Series Solutions of PT-Symmetric Schr\"odinger Equations

We consider series solutions of the Schr\"odinger equation for the Bender-Boettcher potentials V(x)=-(ix)^N with integer N. A simple truncation is introduced which provides accurate results regarding the ground state and first few excited states for any N. This is illustrated with explicit computations of energy levels, node structure and expectation values for some integer N.

The complex Schrödinger equation is known to have positive spectra if N ≥ 2 [1,2,3]. If N is integer the wave functions are entire functions and the complex plane splits naturally into N + 2 Stokes wedges. Energy quantisation is a consequence of demanding that ψ(z) decays exponentially in a PT -symmetric pair of Stokes wedges. For any energy E, real or complex, there is a solution which decays exponentially in any given wedge. For special values of E one can find solutions that decay in two (non-contiguous) wedges.
Consider the double power series where the a pq are constants. Inserting this into the Schrödinger equation yields the recursion relation Viewing a pq as a matrix, equation (3) expresses any element a pq in terms of the elements directly above and directly to the left. On fixing the top left element a 00 one can, in principle, determine all the other elements. For the convenient choice a 00 = 1 all the a pq are positive rational numbers. With this choice ψ 1 (0) = 1 and ψ ′ 1 (0) = 0. A second solution of the Schrödinger equation is where the coefficients b pq satisfy It is convenient to take b 00 = 1 so that ψ 2 (0) = 0 and ψ ′ 2 (0) = i. Consider a linear combination of the two solutions where c is a complex constant. By a suitable choice of c one can ensure that ψ(z) decays exponentially in any one of the N + 2 Stokes wedges. For example, to obtain decay in the wedge centred at (the anti-Stokes line) θ = − 1 2 π(N − 2)/(N + 2) take Although this works for any E it only gives a decaying wave function in one of the N + 2 wedges. However, if both E and c are real the solution will also decay in the PT image of the wedge. To determine the spectrum in a PT symmetric pair of wedges it suffices to determine the real energies for which c is real. This can be implemented graphically by plotting Im c as a function of E -the roots are the energy levels. In figure 1 this is plotted for N = 3; as E is increased Im c approaches zero in an oscillatory fashion. Im c has no roots for negative E. To produce this plot we made two approximations: i) In the double power series (2) and (4) we retained all terms with p+q ≤ 100. ii) In equation (7) a 'large' finite value of r is taken instead of the r → ∞ limit. We took r = 8.
In the matrix language the truncation is anti-diagonal in that entries below the p + q = 100 line are discarded. Applying a root finding algorithm to our approximation for Im c(E) one can compute the energy levels and associated real c values. For large values of n, c n is approximately − √ E n . To investigate the accuracy of our method one can vary the p + q ≤ 100 truncation and the r value. The values of the first few energy levels is not affected by taking r = 7 instead of r = 8 at least to 20 significant figures. Similarly, upping the truncation to p + q ≤ 150 does not change the first few energy levels (again to 20 significant figures). However the higher energy levels are sensitive to changes in r and the truncation. We have quoted E 4 to 17 rather than 20 significant figures as the missing three digits change when the truncation is improved. For higher n the accuracy drops further. As the double power series are expansions in z and E it is expected that the truncation is less accurate for higher energies. Our energy levels are consistent with the Runge-Kutta based results reported by Bender in [4].
For higher N there is more than one pair of (non-adjacent) PT -symmetric wedges [5]. If N is odd there are 1 2 (N − 1) such pairs. If N is even there are 1 2 (N + 2) pairs; one of these pairs is also P symmetric where our graphical method is not applicable. 1 For example, there are three PT -symmetric pairs if N = 7. Each pair gives a distinct real and positive spectrum; Im c is plotted as a function of E for the three pairs below. Figure 2: Im c in the N = 7 theory for the three PT -symmetric spectra. The upper plot is for the wedges centred at θ = π/6 and θ = 5π/6, the middle plot has wedges centred at θ = −π/18 and θ = −17π/18 and the lower plot has wedges centred at θ = −5π/18 and θ = −13π/18.
For higher n the E n have ratios 1.41 : 1 : 3.52 [5]. Although our method ia adapted to small n such behaviour is evident in the third excited state; E 3 has values 23.702, 16.872, 59.026. Note that the ratios of the ground state energies are different; E 0 has values 1.6047, 1.2247, 3.0686.
For the 'upper' spectrum (with wedges centred at θ = π/6 and θ = 5π/6) the c n values are positive with c n ≈ √ E n for large n. The other two spectra yield negative c n with c n ≈ − √ E n . The plots were produced via the same p + q ≤ 100 truncation but with r = 3 rather than r = 8. 2 Similar results can be obtained for higher N. For example the N = 19 model has 9 distinct spectra (4 giving positive c n , 5 with negative c n ) The truncations considered here can be used to identify the nodes of the energy eigenstates. Although our truncation fails for large enough |z|, at least for the first few energy levels, the nodes are close enough to the origin for them to be located with high precision. Returning to the N = 3 case all energy eigenfunctions have an infinite string of zeros on the positive imaginary axis; for each energy level these are above the classical turning point iE 1/3 n . In addition, the nth excited state has n nodes below the real axis (the first excited state has a node at z = −0.661296226442715413308i). The n nodes arch above and between the classical turning points at E 1/3 n e −iπ/6 and E 1/3 n e −5πi/6 [6]. An interesting question considered in [7] is what is the precise form of the arch for large n? Unlike for the N = 2 harmonic oscillator the nodes do not lie on the classical trajectory joining two turning points. This trajectory is exactly circular with its centre at the turning point on the imaginary axis [8].
Our approximations are also useful in computing expectation values. If ψ(z) is an energy eigenstate then the expectation value of z m can be written as a ratio of contour integrals 3 where C is any curve that splits the complex plane into two and starts in one wedge and ends in the PT -symmetric wedge. For N = 3 one can simply choose C to be the real line: where ψ n is the nth energy eigenstate (n = 0, 1, 2, 3, ...) As the wave functions decay exponentially these integrals over the real line are well approximated by integrals over a finite range [−λ, λ] for sufficiently large λ. We have computed expectation values for the first few energy eigenstates in the N = 3 model. Here we cut off the integrals at λ = 5 and approximated the ψ 1 and ψ 2 with the truncation (p + q ≤ 100) described above. Plots of the wave functions indicate that the cut off λ = 5 is good approximation for the first few eigenstates; a plot of ψ 3 (x) is given in Figure 3. Within our approximation z 2 m is very small (eg. z 2 0 ≈ 10 −11 ) which suggests that z 2 n is exactly zero. For general N similar calculations indicate that z N −1 n is zero for all PT -symmetric pairs of Stokes wedges. This is a PT -symmetric example of Ehrenfest's theorem. One can also see a PTsymmetric virial theorem in the N = 3 expectation values; z 3 n appears to be exactly − 2 5 iE n . This can be written as V n = 2 5 E n .