum or the S-matrix of the theory. Yet it provides invaluable insight in the physical meaning and potential applications of pseudo-Hermitian andPT-symmetric Hamiltonians and is indispensable for the determination of the other observables of the corre-sponding quantum systems.

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