On the pseudo-norm in some PT-symmetric potentials

Complete text of publication follows. PT-symmetric quantum mechanical systems possess non-hermitian Hamiltonian, still they have some characteristics similar to hermitian problems. The most notable of these is their discrete energy spectrum, which can be partly or completely real. These systems are invariant under the simultaneous action of the P space and T time inversion operations. Perhaps the simplest PT-symmetric Hamiltonian contains a one-dimensional Schroedinger operator with a complex potential satisfying the V*(-x) = V (x) relation. Another typical feature PT-symmetric systems have in common with hermitian problems is that their basis states form an orthogonal set provided that the inner product is redefined as (ψ φ)PT ≡ (ψ Pφ). However, the norm defined by this inner product, the pseudo-norm turned out to possess indefinite sign, and this raised the question of the probabilistic interpretation of PT-symmetric systems. This problem was later put into a more general context when it was found that PT symmetry is a special case of pseudo-hermiticity, and this explains most of the peculiar features of PT-symmetric systems. There have been several attempts to link PT-symmetric, and in general, pseudo- hermitian systems with equivalent hermitian ones, and the sign of the pseudo-norm was found to play an important role in this respect. It is thus essential to evaluate the pseudo- norm for various potentials, especially considering the fact that there are some inconsistencies in the available results. Numerical studies indicated that the sign of the pseudo-norm typically alternates according to the n principal quantum number as (-1)n, and this was later proven for a class of potentials that are written in a polynomial form of ix. However, some potentials of other type did not fit into this line: this was the case for the Scarf II potential, the most well-known exactly solvable PT-symmetric potential. In contrast with the other examples, this potential is finite at the boundaries (x = ±∞) and it has finite number of discrete levels. Considering these circumstances it seemed worthwhile to study the Scarf I potential, V (x) = (α2+β2 / 2 - 1/4) 1/cos2 x + α2 - β2/2 sin x/cos2x (x ε [-π/2, π/2]), which is PT-symmetric and has real energy eigenvalues if α* = β holds. The Scarf II potential has similar structure, except for some constant factors and that it contains hyperbolic, rather than trigonometric functions. We found a closed expression for the pseudo-norm of the Scarf I potential and it turned out that it varies as (-1)n similarly to other potentials that are infinite at the boundaries and have infinite number of discrete levels. This potential has some further remarkable features. First, it contains the infinite square well as a special case, together with a specific PT-symmetric extension. Some other PT-symmetric extensions of the infinite square well have been analysed in terms of (semi- ) numerical methods, so comparison with these is certainly an interesting task. Second, since the Scarf I potential is singular at the boundaries, the boundary conditions play an especially important role in this case. It turned out that the solutions are regular at the boundaries if Re(α) < 1/2 holds, however, PT-normalizability has a less strict condition: Re(α) < 1. This is especially interesting considering the fact that similarly to other PT-symmetric potentials a second set of solutions is also possible with opposite quasi-parity, and these solutions are obtained from the (α,β) → (-α, -β) transformation (which, of course, leaves the potential invariant). A novel feature of the Scarf I potential is that although states with the same quasi-parity form an orthogonal set, there is non-orthogonality between states with opposite quasi-parity. (author)