This paper builds on our earlier proposal for construction of a positive inner product for pseudo-Hermitian Hamiltonians and we give several examples to clarify our method. We show through the example of the harmonic oscillator how our construction applies equally well to Hermitian Hamiltonians which form a subset of pseudo-Hermitian systems. For finite dimensional pseudo-Hermitian matrix Hamiltonians we construct the positive inner product (in the case of 2×2 matrices for both real as well as complex eigenvalues). When the quantum mechanical system cannot be diagonalized exactly, our construction can be carried out perturbatively and we develop the general formalism for such a perturbative calculation systematically (for real eigenvalues). We illustrate how this general formalism works out in practice by calculating the inner product for a couple of PT symmetric quantum mechanical theories.

Within the context of non-Hermitian quantum mechanics, we use the generators of eigenvectors of the Hamiltonian to construct a unitary inner product space. Such models have been of interest in recent years, for instance, in the context of PT symmetry, although our construction extends to the larger class of so-called pseudo-Hermitian Operators. We provide a detailed example to illustrate the concept and compare with known results.Let us consider a non-Hermitian quantum mechanical Hamiltonian operator, H, which is related to its adjoint through a similarity transformation(1)An operator with the above relation is sometimes called pseudo-Hermitian [1]. The adjoint, H † , is defined with respect to an auxilary Hilbert space, H, equipped with an inner product ·|· (conventionally taken to be the Dirac inner product), such that φ|Hψ = H † φ|ψ , with |ψ , |φ ∈ H. Relation (1) implies that S can be chosen to be self-adjoint.Since H is not Hermitian, it follows that e −itH (for t ∈ R, = 1) is not unitary and, therefore, it is not possible to construct a unitary quantum theory on H. To construct a unitary quantum theory we must define a new Hilbert space, H ′ , with a modified inner product leading to a modified adjoint, H ‡ , for which the Hamiltonian H is self-adjoint.By choosing H ′ so that H ‡ = H, we can guarantee that e −itH is a unitary time evolution operator in H ′ .

The structure of supersymmetry is analyzed systematically in PT symmetric quantum mechanical theories. We give a detailed description of supersymmetric systems associated with one dimensional PT symmetric quantum mechanical theories. We show that there is a richer structure present in these theories compared to the conventional theories associated with Hermitian Hamiltonians. We bring out various properties associated with these supersymmetric systems and generalize such quantum mechanical theories to higher dimensions as well as to the case of one dimensional shape invariant potentials.

We investigate the motion of a test particle in higher dimensions due to the presence of extended sources like Dp-branes by studying the motion in the transverse space of the brane. This is contrasted with the motion of a point particle in the Schwarzschild background in higher dimensions. Since Dp-branes are specific to 10-dimensional space time and exact solutions of geodesic equations for this particular space time has not been possible so far for the Schwarzschild background, we focus here to find the leading order solution of the geodesic equation (for motion of light rays). This enables us to compute the bending of light in both the backgrounds. We show that contrary to the well known result of no noncircular bound orbits for a massive particle, in Schwarzschild background, for d ≥ 5, the Dp-brane background does allow bound elliptic motion only for p = 6 and the perihelion of the ellipse regresses instead of advancement. We also find that circular orbits for photon are allowed only for p ≤ 3.