SUSY Quantum Mechanics with Complex Superpotentials and Real Energy Spectra

We extend the standard intertwining relations used in Supersymmetrical (SUSY) Quantum Mechanics which involve real superpotentials to complex superpotentials. This allows to deal with a large class of non-hermitean Hamiltonians and to study in general the isospectrality between complex potentials. In very specific cases we can construct in a natural way "quasi-complex" potentials which we define as complex potentials having a global property such as to lead to a Hamiltonian with real spectrum. We also obtained a class of complex transparent potentials whose Hamiltonian can be intertwined to a free Hamiltonian. We provide a variety of examples both for the radial problem (half axis) and for the standard one-dimensional problem (the whole axis), including remarks concerning scattering problems.Comment: 22 pages, Late

Isoresonant complex-valued potentials and symmetries

Let $X$ be a connected Riemannian manifold such that the resolvent of the free Laplacian (\Delta-z)^{-1}, z\in\C\setminus\R^{+}, has a meromorphic continuation through $\R^{+}$. The poles of this continuation are called resonances. When $X$ has some symmetries, we construct complex-valued potentials, $V$, such that the resolvent of $\Delta+V$, which has also a meromorphic continuation, has the same resonances with multiplicities as the free Laplacian.Comment: 32 page

Analytically Solvable PT-Invariant Periodic Potentials

Associated Lam\'e potentials V(x)=a(a+1)m\sn^2(x,m)+b(b+1)m{\cn^2 (x,m)}/{\dn^2(x,m)} are used to construct complex, PT-invariant, periodic potentials using the anti-isospectral transformation $x \to ix+\beta$, where $\beta$ is any nonzero real number. These PT-invariant potentials are defined by $V^{PT}(x) \equiv -V(ix+\beta)$, and have a different real period from $V(x)$. They are analytically solvable potentials with a finite number of band gaps, when $a$ and $b$ are integers. Explicit expressions for the band edges of some of these potentials are given. For the special case of the complex potential V^{PT}(x)=-2m\sn^2(ix+\beta,m), we also analytically obtain the dispersion relation. Additional new, solvable, complex, PT-invariant, periodic potentials are obtained by applying the techniques of supersymmetric quantum mechanics.Comment: 12 pages, 3 figure

Pseudo Hermitian interactions in the Dirac Equation

We consider $(2+1)$ dimensional massless Dirac equation in the presence of complex vector potentials. It is shown that such vector potentials (leading to complex magnetic fields) can produce bound states and the Dirac Hamiltonians are $\eta$-pseudo Hermitian. Some examples have been explicitly worked out.Comment: 8 pages, NO figure

Complex periodic potentials with real band spectra

This paper demonstrates that complex PT-symmetric periodic potentials possess real band spectra. However, there are significant qualitative differences in the band structure for these potentials when compared with conventional real periodic potentials. For example, while the potentials V(x)=i\sin^{2N+1}(x), (N=0, 1, 2, ...), have infinitely many gaps, at the band edges there are periodic wave functions but no antiperiodic wave functions. Numerical analysis and higher-order WKB techniques are used to establish these results.Comment: 8 pages, 7 figures, LaTe