Acta Polytechnica doi:10.14311/AP.2017.57.0412 Acta Polytechnica 57(6):412–417, 2017 © Czech Technical University in Prague, 2017 available online at http://ojs.cvut.cz/ojs/index.php/ap ON THE COMMON LIMIT OF THE PT -SYMMETRIC ROSEN–MORSE II AND FINITE SQUARE WELL POTENTIALS József Kovács, Géza Lévai∗ Institute for Nuclear Research, Hungarian Academy of Sciences (MTA Atomki), Debrecen, Pf. 51, Hungary 4001 ∗ corresponding author: levai@atomki.mta.hu Abstract. Two PT -symmetric potentials are compared, which possess asymptotically finite imaginary components: the PT -symmetric Rosen–Morse II and the finite PT -symmetric square well potentials. Despite their different mathematical structure, their shape is rather similar, and this fact leads to similarities in their physical characteristics. Their bound-state energy spectrum was found to be purely real, an this finding was attributed to their asymptotically non-vanishing imaginary potential components. Here the V (x) = γδ(x) + i2Λ sgn(x) potential is discussed, which can be obtained as the common limit of the two other potentials. The energy spectrum, the bound-state wave functions and the transmission and reflection coefficients are studied in the respective limits, and the results are compared. Keywords: PT -symmetric potential; bound states; scattering; Dirac-δ limit. 1. Introduction The introduction of PT -symmetric quantum mechan- ics [1] gave strong impetus to the investigation of non-hermitian quantum mechanical systems (for a review, see [2]). In most cases these systems represent one-dimensional complex potentials that are invariant with respect simultaneous space (P) and time (T ) reflection, where the latter corresponds to complex conjugation. Although these potentials are manifestly non-hermitian, they possess several features that are characteristic of hermitian systems, i.e. real potentials. Perhaps the most spectacular one among these is that their discrete energy spectrum is partly or completely real. This feature was first attributed to PT symme- try, but later it soon turned out that PT symmetry is neither a necessary, nor a sufficient condition for the presence of real energy eigenvalues. It was found that in many such systems the energy eigenvalues merge pairwise with increasing non-hermiticity, and reappear as complex conjugate pairs. Since at the same time the energy eigenstates cease to be eigen- functions of the PT operator, this phenomenon was interpreted as the breakdown of PT symmetry. It was shown that from the mathematical point of view PT symmetry is a particular case of pseudo-hermiticity [3]. More recently, after a decade of theoretical inves- tigations the existence of PT symmetry, as well as that of its breakdown was verified in quantum optical experiments [4]. Although the first PT -symmetric potentials were solved by numerical methods, it was soon realized that most exactly solvable potentials can be cast into a PT -symmetric form, and the usual techniques applied to their hermitian version can be used in the PT - symmetric setting too (see [5–7] for reviews). The PT -symmetrization of shape-invariant [5, 6, 8] and of the more general Natanzon-class potentials [7] that are solved in terms of the (confluent) hypergeometric function [9] revealed that the characteristic features of PT -symmetric potentials can conveniently be studied using the exact analytical solutions of these potentials. A particularly interesting issue was the study of the breakdown of PT symmetry: the transition through the critical point could be reached by fine tuning of some potential parameter, and the whole process could be kept under control. It was found that there are exactly solvable potentials that do not exhibit this feature [10–13], while some others do [14–18]. It was also noticed that in most cases the complexification of the energy eigenvalues occurs at the same value of the control parameter (sudden mechanism) [15–17], while in some cases it is a continuous process [18]. Although there are examples for this latter, gradual mechanism among Natanzon-class potentials [18], it seems to be characteristic of potentials not belonging to the Natanzon (and thus, the shape-invariant) class. Ex- amples are the numerically solvable Bender-Boettcher potentials [1], some piecewise constant potentials, like the PT -symmetric infinite square well [19], and the PT -symmetric exponential potential [20]. The asymptotic behaviour of the imaginary poten- tial component was found to play an important role in determining the characteristics of the energy spectrum. The PT -symmetric Scarf II and Rosen–Morse II po- tentials share the same real component, cosh−2(x), while their imaginary components are different. In the case of the Scarf II potential the imaginary po- tential component vanishes asymptotically, while in the case of the Rosen–Morse II potential it is the i tanh(x) function, reaching finite values for x →±∞. 412 http://dx.doi.org/10.14311/AP.2017.57.0412 http://ojs.cvut.cz/ojs/index.php/ap vol. 57 no. 6/2017 Common Limit of Two PT -Symmetric Potentials In the former case the breakdown of PT symmetry can occur [15], while in the latter the discrete energy spectrum is purely real [10]. This latter finding was later proven for all three PII-class shape-invariant po- tentials (Rosen–Morse I, II, Eckart) using a thorough analysis of PT -symmetric Natanzon-class potentials [7]. This clear difference can obviously be attributed to the different asymptotic behaviour of the two po- tentials [21]. This peculiar character of the asymptotically con- stant imaginary potential component characterising the PT -symmetric Rosen–Morse II potential inspired the investigation of further potentials with similar structure. A natural candidate was the finite PT - symmetric square well potential [22], which is essen- tially the finite real square well potential supplemented outside the well by a constant imaginary component with opposite sign on the two sides. In a way, this po- tential can be considered an approximation of the PT - symmetric Rosen–Morse II potential: the cosh−2(x) and i tanh(x) terms are mimicked by the finite real square well and the constant imaginary terms, respec- tively. It was supposed [22] that given the similar shapes, the main physical features of the two poten- tials would also be close to each other. It was found that the energy spectrum of the finite PT -symmetric square well potential is purely real, similarly to that of the PT -symmetric Rosen–Morse II potential. An- other similarity was that by increasing non-hermiticity, i.e. the coupling coefficient of the imaginary potential component, the energy eigenvalues are rapidly lifted to the positive domain E > 0. There was, however, an important difference: while the number of bound states was fixed for the Rosen–Morse II potential, it was infinite for the finite PT -symmetric square well potential. The additional states were found to be the equivalents of transmission resonances of the real finite square well potential [22]. These results naturally inspire further investiga- tion of PT -symmetric potentials with asymptotically constant imaginary component. Here a potential of this kind is investigated, which, furthermore, can be obtained as the common limit of the PT -symmetric versions of the Rosen–Morse II and finite square well potentials: V (x) = γδ(x) + i2Λ sgn x. (1) The purpose of this work is to explore how the physical quantities of the PT -symmetric Rosen–Morse II and finite square well potentials behave when the limit as above is implemented. The paper is organized as follows. Sections 2 and 3 discuss the specific limits of the PT -symmetric Rosen– Morse II and finite square well potentials, respectively. In Section 4 the results are summarized and are com- pared with those obtained for other potentials with various asymptotic behaviour. 2. The PT -symmetric Rosen–Morse II potential Let us consider the potential V (x) = − s(s + 1)a2 cosh2(ax) + 2iλa2 tanh(ax). (2) Noting that the s →−s− 1 replacement leaves V (x) invariant, we may chose s ≥ −1/2. Following the discussion of [10] with the difference that x is rescaled by the positive real constant a as ax, the bound-state eigenvalues are En = −a2(s−n)2 + λ2a2 (s−n)2 , (3) while the corresponding wave functions are expressed in terms of Jacobi polynomials [23] ψn(x) = Cn(1 − tanh(ax)) α 2 × (1 + tanh(ax)) β 2 P (α,β)n (tanh(ax)). (4) Here Cn is the normalization constant Cn = in2n−s |Γ(s + 1 + iλ/(s−n))| × (an!Γ(2s−n + 1)((s−n)2 + λ2/(s−n)2) s−n )1/2 (5) while αn = s−n + iλ s−n , βn = s−n− iλ s−n . (6) It was shown in [10] that the number of bound states is always finite, and the upper limit does not depend on the parameter λ: n < s. (7) It may be noted that for −1 ≤ s ≤ 0 the real compo- nent of (2) turns into a barrier with Vmax(x) ≤ a2/4, and there are no bound states in this case. Let us reparametrize the potential in the following way: γ = −2as(s + 1), Λ = a2λ. (8) Considering then the following limits: δ(x) = lim a→∞ a 2 cosh2(ax) (9) and sgn(x) = lim a→∞ tanh(ax). (10) the potential in (2) can be transformed into V (x) = γδ(x) + 2iΛ sgn(x). (11) Let us now clarify the effect of this limit on the energy eigenvalues (3) and wave functions (4). From (8) it follows that s = − 1 2 ( 1 − (1 − 2γ/a)1/2 ) , (12) 413 József Kovács, Géza Lévai Acta Polytechnica where the “−” sign inside the square brackets follows from the requirement s > 0. s can be expressed in a series involving the powers of 1/a as s = − γ 2a − γ2 4a2 + O(a−3). (13) Note that for s > 0 (8) implies that γ < 0. Recalling (7) this result also leads to the finding that only the ground state remains normalizable. Substituting s from (13) and Λ from (8) one finds that in the a →∞ limit the ground-state eigenvalue becomes E0 = − γ2 4 + 4Λ2 γ2 . (14) The corresponding wave function can be calculated after substituting s from (13) and Λ from (8) and n = 0 into (4) and (5), and taking the a →∞ limit: ψ0(x) = { C0 exp(−κ+x), x > 0, C0 exp(−κ−x), x < 0, (15) where κ± = −ik± = ∓ γ 2 − i 2Λ γ , (16) and C0 = ( − 2 γ (γ2 4 + 4Λ2 γ2 ))1/2 . (17) It can be seen that (15) satisfies PT symmetry, i.e. PT ψ0(x) = ψ∗0 (−x) = ψ0(x). It is also found that E0 = −κ2± ± 2iΛ, (18) as expected. It is seen that the i sgn(x) potential alone cannot support any bound state, rather the Dirac delta is also required for it. This is similar to the case of the PT -symmetric Rosen–Morse II potential [10]: there the imaginary 2iΛ tanh(ax) potential cannot support bound states without the presence of the real −s(s + 1)a2/ cosh2(ax) potential component. It is worthwhile to check some special cases against already known results. For Λ = 1/2 the potential in (11) reduces to the one considered in [24], and (14) confirms the result E0 = −γ2/4+1/γ2 discussed there. Furthermore, the Λ = 0 choice recovers the simple Dirac delta potential [25]. Finally, the hermitian case with a real step function can also be considered after replacing Λ → ±iΛ. In this case κ± in (16) become real, and a bound state can appear only for γ 2 + 2|Λ| γ < 0, (19) i.e. for a negative γ satisfying γ < −2(|Λ|)1/2. The single real energy eigenvalue can also be ob- tained from the transmission coefficient. Adapting the corresponding formulas from [10] one obtains for an incoming wave from the left TL→R = −ik− a −s− ik−2a − i k+ 2a × Γ(1 − ik−2a − i k+ 2a −s)Γ(1 − i k− 2a − i k+ 2a + s) Γ(1 − ik+ a )Γ(1 − ik− a ) (20) and RL→R = TL→R −s + ik−2a − i k+ 2a ik− a × Γ(1 − ik+ a )Γ(1 + ik− a ) Γ(1 + ik−2a − i k+ 2a −s)Γ(1 + i k− 2a − i k+ 2a + s) , (21) where k2± = E ∓ 2iΛ (22) are the squared wave numbers obtained form the asymptotic limits x →∞ and x →−∞, respectively. For the sake of consistency the original notation k and k ′ was replaced by k− and k+, and an error in the sign of k ′ was corrected in [10, eqs. (38)–(41)]. Recalling (13) and taking the a → ∞ limit the terms with the gamma functions reduce to unity, so (20) and (21) turn into TL→R(k−,k+) = 2ik− ik− + ik+ −γ (23) and RL→R(k−,k+) = ik− − ik+ + γ ik− + ik+ −γ . (24) The transmission and reflection coefficients for the reverse direction are obtained by the k− ↔ k+ replacement. It is found that TR→L(k−,k+) = TL→R(k−,k+)k+/k−, so the difference is represented by a phase factor (as can be seen from (22)), while the reflection coefficients are related by RR→L(k−,k+) = RL→R(k−,k+)(k+ − k− − iγ)/(k− − k+ − iγ). This handedness effect is similar to that observed for the PT -symmetric Rosen–Morse II potential [10]. Note that in the case of asymptotically vanishing PT - symmetric potentials the two transmission coefficients are strictly identical [26, 27]. The connection of the transmission and reflection coefficients to the bound state (15) will be discussed in Section 3, together with the corresponding results obtained there. 3. The finite PT -symmetric square well potential Let us consider the potential V (x) =   −iv, x < −�, −V0, |x| < �, iv, x > �, (25) 414 vol. 57 no. 6/2017 Common Limit of Two PT -Symmetric Potentials where � and V0 take on positive real values, and v takes on real value. For the case v = 0 potential (25) reduces to the real square well potential [28]. The sign of v is not significant, because changing the sign of v is practically equivalent with a spatial reflection, i.e. with the P operation. Following the discussion of [22] and using 2m = ~ = 1 units the solution of the time-independent Schrödinger equation can be described as ψ(x) =   F−(p−)eip−x + F−(−p−)e−ip−x, x < −�, α cos(kx) + iβ sin(kx), |x| < �, F +(p+)eip+x + F +(−p+)e−ip+x, x > �, (26) where p± = (E ∓ iv)1/2, k = (E + V0)1/2. (27) E denotes the energy eigenvalue and the complex square root function is understood as in [23]. Note that in this case [p±]∗ = p∓ holds. The coefficients F−(p−) and F +(p+) can be determined from maching the solutions at x = ±� as F−(p−) = 1 2ip− eip−� ( (αp− + βk)i cos(k�) +(αk + βp−) sin(k�) ) , (28) F +(p+) = 1 2ip+ e−ip+� ( (αp+ + βk)i cos(k�) −(αk + βp+) sin(k�) ) , (29) while F−(−p−) and F +(−p+) follow from (28) and (29) by changing the sign of p− and p+. Considering the case v < 0, following [22] the energy eigenvalues correspond to solutions that vanish asymptotically in both directions are searched as roots of equation 2kp+I cos(2k�) + (p2+R + p 2 +I −k 2) sin(2k�) = 0 (30) on the real axis, where p+R and p+I are definied as the real and imaginary part of p+, i. e. p+ = p+R + ip+I. Note that (30) establishes a connection between p+R and p+I when E is real. Taking into consideration (27) and separating the real and imaginary components of it, it turns out that the latter occurs only in the second term in the form −i(v + 2p+Rp+I) sin(2k�). Since this expression has to be zero in general (irrespective of �), it follows that p+R = − v 2p+I . (31) To reproduce (11) let us reparametrize the potential (25) by introducing γ = −2�V0, Λ = v 2 . (32) Then keeping γ fixed and considering the � → 0 limit the potential in (25) can be transformed into V (x) = γδ(x) + 2iΛ sgn(x) (33) as well. In this limit, after applying the l’Hospital rule, equa- tion (30) transform into 2p+I + γ = 0. Together with (31) and (32) this means that there is a single bound state with p± = 2Λ γ ∓ i γ 2 , (34) with the energy eigenvalue E0 = − γ2 4 + 4Λ2 γ2 , (35) which coincides with (14). The transmission and reflection coefficients can be obtained form the corresponding limit of those in [22]. These coefficient for an incoming wave from the left are given by TL→R = 2ip−ke−ip−�e−ip+� ik cos(2k�)(p++p−) + sin(2k�)(p+p−+k2) (36) and RL→R = e−2ip−� × ik(p− −p+) cos(2k�) + (p+p− −k2) sin(2k�) ik(p+ + p−) cos(2k�) + (p+p− + k2) sin(2k�) , (37) respectively. In the � → 0 limit they turn into TL→R = 2ip− ip− + ip+ −γ (38) and RL→R = ip− − ip+ + γ ip− + ip+ −γ , (39) respectively. The equivalent coefficients for a wave incoming from the right are obtained by the p+ ↔ p− change, similarly to the results of Section 2. Note that PT symmetry, and in particular, the asymptotically non-vanishing potential component has strong influence on the asymptotic properties of the wave functions, and this fact manifests itself in the structure of the transmission and reflection coef- ficients too, in accordance with the findings of [22]. It turns out that for real E (as is the case here) the asymptotically vanishing (i.e. bound) states can be identified with the zeros of the reflection coefficents, rather than with the poles of the transmission (and reflection) coefficients. The reason is that in these solutions exp(±ip±x) occur with the same sign in the exponent for both x > 0 and x < 0, corresponding to a transmitting wave. In particular, for the potential (33) the bound state occurs for 2p+I + γ = 0, which is the zero of (39), while for the reverse direction, the zero of RR→L occurs at 2p−I + γ = 0 = −2p+I + γ corresponding to the interchange of p− and p+ or spatial reflection. The same results are obtained from the discussion of Section 2 too. 415 József Kovács, Géza Lévai Acta Polytechnica 4. Conclusions We investigated the PT -symmetric Rosen–Morse II and finite square well potentials in the limit when their real even potential component turns into the Dirac delta, while their imaginary odd component tend to the sign function, respectively. The energy spectrum was found to contain a single real eigenvalue for γ < 0 and arbitrary Λ, depending on both parameters. The transmission and reflection coefficients were also determined, and it was found that they exhibit the expected handedness effect. The results of [24] were recovered for the bound-state en- ergy after setting Λ = 1/2, while for Λ = 0 the Dirac delta potential was obtained. The results were also derived for the hermitian version of this potential with an imaginary Λ. The transmission and reflection coefficients were also considered in the appropriate limit for the two potentials. It was confirmed that the handedness ef- fect occurs in this case too, i.e. in contrast with real potentials, the reflection coefficients differ essentially for waves arriving from the two directions, while the transmission coefficients differ only in a phase. (Note that for potentials with asymptotically vanishing imag- inary component even this phase is missing.) It was shown that the only bound state that occurs in the limiting case from both potentials is obtained as the zero of the reflection coefficient, rather than as the pole of the transmission coefficient, in accordance with the findings of [22]. The present study confirms the importance of the asymptotically non-vanishing imaginary potential component, which was already pointed out in connec- tion with the PT -symmetric Rosen–Morse II potential [10, 21]. It is notable that supplementing the same real even asymptotically vanishing potential compo- nent with an asymptotically vanishing odd imaginary potential component (Scarf II [15, 29]) leads to com- plex conjugate energy eigenvalues when the relative intensity of the imaginary component is increased, but this phenomenon does not occur when the imaginary potential component is chosen asymptotically non- vanishing (Rosen–Morse II). In this case increasing the intensity of the imaginary component leads to lift- ing the energy spectrum to higher energies, such that even the ground-state energy can be tuned to positive values. This was also found for the PT -symmetric finite square well [22] potential. It is instructive to consider further PT -symmetric potentials with various asymptotic behaviour. The energy spectrum of the Bender–Boettcher potentials V (x) = x2(ix)ε [1] contains complex conjugate energy eigenvalues for ε < 2. In this case complexification is a gradual process, starting from higher energies. For ε ≥ 0 the energy eigenvalues are all real, similarly to the case of the PT -symmetric Rosen–Morse II potential. The ε = 1 choice recovers the purely imaginary ix3 potential. Another interesting case is the PT -symmetric ex- ponential potential [20]. Its solutions are expressed in terms of Bessel functions, so it is also outside the Natanzon class. This two-parameter potential is purely imaginary, and it tends to infinity asymp- totically stronger than the imaginary component of the Bender–Boettcher potential. It has the unusual feature that its energy spectrum generally contains both real and complex energy eigenvalues such that it is not possible to separate parametric domains where only imaginary energy eigenvalues occur. Increasing non-hermiticity leads to the gradual complexification of the energy spectrum from above, however, the ground-state energy always remains real. All these findings indicate that the breakdown of PT symmetry occurs in potentials with rather dif- ferent patterns of the imaginary component. For asymptotically strongly divergent imaginary poten- tial components the complexification of the energy eigenvalues generally occurs gradually, starting from above. For potentials with asymptotically vanishing imaginary component the same procedure occurs from below either suddenly [15–17] or gradually [18], but in these cases a non-vanishing real potential component is also necessary to obtain bound states. 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Gen. 35:8793, 2002. doi:10.1088/0305-4470/35/41/311 417 http://dx.doi.org/10.1088/1751-8113/42/19/195302 http://dx.doi.org/10.1016/j.physleta.2004.01.009 http://dx.doi.org/10.1088/0305-4470/39/32/S17 http://dx.doi.org/10.1016/j.physleta.2008.08.073 http://dx.doi.org/10.1016/S0375-9601(99)00805-1 http://dx.doi.org/10.1016/S0375-9601(01)00218-3 http://dx.doi.org/10.1088/0305-4470/36/27/313 http://dx.doi.org/10.1007/s12043-009-0125-5 http://dx.doi.org/10.1088/1751-8113/45/44/444020 http://dx.doi.org/10.1142/S0217732301005722 http://dx.doi.org/10.1016/j.physleta.2015.04.032 http://dx.doi.org/10.1007/s10773-010-0595-8 http://dx.doi.org/10.1016/j.physleta.2004.03.002 http://dx.doi.org/10.1016/j.aop.2006.05.011 http://dx.doi.org/10.1088/0305-4470/35/41/311 Acta Polytechnica 57(6):412–417, 2017 1 Introduction 2 The PT-symmetric Rosen–Morse II potential 3 The finite PT-symmetric square well potential 4 Conclusions Acknowledgements References