Pseudo-Hermiticity, $$\mathcal{P}\mathcal{T}$$ symmetry,and the metric operator

The main achievements of Pseudo-Hermitian Quantum Mechanics and its distinction from the indefinite-metric quantum theories are reviewed. The issue of the non-uniqueness of the metric operator and its consequences for defining the observables are discussed. A systematic perturbative expression for the most general metric operator is offered and its application for a toy model is outlined. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

Point Interactions: $$\mathcal{P}\mathcal{T}$$ -Hermiticity and Reality of the Spectrum

General point interactions for the second derivative operator in one dimension are studied. In particular, -self-adjoint point interactions with the support at the origin and at points ±l are considered. The spectrum of such non-Hermitian operators is investigated and conditions when the spectrum is pure real are presented. The results are compared with those for standard self-adjoint point interactions.     

Comparative analysis of real and \mathcal{P}\mathcal{T}-symmetric Scarf potentials

The Scarf I and Scarf II potentials are discussed within a common mathematical framework, which is then specified to handle the two potentials separately both in the conventional Hermitian and in the -symmetric setting. The physically admissible solutions are identified in each case together with the corresponding energy eigenvalues. Several main differences between the -symmetric Scarf I and II potentials are pointed out. These include the presence and absence of the quasi-parity quantum number, the sign of the pseudo-norm, the mechanism of the spontaneous breakdown of symmetry and the non- orthogonality of otherwise admissible solutions in the Scarf I potential. Similarities and differences with respect to the corresponding Hermitian systems are also pointed out.     

Parity anomaly in a \mathcal{P}\mathcal{T}-symmetric quartic Hamiltonian

In this paper, two independent methods are used to show that the non-Hermitian -symmetric wrong-sign quartic Hamiltonian H = (1/2m)p ² − gx ⁴ is exactly equivalent to the conventional Hermitian Hamiltonian . First, this equivalence is demonstrated by using elementary differential-equation techniques and second, it is demonstrated by using functional-integration methods. As the linear term in the Hermitian Hamiltonian is proportional to ℏ, this term is anomalous; that is, the linear term in the potential has no classical analog. The anomaly is a consequence of the broken parity symmetry of the original non-Hermitian -symmetric Hamiltonian. The anomaly term in remains unchanged if an x ² term is introduced into H. When such a quadratic term is present in H, this Hamiltonian possesses bound states. The corresponding bound states in are a direct physical measure of the anomaly. If there were no anomaly term, there would be no bound states.     

The He scattering and reactions on C and cluster states of C

The ⁶He+¹²C elastic and inelastic scattering and the ⁶He+¹²C→α+¹⁴C reaction have been measured using a 18.0 MeV ⁶He beam. Experimental results for the elastic scattering are in fair agreement with optical model predictions, using the potentials found in the analysis of ⁶Li scattering on ¹²C at similar energies. In triple coincidences, the ⁶He+¹²C→¹⁰Be+2α reactions were clearly seen, with the ¹⁰Be nucleus left in ground and several excited states. The dominant mechanism of this reaction is sequential decay through cluster states of ¹⁴C.     

High spin states in 63Cu

Excited states of ⁶³Cu were populated via the ⁵²Cr+¹⁰O (65 MeV) reaction using the gamma detector array equipped with charged particle detector array for reaction channel separation. On the basis of γ-γ coincidence relations and angular distribution ratios, a level scheme was constructed up to E _n=7 MeV and J ^π=23/2⁽⁺⁾. The decay scheme deduced was interpreted in terms of shell model calculations, with a restricted basis of the f _5/2, p _3/2, p _1/2, g _9/2 orbitals outside a ⁵⁶ ₂₈Ni core.     

Microscopic study of low-lying yrast spectra in <Superscript>100–108</Superscript>Mo isotopes

Variation-after-projection (VAP) calculations in conjunction with Hartree-Bogoliubov (HB) ansatz have been carried out for A=100−108 molybdenum (Mo) isotopes. In this framework, the yrast spectra with J _max ^π ≥10⁺, B(E2) transition probabilities, quadrupole (β₂) and hexadecapole (β₄) deformation parameters, moment of inertia (I) and square of cranking frequency (ω²) for even-even Mo isotopes have been obtained. The results of the calculation give an indication that it is important to include the hexadecapole-hexadecapole component of the two-body interaction for obtaining various nuclear structure quantities in these Mo isotopes.     

() reaction studies of Dy and Hf

Angular distributions for the ¹⁶³Dy(t,p) and ¹⁷⁷Hf(t,p) reactions were measured using 17 MeV tritons from the McMaster University Tandem Van de Graaff accelerator. Reaction products were analyzed with a magnetic spectrograph and detected with photographic emulsions. Favored L=0 transitions confirmed assignments of the 5/2⁻[523] band in ¹⁶⁵Dy and 7/2⁻[514] band in ¹⁷⁹Hf. Additional L=0 transitions in each nuclide identified previously unknown 5/2⁻ levels in ¹⁶⁵Dy and 7/2⁻ ones in ¹⁷⁹Hf. Overall trends of L=0 strengths support the existence of subshell closures at neutron numbers 98 and 108. On the basis of a relatively strong L=2 transition, the K^π=11/2⁻ γ-vibration based on the 7/2⁻[514] state is identified at 1689 keV in ¹⁷⁹Hf, about 440 keV above its previously-assigned K^π=3/2⁻ partner.     

Observation of the hot GDR in neutron-deficient thorium evaporation residues

The giant dipole resonance built on excited states was observed in very fissile nuclei in coincidence with evaporation residues. The reaction ⁴⁸Ca + ¹⁷⁶Yb populated evaporation residues of mass A=213–220 with a cross section of 200 μb at 259 MeV. The extracted giant dipole resonance parameters are in agreement with theoretical predictions for this mass region.     

Nuclear reactions of C on a liquid hydrogen target measured with the superconducting TOF spectrometer

Reaction cross sections with various kinds of breakup channels for neutron-rich carbon isotopes ^18-20C and for ⁹Be impinging on a liquid hydrogen target were investigated at 40 MeV/nucleon. The nuclides of interest were produced via projectile fragmentation from a 63 MeV/nucleon ⁴⁰Ar beam and were separated in flight at the RIKEN projectile fragment separator (RIPS). The combination of the large-acceptance superconducting TOF spectrometer, TOMBEE (TOF Mass analyzer for exotic BEam Experiment), with a liquid hydrogen target, CRYPTA (CRYogenic ProTon and Alpha target system), enables simultaneous measurements of several reaction channels: the reaction cross sections (σR), individual elemental fragmentation cross sections (σΔZ), charge-changing cross sections (σcc), neutron-removal cross sections (σ−xn), and charge-pickup cross sections (σΔZ+1) for ^19,20C; σΔZ, σ−xn, and σΔZ+1 for ¹⁸C; and σR for ⁹Be. The present σR of ⁹Be on proton, σR=397±23 mb, measured in the inverse kinematics, was consistent with the previous measurements using proton beams at different laboratories. The σR of ¹⁹C and ²⁰C on proton were determined to be σR=754±22 mb and σR=791±34 mb, respectively. Taking into account the beam energy and target dependence of σR, the present σR are found to be considerably enhanced compared with those measured at around 1 GeV/nucleon. The σΔZ+1 appears to increase with the mass number of the projectiles, and it significantly contributes to σR in the present energy range. The finite-range optical-limit and few-body Glauber model analyses were performed for σR to study the nuclear matter density distributions and to derive the relative strength of the s-wave components of the valence neutrons in ¹⁹C and ²⁰C. A neutron halo structure of ¹⁹C is confirmed with an s-wave dominance of the valence neutron when the effect of the charge-pickup reaction is taken into account. The large σ−n of ¹⁹C and σ−2n of ²⁰C also support the decoupled structures of ¹⁸C +n and ¹⁸C+2n, respectively. The σcc of ¹⁹C and ²⁰C agree with each other within their experimental uncertainties, which might indicate a similar proton density distribution in ¹⁹C and ²⁰C. The σΔZ decreases monotonically without the even-odd effect as the number of removed protons increases.     

Analysis of alpha-cluster transfer in O + C and C + O at energies near Coulomb barrier

The aim of this work is to track the phenomenon of α-cluster transfer mechanism at low energies 1.25, 1.5 and 1.75 MeV/n, close to the Coulomb barrier energy for ¹²C(¹⁶O, ¹²C)¹⁶O and ¹⁶O(¹²C, ¹⁶O)¹²C nuclear systems. The measurements of the angular distribution show a significant increase in the differential cross section at large angles due to alpha-transfer mechanism. The optical model code SPI-GENOA could be used effectively for fitting the experimental data with the theoretical predictions nearly up to angle 90°, where the differential cross section decreases steadily with increasing the scattering angle. For the second hemisphere, at angles greater than 100°, there is a large increase in cross section due to the contribution of α-transfer mechanism, and the Distorted Wave Born Approximation (DWBA) method could be used effectively for fitting the experimental data with the theoretical predictions at this region using (DWUCK5) code.