AP07_2-3.vp 1 Introduction The most widely studied quantum mechanical potentials are formulated as one-dimensional problems. These include potentials defined on a finite domain (e.g. the infinite square well) or on the full x axis (e.g. the Pöschl-Teller potential). Po- tentials defined on the positive semi-axis also occur as radial problems obtained after the separation of the angular vari- ables in centrally symmetric potentials. Potentials in higher dimensions are less frequently discussed, and mainly in cases when they can be reduced to one-dimensional problems by the separation of the variables in some coordinates (Carte- sian, polar, etc.). These potentials differ from the one-dimen- sional ones in several respects: their spectrum can be richer due to the more degrees of freedom, and this can be mani- fested in the occurrence of degeneracies, for example. An interesting recent development in quantum mechanics was the introduction of �� symmetry [1]. Quantum systems with this symmetry are invariant under the simultaneous ac- tion of the � space and � time inversion operations, where the latter is represented by complex conjugation. It has been found that although these ��-symmetric problems are mani- festly non-Hermitian, as they possess an imaginary potential component too, they have several features in common with traditional self-adjoint systems. The most striking of these is the presence of real energy eigenvalues in the spectrum, but the orthogonality of the energy eigenstates and the time-in- dependence of their norm is also non-typical for complex potentials. There are, however, important differences too, with respect to conventional problems. The energy spectrum can turn into complex conjugate pairs as the non-Hermiticity increases, and this can be interpreted as the spontaneous breakdown of �� symmetry in that the energy eigenstates cease to be eigenstates of the �� operator then. Also, the pseudo-norm defined by the modified inner product � � � � �� �� turned out to have indefinite sign. �� sym- metry was later identified as the special case of pseudo- -Hermiticity, and this explained much of the unusual results. The proceedings volumes of recent workshops [2], [3] give a comprehensive account of the status of ��-symmetric quan- tum mechanics and related fields. With only a few exceptions the study of ��-symmetric systems has been restricted to the bound states of one-dimen- sional non-relativistic problems, where �� symmetry amounts to the requirement V x V x*( ) ( )� � . Here we extend the scope of these investigations by considering ��-symmetric prob- lems in higher spatial dimensions. In particular, we employ a simple method of generating solvable non-central potentials by the separation of the variables and combine it with the requirements of �� symmetry [4]. 2 Non-central potentials in polar coordinates Let us consider the Schrödinger equation with constant mass p r r r r r 2 2 2 2m V m V� � � � � � � � �( ) ( ) ( ) ( ) ( )� � � � � , (1) where the potential V(r) is a general function of the position r. Although in this section we implicitly assume that V(r) is real, so the Hamiltonian describing the quantum system is self- -adjoint, the procedure we follow here can be applied to complex potentials too. In what follows we choose the units as 2 1m � �� . Specifying (1) for d � 3 dimensions and using po- lar coordinates we obtain 1 1 1 1 2 2 2 2 2 2 2 2 r r r r r r r � � �� � � � �� � �� �� � � � � � � � cot( ) sin � � � �� � � � � 2 2 0� � �V r E( , , ) . (2) 40 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 47 No. 2–3/2007 ��-symmetry and Non-Central Potentials G. Lévai We present a general procedure by which solvable non-central potentials can be obtained in 2 and 3 dimensions by the separation of the angular and radial variables. The method is applied to generate solvable non-central ��-symmetric potentials in polar coordinates. General considerations are presented concerning the �� transformation properties of the eigenfunctions, their pseudo-norm and the nature of the energy eigenvalues. It is shown that within the present framework the spontaneous breakdown of �� symmetry can be implemented only in two dimensions. Keywords: �� symmetry, angular and radial variables, non-central potentials Assuming that the separation of the variables is possible, we search for the solution as � � � � � � �( , , ) ( ) ( ) ( )r r r� �1 , (3) where � r � �0, , � �� � 0, and � �� � 0 2, . Then (2) turns into � �� � �� � � � �� � � � � � � � � � � � � � 1 1 2 2 2 r r V r E ( cot( ) ) sin ( , , ) � � � 0, (4) where prime denotes the derivative with respect to the appro- priate variable. Next we assume that � �( ) and �( ) satisfy the second-order differential equations � �� � � � �� � � � �cot( ) ( )Q q , (5) � �� � � � K k( ) . (6) It is seen that (6) can be considered a one-dimensional Schrödinger equation defined in the finite domain � �0 2, with periodic boundary conditions ( ) ( )0 2� and � � � ( ) ( )0 2 . Note that in the case of one-dimensional po- tentials defined within a finite domain the wavefunction is usually required to vanish at the boundaries, however, consid- ering periodic boundary conditions, this is not a necessary requirement: it can also be finite there. Equation (5) is solvable for the choice Q( ) sin ( )� � �� �2 2 , q � �� �( )1 , (7) when the solutions are given by the associated Legendre func- tions � P� � �cos( ) [5]. Normalizability requires � and � to be non-negative integers such that � � l, � � �m l. Then � � � �lm m l mi l l m l m P( ) ( ) ! ( ) ! cos( )� �� � � � � � � � � � � � 1 2 1 2 . (8) Substituting (6), (5) and (7) in (4) the angular part can be separated, and a radial Schrödinger equation is obtained � �� � � �� �� � �� � �� � �V r l l r E0 2 1 0( ) ( ) , (9) where the central potential V r0( ) is related to V r( , , )� � as � V r V r r K k m( , , ) ( ) sin ( ) ( )� � � �� � � �0 2 2 21 . (10) In its most general form, (10) is a non-central potential that depends on the states through k and m. In order to eliminate the state-dependence one can apply the prescrip- tion k m a� �2 , where a is a constant. Since m has to be an in- teger, this prescription represents a restriction on the solu- tions of equation (6). A special case occurs for K a( )� � , i.e. for the free motion on a circle (or an infinite square well with peri- odic boundary conditions), which reduces (10) to a central po- tential, and takes the angular wavefunctions � � �( ) ( ) into spherical harmonics Ylm( , )� � [5]. Exact solutions of the radial Schrödinger equation (9) are known for the harmonic oscillator, Coulomb and square well potentials for arbitrary value of l, while for l � 0 (i.e. for s waves), it is solvable for many more potentials. Some solutions can also be obtained for arbitrary l for quasi-exactly solvable (QES) potentials [6] in the sense that the first few solutions (up to a given principal quantum number) can be determined exactly then. Specifying (1) to d � 2 dimensions the whole procedure can essentially be repeated. The equivalents of equations (2) and (3) are then 1 1 02 2 2 � � �� � � � �� � � � � � �� �� � � � �V E( , ) (11) and � � � �( , ) ( ) ( )� � 1 2 (12) The separation of the angular variable � is again possible if (6) holds, and the solutions are required to satisfy periodic boundary conditions. The radial Schrödinger equation is now � �� � � � � � � � � � � � � � � �� � �V k E0 2 1 4 1 0( ) , (13) where V V K( , ) ( ) ( ) � �� �0 2 1 . (14) Equation (13) can be solved exactly for the same potentials as in three dimensions. 3 Non-central ��-symmetric potentials Let us now specify the procedure outlined in the previous section to ��-symmetric potentials. Since the kinetic term in (1) is ��-symmetric, we have to take care separately only of the �� symmetry of the potential term. The effect of the � operation is � : r r� � , so the condition for the �� symmetry of a general potential in d � 3 dimensions is V r V r( , , ) ( , , )*� � � � � � � . (15) It is obvious that central potentials V V V r( ) ( ) ( )r r� � can be ��-symmetric only if they are real: V r V r( ) ( )*� , so the angular variables play an essential role in introducing an imaginary potential component. Applying condition (15) to the general potential form (10), the prescriptions V r V r0 0 *( ) ( )� , K K*( ) ( )� �� � , k k* � (16) are obtained, i.e. V0( ) is real, K ( )� is ��-symmetric and the eigenvalue of equation (6) is real. Note that the reality of the potential V r0( ) implies that (9) has the same form as the radial Schrödinger equation of a centrally symmetric self-adjoint quantum system, therefore the eigenvalues E also have to come out real. This means that the spontaneous breakdown © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 41 Acta Polytechnica Vol. 47 No. 2–3/2007 of �� symmetry cannot be implemented in the present ap- proach for non-central potentials in d � 3 dimensions. According to (15) the � operator can be factorized as � � � �� r � �, (where, obviously, �r � 1), so the �� transformation properties of the functions �(r), �(�) and (�) can also be studied. Due to the arguments concern- ing (9) above, �(r) can be chosen real, and in this case it obvi- ously satisfies � �r r r� �( ) ( )� . Introducing an extra phase factor il in (8), it is possible to make �(�) the eigenfunction of the ��� operator with eigenvalue 1. A similar procedure also has to be applied to the (�) functions and the �� operator, but this can be done only in the exact knowledge of the K(�) function. This guarantees that the full wavefunction �(r, �, �) (3) is also the eigenfunction of the �� operator with unit eigenvalue. These phase choices are also reflected in the sign of the pseudo-norm of the eigenstates �(r, �, �), since � �� can be determined by the inner products calculated with the con- stituent functions �(r), �(�) and (�), using the appropriate �i (i r� , �, �) operators. Obviously, the contribution of the radial component will be 1, while it can be shown that with the phase convention described above, � ��� � � �( )1 l m holds. Although the corresponding inner product for the (�) func- tions can be evaluated only in the knowledge of K(�), similar inner products are known to exhibit oscillatory behaviour (�1)n with respect to the principal quantum number for wide classes of ��-symmetric potentials with infinite number of eigenstates [7], [8], [9]. (Note that for some potentials with finite number of eigenstates this is not necessarily the case [10].) The sign of the pseudo-norm is thus indefinite for three-dimensional non-central ��-symmetric potentials too, and it depends on the quantum numbers associated with the angular component of the eigenfunctions. Let us now discuss the conditions under which non-central potentials can be ��-symmetric in d � 2 dimensions. The equivalent of (15) is now V V( , ) ( , )* � � � � , (17) and from (14) the conditions V V0 0 *( ) ( ) � , K K*( ) ( )� �� � (18) follow from the �� symmetry of the V( , �). The arguments on the �� symmetry of the wavefunction and the constituent functions are the same as in the three-dimensional case, as are those concerning the sign of the pseudo-norm. A major difference with respect to the three-dimensional case is that now k can be complex too. Since k is the eigenvalue of (6), which itself can be considered a Schrödinger equation with a ��-symmetric potential (K(�)), its complex eigenvalues occur in complex conjugate pairs. Substituting k and k* into the radial Schrödinger equation (13) one finds that the two equations are each other’s complex conjugate, so their energy eigenvalues will also appear as each other’s complex conju- gates. This indicates that similarly to the one-dimensional case, the spontaneous breakdown of �� symmetry leads to complex conjugate energy eigenvalues for d � 2 too. 4 Summary The most important results of this work are presented in the table below. d � 2 d � 3 State-independent potential always k m a� �2 Central potential K const( ) .� � K a( )� � ��-symmetric potential V V0 0 *( ) ( ) � K K*( ) ( )� �� � V r V r0 0 *( ) ( ),� k k� * K K*( ) ( )� �� � Energy eigenvalues real or complex conjugate pairs real Sign of pseudo-norm indefinite indefinite 5 Acknowledgment This work was supported by the OTKA grant No. T49646 (Hungary). References [1] Bender, C. M., Boettcher, S.: Real Spectra in Non- -Hermitian Hamiltonians Having �� Symmetry. Phys. Rev. Lett., Vol. 80 (1998), No. 24, p. 5243–5246. [2] J. Phys. A: Math. Gen., Vol. 39 (2006), No. 32. [3] Czech. J. Phys., Vol. 56 (2006), No. 9. [4] Lévai, G.: Solvable ��-symmetric Potentials in Higher Dimensions. J. Phys. A: Math. Theor., Vol. 40 (2007), No. 15, p. F273–F280. [5] Edmonds, A. R.: Angular Momentum in Quantum Mechan- ics. Princeton University Press, Princeton, 1957. [6] Ushveridze, A. G.: Quasi-Exactly Solvable Models in Quan- tum Mechanics. Institute of Physics Publishing, Bristol, 1994. [7] Trinh, T. H: Remarks on the ��-pseudo-norm in ��-symmetric Quantum Mechanics. J. Phys. A: Math. Gen., Vol. 38 (2005), No. 16, p. 3665–3678. [8] Bender, C. M., Tan, B.: Calculation of the Hidden Sym- metry Operator for a ��-symmetric Square Well. J. Phys. A: Math. Gen., Vol. 39 (2006), No. 8, p. 1945–1954. [9] Lévai, G.: On the Pseudo-Norm and Admissible Solu- tions of the ��-symmetric Scarf I Potential. J. Phys. A: Math. Gen., Vol. 39 (2006), No. 32, p. 10161–10170. 42 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 47 No. 2–3/2007 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 43 Acta Polytechnica Vol. 47 No. 2–3/2007 [10] Lévai, G., Cannata, F., Ventura, A.: �� Symmetry Break- ing and Explicit Expressions for the Pseudo-Norm in the Scarf II Potential. Phys. Lett. A, Vol. 300 (2002), No. 2–3, p. 271–281. Dr. Géza Lévai phone: +36 52 509 298 fax: +36 52 416 181 e-mail: levai@namafia.atomki.hu Institute of Nuclear Research of the Hungarian Academy of Sciences (ATOMKI) Bem ter 18/c Debrecen, 4026 Hungary