Complex WKB Analysis of a PT Symmetric Eigenvalue Problem

The spectra of a particular class of PT symmetric eigenvalue problems has previously been studied, and found to have an extremely rich structure. In this paper we present an explanation for these spectral properties in terms of quantisation conditions obtained from the complex WKB method. In particular, we consider the relation of the quantisation conditions to the reality and positivity properties of the eigenvalues. The methods are also used to examine further the pattern of eigenvalue degeneracies observed by Dorey et al. in [1,2].Comment: 22 pages, 13 figures. Added references, minor revision

D3 instantons in Calabi-Yau orientifolds with(out) fluxes

We investigate the instanton effect due to D3 branes wrapping a four-cycle in a Calabi-Yau orientifold with D7 branes. We study the condition for the nonzero superpotentials from the D3 instantons. For that matter we work out the zero mode structures of D3 branes wrapping a four-cycle both in the presence of the fluxes and in the absence of the fluxes. In the presence of the fluxes, the condition for the nonzero superpotential could be different from that without the fluxes. We explicitly work out a simple example of the orientifold of $K3 \times T^2/Z_2$ with a suitable flux to show such behavior. The effects of D3-D7 sectors are interesting and give further constraints for the nonzero superpotential. In a special configuration where D3 branes and D7 branes wrap the same four-cycle, multi-instanton calculus of D3 branes could be reduced to that of a suitable field theory. The structure of D5 instantons in Type I theory is briefly discussed.Comment: 17 pages; Typos corrected, arguments improved and references adde

Identification of observables in quantum toboggans

Quantum systems with real energies generated by an apparently non-Hermitian Hamiltonian may re-acquire the consistent probabilistic interpretation via an ad hoc metric which specifies the set of observables in the updated Hilbert space of states. The recipe is extended here to quantum toboggans. In the first step the tobogganic integration path is rectified and the Schroedinger equation is given the generalized eigenvalue-problem form. In the second step the general double-series representation of the eligible metric operators is derived.Comment: 25 p

Spectral zeta functions of a 1D Schr\"odinger problem

We study the spectral zeta functions associated to the radial Schr\"odinger problem with potential V(x)=x^{2M}+alpha x^{M-1}+(lambda^2-1/4)/x^2. Using the quantum Wronskian equation, we provide results such as closed-form evaluations for some of the second zeta functions i.e. the sum over the inverse eigenvalues squared. Also we discuss how our results can be used to derive relationships and identities involving special functions, using a particular 5F_4 hypergeometric series as an example. Our work is then extended to a class of related PT-symmetric eigenvalue problems. Using the fused quantum Wronskian we give a simple method for calculating the related spectral zeta functions. This method has a number of applications including the use of the ODE/IM correspondence to compute the (vacuum) nonlocal integrals of motion G_n which appear in an associated integrable quantum field theory.Comment: 15 pages, version

Scattering in the PT-symmetric Coulomb potential

Scattering on the ${\cal PT}$-symmetric Coulomb potential is studied along a U-shaped trajectory circumventing the origin in the complex $x$ plane from below. This trajectory reflects ${\cal PT}$ symmetry, sets the appropriate boundary conditions for bound states and also allows the restoration of the correct sign of the energy eigenvalues. Scattering states are composed from the two linearly independent solutions valid for non-integer values of the 2L parameter, which would correspond to the angular momentum in the usual Hermitian setting. Transmission and reflection coefficients are written in closed analytic form and it is shown that similarly to other ${\cal PT}$-symmetric scattering systems the latter exhibit handedness effect. Bound-state energies are recovered from the poles of the transmission coefficients.Comment: Journal of Physics A: Mathematical and Theoretical 42 (2009) to appea

Reflectionless Potentials and PT Symmetry

Large families of Hamiltonians that are non-Hermitian in the conventional sense have been found to have all eigenvalues real, a fact attributed to an unbroken PT symmetry. The corresponding quantum theories possess an unconventional scalar product. The eigenvalues are determined by differential equations with boundary conditions imposed in wedges in the complex plane. For a special class of such systems, it is possible to impose the PT-symmetric boundary conditions on the real axis, which lies on the edges of the wedges. The PT-symmetric spectrum can then be obtained by imposing the more transparent requirement that the potential be reflectionless.Comment: 4 Page

From Marginal Deformations to Confinement

We consider type IIB supergravity backgrounds which describe marginal deformations of the Coulomb branch of N=4 super Yang-Mills theory with SO(4) x SO(2) global symmetry. Wilson loop calculations indicate that certain deformations enhance the Coulombic attraction between quarks and anti-quarks at the UV conformal fixed-point. In the IR region, these deformations can induce a transition to linear confinement.Comment: 14 pages, 4 figures, minor corrections, comments and references adde

PTPT symmetric non-selfadjoint operators, diagonalizable and non-diagonalizable, with real discrete spectrum

Consider in $L^2(R^d)$, $d\geq 1$, the operator family $H(g):=H_0+igW$. \ds H_0= a^\ast_1a_1+... +a^\ast_da_d+d/2 is the quantum harmonic oscillator with rational frequencies, $W$ a $P$ symmetric bounded potential, and $g$ a real coupling constant. We show that if $|g|<\rho$, $\rho$ being an explicitly determined constant, the spectrum of $H(g)$ is real and discrete. Moreover we show that the operator \ds H(g)=a^\ast_1 a_1+a^\ast_2a_2+ig a^\ast_2a_1 has real discrete spectrum but is not diagonalizable.Comment: 20 page

Negative-energy PT-symmetric Hamiltonians

The non-Hermitian PT-symmetric quantum-mechanical Hamiltonian $H=p^2+x^2(ix)^\epsilon$ has real, positive, and discrete eigenvalues for all $\epsilon\geq 0$. These eigenvalues are analytic continuations of the harmonic-oscillator eigenvalues $E_n=2n+1$ (n=0, 1, 2, 3, ...) at $\epsilon=0$. However, the harmonic oscillator also has negative eigenvalues $E_n=-2n-1$ (n=0, 1, 2, 3, ...), and one may ask whether it is equally possible to continue analytically from these eigenvalues. It is shown in this paper that for appropriate PT-symmetric boundary conditions the Hamiltonian $H=p^2+x^2(ix)^\epsilon$ also has real and {\it negative} discrete eigenvalues. The negative eigenvalues fall into classes labeled by the integer N (N=1, 2, 3, ...). For the Nth class of eigenvalues, $\epsilon$ lies in the range $(4N-6)/3<\epsilon<4N-2$. At the low and high ends of this range, the eigenvalues are all infinite. At the special intermediate value $\epsilon=2N-2$ the eigenvalues are the negatives of those of the conventional Hermitian Hamiltonian $H=p^2+x^{2N}$. However, when $\epsilon\neq 2N-2$, there are infinitely many complex eigenvalues. Thus, while the positive-spectrum sector of the Hamiltonian $H=p^2+x^2(ix)^\epsilon$ has an unbroken PT symmetry (the eigenvalues are all real), the negative-spectrum sector of $H=p^2+x^2(ix)^\epsilon$ has a broken PT symmetry (only some of the eigenvalues are real).Comment: 12 pages, 8 figure