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The Coulomb – harmonic oscillator correspondence in PT symmetric quantum mechanics
10 July 2000
Physics Letters A 271 Ž2000. 327–333 www.elsevier.nlrlocaterpla
The Coulomb – harmonic oscillator correspondence in PT symmetric quantum mechanics Miloslav Znojil a,) , Geza ´ Levai ´ b a
b
ˇ ˇ Czech Republic Nuclear Physics Institute of Academy of Sciences of the Czech Republic, 250 68 Rez, Institute of Nuclear Research of the Hungarian Academy of Sciences, PO Box 51, H-4001 Debrecen, Hungary Received 20 March 2000; received in revised form 30 May 2000; accepted 5 June 2000 Communicated by P.R. Holland
Abstract We show that and how the Coulomb potential V Ž x . s Z e 2rx can be regularized and solved exactly at the imaginary coupling Z e 2. The new spectrum of energies is real and bounded as expected, but its explicit form proves totally different from the usual real-coupling case. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Quantum mechanics often works with exactly solvable simplified models. For precise fits of data or for some more subtle quantitative analyses, unfortunately, the number of solvable models is too limited. In D dimensions, the only really useful and easily tractable interactions are harmonic oscillators andror the central Coulomb well V Ž Z . Ž r . s yZ e 2r< r <. A new way out of this deadlock emerges within the framework of the alternative, so called ‘ PT symmetric’ quantum mechanics. With its complex Hamiltonians H breaking both parity P and timereflection symmetry T and commuting only with their product PT , this formalism was proposed by Bessis w1x and by Bender et al. w2–4x as a possible way towards weakening the standard requirements of Hermiticity. )
Corresponding author. E-mail addresses: [email protected] ŽM. Znojil., [email protected] ŽG. Levai ´ ..
Several new exactly solvable PT symmetric models have been proposed recently w5–7x. This is a promising development, with possible applications ranging from field theories w8x to supersymmetric models w9x and from quasi-classical methods w3x to perturbation theory w10–13x. Even the solvable harmonic oscillator itself acquires a richer spectrum after its consequent PT symmetric regularization in D dimensions w14x. The detailed structure of spectrum of this prominent example does not in fact offer any really serious surprise. A manifest violation of the parity P is compensated by the emergence of the so called quasiparity q s "1. It distinguishes between the two equidistant subspectra. The new quantum number q degenerates back to the eigenvalue of parity after a return to the standard Hermitean and one-dimensional oscillator. No immediate surprise emerges either for the quartic anharmonic oscillator w15–17x. The situation only becomes less clear after one moves towards asymptotically vanishing models. They exhibit sev-
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 4 0 0 - X
M. Znojil, G. LeÕair Physics Letters A 271 (2000) 327–333 ´
328
eral counterintuitive properties and open new mathematical challenges w18x. In particular, the popular Coulomb potential did not even seem particularly suitable for any immediate PT symmetric regularization w19x. A psychological barrier has been created by numerical and semiclassical studies of the general power-law forces V Ž x . ; yŽ ix . d. They may be well defined everywhere near the harmonic exponents d s 2, d s 6 etc. w20,21x. At the same time, related analyses hinted that it is apparently difficult to move beyond the Herbst singularity located at d s 1 w22x. In what follows, we intend to employ a slightly different strategy and try to study the Coulomb problem directly, via its well known correspondence with the harmonic oscillator. This correspondence is based on an elementary change of variables. Its background dates back to nineteenth century mathematics and, in particular, to the work of Liouville w23–25x. Newton’s monograph w26x cites Fivel w27x as a newer source of the idea. In the contemporary literature Žcf., e.g., w28x for further references. people usually speak about the Kustaanheimo–Steifel ŽKS. transformation w29x. In all the implementations of this idea the parameters appearing in the Coulombic and oscillator problems are interrelated, of course. Details will be mentioned below. Preliminarily, let us only warn the reader that all the KS-type mappings can change the dimensions and angular momenta and that the energies of one problem are related to the coupling constants of the other one and vice versa. Within ‘normal’ quantum mechanics, all this has already been thoroughly discussed elsewhere: in Ref. w30x for D s 3 and in Ref. w31x for the continuous transformation between Coulomb problems and harmonic oscillators in various dimensions.
of Ref. w14x, using the complex coordinate r s x y i c with real x g Žy`,`. and with, say, positive c ) 0. This means that the integration path has been shifted down from the position where it would cross the strong centrifugal singularity. Such a regularization preserves the asymptotic decrease of the normalizable solutions. Only in the limit c 0 and beyond the trivial one-dimensional case does one have to omit all the so called irregular solutions w32x defined by their ‘physically unnacceptable’ x Ž r . ; ryl behaviour near the origin. Within the framework of the general Liouville method the change of variables mediates a transition to a different potential V Ž t .. It is easy to show that once we forget about boundary conditions one merely has to demand the existence of an invertible function r s r Ž t . and a few of its derivatives r X Ž t ., r XX Ž t ., . . . . Then, the explicit correspondence between the two bound state problems may be explicitly given by elementary formulae. From our original Eq. Ž1. Ži.e., in our case, the harmonic oscillator. one obtains the new Ži.e., in our case, Coulombic. Schrodinger equa¨ tion
™
d2 y dt
2
q
L Ž L q 1.
C Ž t . s x r Ž t . r rX Ž t .
(
y dr
2
q
l Ž l q 1. r
2
Ž 3.
and with the new interaction and the new energies w25x, L Ž L q 1. t2
2. Liouvillean changes of variables
d2
Ž 2.
with the new wave functions
qV Ž t. yE
s rX Ž t .
The change-of-variable approach to the Coulombic bound-state problem enables us to start directly from the harmonic oscillator potential W Ž r . s r 2 or, in the present less traditional context, from the PT symmetric radial Schrodinger equation ¨
q V Ž t . C Ž t . s EC Ž t .
t2
q
2
½
l Ž l q 1. r2Ž t.
3 r XX Ž t . 4
rX Ž t .
2
y
qW r Ž t. y´ 2
1 r XXX Ž t . 2
rX Ž t .
5
.
Thus, it only remains for us to re-analyse the boundary conditions.
3. PT symmetric KS transformation
qW Ž r . x Ž r . s´ 2 x Ž r .
Ž 1.
Without any serious formal difficulties let us extend the scope of the present considerations to all the
M. Znojil, G. LeÕair Physics Letters A 271 (2000) 327–333 ´
329
Fig. 1. PT symmetric integration paths.
singular forces Wˆ Ž r . s W Ž r . q frr 2 andror VˆŽ t . s V Ž t . q Frt 2 . Both these central forces may act in the respective d and D dimensions. This means that l Ž l q 1 . s j q Ž d y 3 . r2
j q Ž d y 1 . r2 q f 4
or rather 2 2 Ž l q 1r2. Ž s a 2 . s j q Ž d y 2 . r2 q f
Žwith Re a ) 0. and 2 2 Ž L q 1r2. Ž s A2 . s J q Ž D y 2 . r2 q F
Žwith Re A ) 0. where the partial waves are numbered by the respective integers j s 0,1, . . . and J s 0,1, . . . . An important simplification of our effort is provided by our knowledge of the complete harmonic oscillator solution as derived in Ref. w14x. Its two equidistant subsets of energies
´ 2 s ´Ž2n , q . s 4 n q 2 y 2 q a ,
q s "1,
n s 0,1, . . . correspond to the two families of the Laguerre-polynomial wave functions
xŽ n , q . Ž r . s N r 1r2yq a eyr
2
r2
a. LŽyq Ž r2. . n
The integration path r s x y i c is a straight line. It lies in the lower half of the complex plane and does not change its form and position after the subsequent P transformation r s r Ž x . yr and T transformation yr yr ) s r Žyx .. In the spirit of the above-mentioned KS mapping of harmonic oscillators on Coulombic bound states we now have to define a complex variable t as a re-scaled square of r Ž x . such that the resulting path t Ž x . remains PT invariant. Our requirement implies that the complex plane of r will twice cover the complex plane of t. In such an arrangement, our lower half plane of r should cover the Riemann sheet given as a whole plane of t which is cut upwards from the origin. In the polar representation, one has r ; expŽyi w . mapped upon t ; expŽy2 i w . with w g Ž0, p .. Once we introduce a suitable free parameter k ) 0 we can put, say, r 2 s 2 k 2 z and then rotate the z-plane Žwhich is cut, by construction, along the real and positive semi-axis. by the angle pr2 giving t s i z. Our final recipe
™
r 2 s y2 i k 2 t
™
Ž 4.
maps the above-mentioned straight line r Ž x . s x y i c upon a curve t Ž x . s u q i Õ. Its real part u s uŽ x .
M. Znojil, G. LeÕair Physics Letters A 271 (2000) 327–333 ´
330
s x crk 2 and imaginary part Õ s Õ Ž x . s Ž x 2 y c 2 .r2 k 2 form an upwards-oriented parabola Õ s yc 2r2 k 2 q Ž k 2r2 c 2 . u 2 in the complex plane. The two curves ‘A’ in Fig. 1 illustrate the shape of both these PT symmetric trajectories in more detail. The harmonic-oscillator straight line r Ž x . s x y ic with c s 1 is denoted by the symbol AŽHO. there. Its Coulombic transform AŽCoul. is defined by formula Ž4. and displayed in Fig. 1 at k s 1. Many alternative choices of the original coordinates are admissible and remain compatible with the overall analyticity constraints w14x. Their sample BŽHO. in Fig. 1 is characterized by an asymptotic deformation of our original integration path r Ž x . given by an x-dependent shift c s c Ž x . ; 1r '2 q x 4 . This produces a different curve BŽCoul. Žwith k s 1 again.. We see that the Coulombic coordinates BŽCoul. may in fact lie quite close to the cut in the asymptotic domain of < x < 4 1. Having achieved a PT symmetry in the complex plane of t, we may move to the Žtrivial. insertions and conclude that all the above-mentioned harmonic oscillator bound-state solutions are in a one-to-one correspondence with the solutions of the Coulombic Schrodinger Eq. Ž2., ¨ d2 y
dt 2
q
L Ž L q 1. t2
t s uŽ x . q i Õ Ž x . ,
qi
Z e2 t
C Ž t . s EC Ž t . ,
x g R.
Ž 5.
The underlying assignment of constants is such that a s 2 A, while k itself becomes n-dependent, k 2 s 2 Z e 2r´ 2 s Z e 2rŽ2 n q 1 y 2 q A.. In full detail one gets the new, Laguerre-polynomial wave functions 2
q A. CŽ n , q . Ž t . s M t 1r2yq A e i k t LŽy2 Ž y2 i k 2 t . n
Ž 6. and their energy spectrum specified by the elementary formula EŽ n , q . s kŽ4n , q . s q s "1,
Z 2e4
Ž 2 n q 1 y 2 q A.
n s 0,1, . . . .
This is our main result.
2
4. Discussion 4.1. Consequences of the curÕature of our integration path The latter two formulae exhibit several unusual features. The first concerns the asymptotics of the wave functions which are determined by the decreasing exponential expŽ i k 2 t .. Its form re-confirms the correctness of the above, slightly counter-intuitive KS-dictated choice of our PT symmetric path of integration. Asymptotically, it encircles more or less closely the positive imaginary axis in the t plane. This clarifies the apparent paradox. The second unexpected result is the positivity and unusual n-dependence of the energies. This can be related to the choice of the KS integration path t Ž x . again. In the very vicinity of the origin, one can visualize this path as a circle with radius s , u2 Ž x . q Õ 2 Ž x . s s 2 ,
< x < < 1.
From the appropriate definitions we get the formula
s s c 2 Ž 0 . r2 kŽ2n , q . q O Ž x 2 . and see that this radius is n-dependent and increases with the growth of this principal quantum number, s ; n c 2 Ž0.rZ e 2 , n 4 1. As a consequence, the ‘effective charge’ of our P T symmetric Coulomb potential appears to decrease with n, since i Z e2 t
;
Z e2
s
q O Ž t . s O Ž 1rn . .
This offers a ‘rule-of-thumb’ guide to the unusual and certainly counterintuitive n-dependence of the energy levels Ž7.. Of course, in practice, the preferred integration path will be n-independent. In such a case, the n-dependence re-appears in the small-x deformation of the initial harmonic-oscillator path with c s c nŽ x . s O Ž1rn.. Such a flexible transfer of the excitation-dependence throws also a new light on the complexified KS transformation itself.
, 4.2. ‘Flown-away’ energies and unaÕoided crossings
Ž 7.
Let us return in more detail to the A-dependence of our energies Ž7.. Firstly, we notice their power-law
M. Znojil, G. LeÕair Physics Letters A 271 (2000) 327–333 ´
dependence on n and A Žwith exponent s -2. as somewhat similar to the spectra in the PT symmetric oscillator well w14x Žwith exponent s q1. and in the Morse potential of Ref. w33x Žwith exponent s q2.. In the present case, obviously, we have to distinguish between the two separate families EŽ n, q . with q s q1 Žcf. Fig. 2. and q s y1 Žcf. Fig. 3.. The latter set is, up to its sign, analogous to the ordinary Coulombic spectrum. By far not so the former one. Its energies enrich and dominate the spectrum. The n div-th energy ‘flies away’ and disappears from the spectrum at A div s n div q 1r2. Moreover, at all the positive integers and half-integers A one encounters unaÕoided level crossings. In contrast to the harmonic case, they appear at both the opposite and identical Žviz., positive. quasiparities q. The former
331
case takes place at A s A crit s Ž n y nX .r2 while in the latter case we must fulfill the condition A s A crit s Ž n q nX q 1.r2. A sample of this phenomenon is given in Fig. 4. Formally the unavoided crossings generate certain identities which connect different Laguerre polynomials Žcf. a sample in Ref. w14x.. In applications, these ‘critical’ cases are not exceptional at all. For F s 0 forces without a spike, the critical integer or half-integer coordinates A crit s J y 1 q Dr2 correspond precisely to physical Žnamely, integer. dimensions D and partial waves J. In conclusion, let us not forget about many open questions. Pars pro toto, one could mention a not yet clear possibility of re-interpretation of our bound states, say, in the limit t real, i.e., beyond the mathematical and apparently natural boundaries of
Fig. 2. Quasi-even levels Žn,q. Žunits Z s e s 1..
™
332
M. Znojil, G. LeÕair Physics Letters A 271 (2000) 327–333 ´
Fig. 3. Quasi-odd levels Ž n s 0–5, units Z s e s 1..
Fig. 4. Sample of unavoided crossings.
M. Znojil, G. LeÕair Physics Letters A 271 (2000) 327–333 ´
our present approach. Moreover, we must keep in mind that via our complexification of the coordinates we of course broke their immediate connection to any standard D-dimensional problem. In this sense, our F s 0 and F / 0 Hamiltonians differ just in an inessential way. One could even prefer the latter option as a model which is formally simpler, due to the generic absence of the puzzling unavoided crossings. At F / 0 the structure of the spectrum of our present Coulomb model becomes also richer and, in this sense, more interesting.
Acknowledgements M.Z. thanks for the hospitality of the Institute of Nuclear Research of the Hungarian Academy of Sciences, in Debrecen, and appreciates the support by the grant Nr. A 1048004 of GA AS CR. G.L. acknowledges the OTKA grant no. T031945 and the support of the Janos Bolyai Research Fellowship. ´
References w1x w2x w3x w4x
D. Bessis, private communication, 1992. C.M. Bender, S. Boettcher, Phys. Rev. Lett. 24 Ž1998. 5243. E. Delabaere, F. Pham, Phys. Lett. A 250 Ž1998. 25. C.M. Bender, S. Boettcher, P.N. Meisinger, J. Math. Phys. 40 Ž1999. 2201. w5x F. Cannata, G. Junker, J. Trost, Phys. Lett. A 246 Ž1998. 219. w6x B. Bagchi, R. Roychoudhury, J. Phys. A: Math. Gen. 33 Ž2000. L1. w7x M. Znojil, J. Phys. A: Math. Gen. 33 Ž2000. L61.
333
w8x C.M. Bender, K.A. Milton, Phys. Rev. D 55 Ž1997. R3255; Phys. Rev. D 57 Ž1998. 3595. w9x A.A. Andrianov, M.V. Ioffe, F. Cannata, J.P. Dedonder, Int. J. Mod. Phys. A 14 Ž1999. 2675. w10x E. Caliceti, S. Graffi, M. Maioli, Commun. Math. Phys. 75 Ž1980. 51. w11x F. Fernandez, R. Guardiola, J. Ros, M. Znojil, J. Phys. A: ´ Math. Gen. 31 Ž1998. 10105. w12x C.M. Bender, G.V. Dunne, J. Math. Phys. 40 Ž1999. 4616. w13x E. Caliceti, arXiv: math-phr9910001. w14x M. Znojil, Phys. Lett. A 259 Ž1999. 220. w15x V. Buslaev, V. Grecchi, J. Phys. A: Math. Gen. 26 Ž1993. 5541. w16x C.M. Bender, K.A. Milton, J. Phys. A: Math. Gen. 32 Ž1999. L87. w17x M. Znojil, J. Phys. A: Math. Gen. 32 Ž1999. 7419. w18x M. Znojil, arXiv: quant-phr9912027v2, to appear in J. Phys. A: Math. Gen. w19x M. Znojil, J. Phys. A: Math. Gen. 32 Ž1999. 4563. w20x C.M. Bender, S. Boettcher, H.F. Jones, V.M. Savage, J. Phys. A: Math. Gen. 32 Ž1999. 6771. w21x C.M. Bender, F. Cooper, P.N. Meisinger, V.M. Savage, Phys. Lett. A 259 Ž1999. 224. w22x I. Herbst, Commun. Math. Phys. 64 Ž1979. 279. w23x J. Liouville, J. Math. Pures. Appl. 1 Ž1837. 16. w24x G. Jaffe, ´ Z. Phys. 87 Ž1933. 535. w25x F.W.J. Olver, Introduction to Asymptotics and Special Functions, Academic, New York, 1974, Chp. VI. w26x R.G. Newton, Scattering Theory of Waves and Particles, Springer, Berlin, 1982, p. 442, problems 17 and 18. w27x D.I. Fivel, Phys. Rev. 142 Ž1966. 1219. w28x E.G. Kalnins, W. Miller Jr., G.S. Pogosyan, arXiv: quantphr9906055. w29x P. Kustaanheimo, E. Steifel, J. Reine Angew. Math. 218 Ž1965. 204. w30x M. Znojil, J. Phys. A: Math. Gen. 27 Ž1994. 4945. w31x G. Levai, B. Konya, Z. Papp, J. Math. Phys. 39 Ž1998. 5811. ´ ´ w32x M. Znojil, arXiv: quant-phr9811088, to appear in Phys. Rev. A. w33x M. Znojil, Phys. Lett. A 264 Ž1999. 108.
Physics Letters A 271 Ž2000. 327–333 www.elsevier.nlrlocaterpla
The Coulomb – harmonic oscillator correspondence in PT symmetric quantum mechanics Miloslav Znojil a,) , Geza ´ Levai ´ b a
b
ˇ ˇ Czech Republic Nuclear Physics Institute of Academy of Sciences of the Czech Republic, 250 68 Rez, Institute of Nuclear Research of the Hungarian Academy of Sciences, PO Box 51, H-4001 Debrecen, Hungary Received 20 March 2000; received in revised form 30 May 2000; accepted 5 June 2000 Communicated by P.R. Holland
Abstract We show that and how the Coulomb potential V Ž x . s Z e 2rx can be regularized and solved exactly at the imaginary coupling Z e 2. The new spectrum of energies is real and bounded as expected, but its explicit form proves totally different from the usual real-coupling case. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Quantum mechanics often works with exactly solvable simplified models. For precise fits of data or for some more subtle quantitative analyses, unfortunately, the number of solvable models is too limited. In D dimensions, the only really useful and easily tractable interactions are harmonic oscillators andror the central Coulomb well V Ž Z . Ž r . s yZ e 2r< r <. A new way out of this deadlock emerges within the framework of the alternative, so called ‘ PT symmetric’ quantum mechanics. With its complex Hamiltonians H breaking both parity P and timereflection symmetry T and commuting only with their product PT , this formalism was proposed by Bessis w1x and by Bender et al. w2–4x as a possible way towards weakening the standard requirements of Hermiticity. )
Corresponding author. E-mail addresses: [email protected] ŽM. Znojil., [email protected] ŽG. Levai ´ ..
Several new exactly solvable PT symmetric models have been proposed recently w5–7x. This is a promising development, with possible applications ranging from field theories w8x to supersymmetric models w9x and from quasi-classical methods w3x to perturbation theory w10–13x. Even the solvable harmonic oscillator itself acquires a richer spectrum after its consequent PT symmetric regularization in D dimensions w14x. The detailed structure of spectrum of this prominent example does not in fact offer any really serious surprise. A manifest violation of the parity P is compensated by the emergence of the so called quasiparity q s "1. It distinguishes between the two equidistant subspectra. The new quantum number q degenerates back to the eigenvalue of parity after a return to the standard Hermitean and one-dimensional oscillator. No immediate surprise emerges either for the quartic anharmonic oscillator w15–17x. The situation only becomes less clear after one moves towards asymptotically vanishing models. They exhibit sev-
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 4 0 0 - X
M. Znojil, G. LeÕair Physics Letters A 271 (2000) 327–333 ´
328
eral counterintuitive properties and open new mathematical challenges w18x. In particular, the popular Coulomb potential did not even seem particularly suitable for any immediate PT symmetric regularization w19x. A psychological barrier has been created by numerical and semiclassical studies of the general power-law forces V Ž x . ; yŽ ix . d. They may be well defined everywhere near the harmonic exponents d s 2, d s 6 etc. w20,21x. At the same time, related analyses hinted that it is apparently difficult to move beyond the Herbst singularity located at d s 1 w22x. In what follows, we intend to employ a slightly different strategy and try to study the Coulomb problem directly, via its well known correspondence with the harmonic oscillator. This correspondence is based on an elementary change of variables. Its background dates back to nineteenth century mathematics and, in particular, to the work of Liouville w23–25x. Newton’s monograph w26x cites Fivel w27x as a newer source of the idea. In the contemporary literature Žcf., e.g., w28x for further references. people usually speak about the Kustaanheimo–Steifel ŽKS. transformation w29x. In all the implementations of this idea the parameters appearing in the Coulombic and oscillator problems are interrelated, of course. Details will be mentioned below. Preliminarily, let us only warn the reader that all the KS-type mappings can change the dimensions and angular momenta and that the energies of one problem are related to the coupling constants of the other one and vice versa. Within ‘normal’ quantum mechanics, all this has already been thoroughly discussed elsewhere: in Ref. w30x for D s 3 and in Ref. w31x for the continuous transformation between Coulomb problems and harmonic oscillators in various dimensions.
of Ref. w14x, using the complex coordinate r s x y i c with real x g Žy`,`. and with, say, positive c ) 0. This means that the integration path has been shifted down from the position where it would cross the strong centrifugal singularity. Such a regularization preserves the asymptotic decrease of the normalizable solutions. Only in the limit c 0 and beyond the trivial one-dimensional case does one have to omit all the so called irregular solutions w32x defined by their ‘physically unnacceptable’ x Ž r . ; ryl behaviour near the origin. Within the framework of the general Liouville method the change of variables mediates a transition to a different potential V Ž t .. It is easy to show that once we forget about boundary conditions one merely has to demand the existence of an invertible function r s r Ž t . and a few of its derivatives r X Ž t ., r XX Ž t ., . . . . Then, the explicit correspondence between the two bound state problems may be explicitly given by elementary formulae. From our original Eq. Ž1. Ži.e., in our case, the harmonic oscillator. one obtains the new Ži.e., in our case, Coulombic. Schrodinger equa¨ tion
™
d2 y dt
2
q
L Ž L q 1.
C Ž t . s x r Ž t . r rX Ž t .
(
y dr
2
q
l Ž l q 1. r
2
Ž 3.
and with the new interaction and the new energies w25x, L Ž L q 1. t2
2. Liouvillean changes of variables
d2
Ž 2.
with the new wave functions
qV Ž t. yE
s rX Ž t .
The change-of-variable approach to the Coulombic bound-state problem enables us to start directly from the harmonic oscillator potential W Ž r . s r 2 or, in the present less traditional context, from the PT symmetric radial Schrodinger equation ¨
q V Ž t . C Ž t . s EC Ž t .
t2
q
2
½
l Ž l q 1. r2Ž t.
3 r XX Ž t . 4
rX Ž t .
2
y
qW r Ž t. y´ 2
1 r XXX Ž t . 2
rX Ž t .
5
.
Thus, it only remains for us to re-analyse the boundary conditions.
3. PT symmetric KS transformation
qW Ž r . x Ž r . s´ 2 x Ž r .
Ž 1.
Without any serious formal difficulties let us extend the scope of the present considerations to all the
M. Znojil, G. LeÕair Physics Letters A 271 (2000) 327–333 ´
329
Fig. 1. PT symmetric integration paths.
singular forces Wˆ Ž r . s W Ž r . q frr 2 andror VˆŽ t . s V Ž t . q Frt 2 . Both these central forces may act in the respective d and D dimensions. This means that l Ž l q 1 . s j q Ž d y 3 . r2
j q Ž d y 1 . r2 q f 4
or rather 2 2 Ž l q 1r2. Ž s a 2 . s j q Ž d y 2 . r2 q f
Žwith Re a ) 0. and 2 2 Ž L q 1r2. Ž s A2 . s J q Ž D y 2 . r2 q F
Žwith Re A ) 0. where the partial waves are numbered by the respective integers j s 0,1, . . . and J s 0,1, . . . . An important simplification of our effort is provided by our knowledge of the complete harmonic oscillator solution as derived in Ref. w14x. Its two equidistant subsets of energies
´ 2 s ´Ž2n , q . s 4 n q 2 y 2 q a ,
q s "1,
n s 0,1, . . . correspond to the two families of the Laguerre-polynomial wave functions
xŽ n , q . Ž r . s N r 1r2yq a eyr
2
r2
a. LŽyq Ž r2. . n
The integration path r s x y i c is a straight line. It lies in the lower half of the complex plane and does not change its form and position after the subsequent P transformation r s r Ž x . yr and T transformation yr yr ) s r Žyx .. In the spirit of the above-mentioned KS mapping of harmonic oscillators on Coulombic bound states we now have to define a complex variable t as a re-scaled square of r Ž x . such that the resulting path t Ž x . remains PT invariant. Our requirement implies that the complex plane of r will twice cover the complex plane of t. In such an arrangement, our lower half plane of r should cover the Riemann sheet given as a whole plane of t which is cut upwards from the origin. In the polar representation, one has r ; expŽyi w . mapped upon t ; expŽy2 i w . with w g Ž0, p .. Once we introduce a suitable free parameter k ) 0 we can put, say, r 2 s 2 k 2 z and then rotate the z-plane Žwhich is cut, by construction, along the real and positive semi-axis. by the angle pr2 giving t s i z. Our final recipe
™
r 2 s y2 i k 2 t
™
Ž 4.
maps the above-mentioned straight line r Ž x . s x y i c upon a curve t Ž x . s u q i Õ. Its real part u s uŽ x .
M. Znojil, G. LeÕair Physics Letters A 271 (2000) 327–333 ´
330
s x crk 2 and imaginary part Õ s Õ Ž x . s Ž x 2 y c 2 .r2 k 2 form an upwards-oriented parabola Õ s yc 2r2 k 2 q Ž k 2r2 c 2 . u 2 in the complex plane. The two curves ‘A’ in Fig. 1 illustrate the shape of both these PT symmetric trajectories in more detail. The harmonic-oscillator straight line r Ž x . s x y ic with c s 1 is denoted by the symbol AŽHO. there. Its Coulombic transform AŽCoul. is defined by formula Ž4. and displayed in Fig. 1 at k s 1. Many alternative choices of the original coordinates are admissible and remain compatible with the overall analyticity constraints w14x. Their sample BŽHO. in Fig. 1 is characterized by an asymptotic deformation of our original integration path r Ž x . given by an x-dependent shift c s c Ž x . ; 1r '2 q x 4 . This produces a different curve BŽCoul. Žwith k s 1 again.. We see that the Coulombic coordinates BŽCoul. may in fact lie quite close to the cut in the asymptotic domain of < x < 4 1. Having achieved a PT symmetry in the complex plane of t, we may move to the Žtrivial. insertions and conclude that all the above-mentioned harmonic oscillator bound-state solutions are in a one-to-one correspondence with the solutions of the Coulombic Schrodinger Eq. Ž2., ¨ d2 y
dt 2
q
L Ž L q 1. t2
t s uŽ x . q i Õ Ž x . ,
qi
Z e2 t
C Ž t . s EC Ž t . ,
x g R.
Ž 5.
The underlying assignment of constants is such that a s 2 A, while k itself becomes n-dependent, k 2 s 2 Z e 2r´ 2 s Z e 2rŽ2 n q 1 y 2 q A.. In full detail one gets the new, Laguerre-polynomial wave functions 2
q A. CŽ n , q . Ž t . s M t 1r2yq A e i k t LŽy2 Ž y2 i k 2 t . n
Ž 6. and their energy spectrum specified by the elementary formula EŽ n , q . s kŽ4n , q . s q s "1,
Z 2e4
Ž 2 n q 1 y 2 q A.
n s 0,1, . . . .
This is our main result.
2
4. Discussion 4.1. Consequences of the curÕature of our integration path The latter two formulae exhibit several unusual features. The first concerns the asymptotics of the wave functions which are determined by the decreasing exponential expŽ i k 2 t .. Its form re-confirms the correctness of the above, slightly counter-intuitive KS-dictated choice of our PT symmetric path of integration. Asymptotically, it encircles more or less closely the positive imaginary axis in the t plane. This clarifies the apparent paradox. The second unexpected result is the positivity and unusual n-dependence of the energies. This can be related to the choice of the KS integration path t Ž x . again. In the very vicinity of the origin, one can visualize this path as a circle with radius s , u2 Ž x . q Õ 2 Ž x . s s 2 ,
< x < < 1.
From the appropriate definitions we get the formula
s s c 2 Ž 0 . r2 kŽ2n , q . q O Ž x 2 . and see that this radius is n-dependent and increases with the growth of this principal quantum number, s ; n c 2 Ž0.rZ e 2 , n 4 1. As a consequence, the ‘effective charge’ of our P T symmetric Coulomb potential appears to decrease with n, since i Z e2 t
;
Z e2
s
q O Ž t . s O Ž 1rn . .
This offers a ‘rule-of-thumb’ guide to the unusual and certainly counterintuitive n-dependence of the energy levels Ž7.. Of course, in practice, the preferred integration path will be n-independent. In such a case, the n-dependence re-appears in the small-x deformation of the initial harmonic-oscillator path with c s c nŽ x . s O Ž1rn.. Such a flexible transfer of the excitation-dependence throws also a new light on the complexified KS transformation itself.
, 4.2. ‘Flown-away’ energies and unaÕoided crossings
Ž 7.
Let us return in more detail to the A-dependence of our energies Ž7.. Firstly, we notice their power-law
M. Znojil, G. LeÕair Physics Letters A 271 (2000) 327–333 ´
dependence on n and A Žwith exponent s -2. as somewhat similar to the spectra in the PT symmetric oscillator well w14x Žwith exponent s q1. and in the Morse potential of Ref. w33x Žwith exponent s q2.. In the present case, obviously, we have to distinguish between the two separate families EŽ n, q . with q s q1 Žcf. Fig. 2. and q s y1 Žcf. Fig. 3.. The latter set is, up to its sign, analogous to the ordinary Coulombic spectrum. By far not so the former one. Its energies enrich and dominate the spectrum. The n div-th energy ‘flies away’ and disappears from the spectrum at A div s n div q 1r2. Moreover, at all the positive integers and half-integers A one encounters unaÕoided level crossings. In contrast to the harmonic case, they appear at both the opposite and identical Žviz., positive. quasiparities q. The former
331
case takes place at A s A crit s Ž n y nX .r2 while in the latter case we must fulfill the condition A s A crit s Ž n q nX q 1.r2. A sample of this phenomenon is given in Fig. 4. Formally the unavoided crossings generate certain identities which connect different Laguerre polynomials Žcf. a sample in Ref. w14x.. In applications, these ‘critical’ cases are not exceptional at all. For F s 0 forces without a spike, the critical integer or half-integer coordinates A crit s J y 1 q Dr2 correspond precisely to physical Žnamely, integer. dimensions D and partial waves J. In conclusion, let us not forget about many open questions. Pars pro toto, one could mention a not yet clear possibility of re-interpretation of our bound states, say, in the limit t real, i.e., beyond the mathematical and apparently natural boundaries of
Fig. 2. Quasi-even levels Žn,q. Žunits Z s e s 1..
™
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Fig. 3. Quasi-odd levels Ž n s 0–5, units Z s e s 1..
Fig. 4. Sample of unavoided crossings.
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our present approach. Moreover, we must keep in mind that via our complexification of the coordinates we of course broke their immediate connection to any standard D-dimensional problem. In this sense, our F s 0 and F / 0 Hamiltonians differ just in an inessential way. One could even prefer the latter option as a model which is formally simpler, due to the generic absence of the puzzling unavoided crossings. At F / 0 the structure of the spectrum of our present Coulomb model becomes also richer and, in this sense, more interesting.
Acknowledgements M.Z. thanks for the hospitality of the Institute of Nuclear Research of the Hungarian Academy of Sciences, in Debrecen, and appreciates the support by the grant Nr. A 1048004 of GA AS CR. G.L. acknowledges the OTKA grant no. T031945 and the support of the Janos Bolyai Research Fellowship. ´
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