Carl M. Bender, Moshe Moshe, Sarben Sarkar
The conventional double-scaling limit of a quartic quantum field theory is inconsistent because the critical coupling constant is negative. Thus, at the critical coupling the Lagrangian appears to define a quantum theory whose energy is complex. Worse yet, the functional integral for the partition function of the theory does not exist. It is shown that one can avoid these difficulties if one approaches this correlated limit in a PT-symmetric fashion. The partition function is calculated explicitly in the double-scaling limit of an zero-dimensional O(N)-symmetric quartic model.
http://arxiv.org/abs/1206.4943
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Yogesh N. Joglekar, Bijan Bagchi
We investigate the effects of competition between two complex, \(\mathcal{PT}\)-symmetric potentials on the \(\mathcal{PT}\)-symmetric phase of a “particle in a box”. These potentials, given by \(V_Z(x)=iZ\mathrm{sign}(x)\) and \(V_\xi(x)=i\xi[\delta(x-a)-\delta(x+a)]\), represent long-range and localized gain/loss regions respectively. We obtain the \(\mathcal{PT}\)-symmetric phase in the \((Z,\xi)\) plane, and find that for locations \(\pm a\) near the edge of the box, the \(\mathcal{PT}\)-symmetric phase is strengthened by additional losses to the loss region. We also predict that a broken \(\mathcal{PT}\)-symmetry will be restored by increasing the strength \(\xi\) of the localized potential. By comparing the results for this problem and its lattice counterpart, we show that a robust \(\mathcal{PT}\)-symmetric phase in the continuum is consistent with the fragile phase on the lattice. Our results demonstrate that systems with multiple, \(\mathcal{PT}\)-symmetric potentials show unique, unexpected properties.
http://arxiv.org/abs/1206.3310
Quantum Physics (quant-ph)
Ali Mostafazadeh
For a general complex scattering potential defined on a real line, we show that the equations governing invisibility of the potential are invariant under the combined action of parity and time-reversal (PT) transformation. We determine the PT-symmetric an well as non-PT-symmetric invisible configurations of an easily realizable exactly solvable model that consists of a two-layer planar slab consisting of optically active material. Our analysis shows that although PT-symmetry is neither necessary nor sufficient for the invisibility of a scattering potential, it plays an important role in the characterization of the invisible configurations. A byproduct of our investigation is the discovery of certain configurations of our model that are effectively reflectionless in a spectral range as wide as several hundred nanometers.
http://arxiv.org/abs/1206.0116
Mathematical Physics (math-ph); Optics (physics.optics); Quantum Physics (quant-ph)
Philip D. Mannheim
In this review we discuss the connection between two seemingly disparate topics, macroscopic gravity on astrophysical scales and Hamiltonians that are not Hermitian but $PT$ symmetric on microscopic ones. In particular we show that the quantum-mechanical unitarity problem of the fourth-order derivative conformal gravity theory is resolved by recognizing that the scalar product appropriate to the theory is not the Dirac norm associated with a Hermitian Hamiltonian but is instead the norm associated with a non-Hermitian but \(PT\)-symmetric Hamiltonian. Moreover, the fourth-order theory Hamiltonian is not only not Hermitian, it is not even diagonalizable, being of Jordan-block form. With \(PT\) symmetry we establish that conformal gravity is consistent at the quantum-mechanical level. In consequence, we can apply the theory to data, to find that the theory is capable of naturally accounting for the systematics of the rotation curves of a large and varied sample of 138 spiral galaxies without any need for dark matter. The success of the fits provides evidence for the relevance of non-diagonalizable but \(PT\)-symmetric Hamiltonians to physics.
http://arxiv.org/abs/1205.5717
High Energy Physics – Theory (hep-th); Cosmology and Extragalactic Astrophysics (astro-ph.CO); General Relativity and Quantum Cosmology (gr-qc); Quantum Physics (quant-ph)
Bikashkali Midya
Using the method of point canonical transformation, we derive some exactly solvable rationally extended quantum Hamiltonians which are non-Hermitian in nature and whose bound state wave functions are associated with Laguerre and Jacobi-type \(X_1\) exceptional orthogonal polynomials. These Hamiltonians are shown, with the help of imaginary shift of co-ordinate: \(e^{-\alpha p} x e^{\alpha p} = x+ i \alpha\), to be both quasi and pseudo-Hermitian. It turns out that the corresponding energy spectra is entirely real.
http://arxiv.org/abs/1205.5860
Mathematical Physics (math-ph); Quantum Physics (quant-ph)
N. Kandirmaz, R. Sever
The wave functions and the energy spectrum of PT-/non-PT-Symmetric and non-Hermitian Hulthen potential are of an exponential type and are obtained via the path integral. The path integral is constructed using parametric time and point transformation.
http://ctn.cvut.cz/ap/download.php?id=572
Amine B Hammou
Scattering from a discrete quasi-Hermitian delta function potential is studied and the metric operator is found. A generalized continuity relation in the physical Hilbert space \({\mathcal H}_{{\rm phys}}\) is derived and the probability current density is defined. The reflection \({\mathcal R}\) and transmission \({\mathcal T}\) coefficients computed with this current are shown to obey the unitarity relation \({\mathcal R}+{\mathcal T}=1\).
http://dx.doi.org/10.1088/1751-8113/45/21/215310
Miloslav Znojil
A new exactly solvable model of a quantum system is proposed, living on an equilateral q-pointed star graph (q is arbitrary). The model exhibits a weak and spontaneously broken form of \({\cal PT}-\)symmetry, offering a straightforward generalization of one of the standard solvable square wells with \(q=2\) and unbroken \({\cal PT}-\)symmetry. The kinematics is trivial, Kirchhoff in the central vertex. The dynamics is one-parametric (viz., \(\alpha-\)dependent), prescribed via complex Robin boundary conditions (i.e., the interactions are non-Hermitian and localized at the outer vertices of the star). The (complicated, trigonometric) secular equation is shown reducible to an elementary and compact form. This renders the model (partially) exactly solvable at any \(q \geq 2\) — an infinite subset of the real roots of the secular equation proves q-independent and known (i.e., inherited from the square-well \(q=2\) special case). The systems with \(q=4m-2\) are found anomalous, supporting infinitely many (or, at m=1, one) additional real m-dependent and \(\alpha-\)dependent roots.
http://arxiv.org/abs/1205.5211
Quantum Physics (quant-ph)
Ali Mostafazadeh
A PT-symmetric optically active medium that lases at the threshold gain also acts as a complete perfect absorber at the laser wavelength. This is because spectral singularities of PT-symmetric complex potentials are always accompanied by their time-reversal dual. We investigate the significance of PT-symmetry for the appearance of these self-dual spectral singularities. In particular, using a realistic optical system we show that self-dual spectral singularities can emerge also for non-PT-symmetric configurations. This signifies the existence of non-PT-symmetric CPA-lasers.
http://arxiv.org/abs/1205.4560
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Optics (physics.optics)
Carl M. Bender, Daniel W. Hook
The PT-symmetric Hamiltonian \(H=p^2+x^2(ix)^\epsilon\) (\(\epsilon\) real) exhibits a phase transition at \(\epsilon=0\). When \(\epsilon\geq0$\) the eigenvalues are all real, positive, discrete, and grow as \(\epsilon\) increases. However, when \(\epsilon<0\) there are only a finite number of real eigenvalues. As \(\epsilon\) approaches -1 from above, the number of real eigenvalues decreases to one, and this eigenvalue becomes infinite at \(\epsilon=-1\). In this paper it is shown that these qualitative spectral behaviors are generic and that they are exhibited by the eigenvalues of the general class of Hamiltonians \(H^{(2n)}=p^{2n}+x^2(ix)^\epsilon\) (\(\epsilon\) real, n=1, 2, 3, …). The complex classical behaviors of these Hamiltonians are also examined.
http://arxiv.org/abs/1205.4425
High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)