Exact solution for Morse oscillator in PT-symmetric quantum mechanics

The recently proposed PT-symmetric quantum mechanics works with complex potentials which possess, roughly speaking, a symmetric real part and an anti-symmetric imaginary part. We propose and describe a new exactly solvable model of this type. It is defined as a specific analytic continuation of the shape-invariant potential of Morse. In contrast to the latter well-known example, all the new spectrum proves real, discrete and bounded below. All its three separate subsequences are quadratic in n.Comment: 8 pages, submitted to Phys. Lett.

A family of complex potentials with real spectrum

We consider a two-parameter non hermitean quantum-mechanical hamiltonian that is invariant under the combined effects of parity and time reversal transformation. Numerical investigation shows that for some values of the potential parameters the hamiltonian operator supports real eigenvalues and localized eigenfunctions. In contrast with other PT symmetric models, which require special integration paths in the complex plane, our model is integrable along a line parallel to the real axis.Comment: Six figures and four table

Classical oscillator with position-dependent mass in a complex domain

We study complexified Harmonic Oscillator with a position-dependent mass, termed as Complex Exotic Oscillator (CEO). The complexification induces a gauge invariance [19,11]. The role of PT -symmetry is discussed from the perspective of classical trajectories of CEO for real energy. Some trajectories of CEO are similar to those for the particle in a quartic potential in the complex domain [10, 32].Comment: 20 pages, 4 figures, title is changed, some new features have been discussed, to appear in Phys. Lett.

Non-Hermitian matrix description of the PT symmetric anharmonic oscillators

Schroedinger equation H \psi=E \psi with PT - symmetric differential operator H=H(x) = p^2 + a x^4 + i \beta x^3 +c x^2+i \delta x = H^*(-x) on L_2(-\infty,\infty) is re-arranged as a linear algebraic diagonalization at a>0. The proof of this non-variational construction is given. Our Taylor series form of \psi complements and completes the recent terminating solutions as obtained for certain couplings \delta at the less common negative a.Comment: 18 pages, latex, no figures, thoroughly revised (incl. title), J. Phys. A: Math. Gen., to appea

Sistemi insediamentali dell’età del Bronzo nei bacini del fiume Secchia e del torrente Dragone

We present an addition to the Atlante dei Beni Archeologici della Provincia di Modena, vol. II (a catalog of archaeological sites in Modena Apennine published in 2006), reporting new field surveys and excavations (2007-2010) carried out on Bronze Age settlements in the Secchia River and Dragone Stream basins. The settlements, especially during the Middle and Recent Bronze Age, were aimed at controlling the tracks along the Secchia River and the Dragone Stream. In these phases, in addition to the mountain-top sites, many settlements were concentrated along the Secchia and Dragone tracks, probably in connection with copper ore mines near Poggio Bianco Dragone and with native copper outcrops in Monte Modino

Continuous monitoring of active rock blocks slide in the northern Apennines

A large and morphologically spectacular rock slope instability is detected in the northern Apennines of Italy. Ten of metres of blocks and columns, in calcareous rock formation, slide and spread on clay shale, dipping down slope 30\ub0. The geological structures and the morphological features of the slope deformation have been identi\ufb01ed with use of high-resolution digital elevation models obtained from ground based laser scanner and airborne LiDAR. The actual morphological setting of the slope is characterized, also far away form the main scarp, by some metres high columns in vertical position. This suggests that the main deformations are parallel to the slope. Inside the landslide, rock falls are also frequent, due to the failure of large blocks and columns through set of joints. The origin of the instability is probably related with the erosion of the clay formation, at the foot of the slope, by the close torrent. From the analysis of aero-photographs of the past \ufb01fty years is possible identi\ufb01ed an enlargement of the mass instability in the up slope direction. A destructive event was recognized in the Seventies, when the road and the bridge below the landslide were destroyed. During the event was documented the passage between slow to very fast moving. In the summer 2009, after the increase of the slope deformations of the last years, a continuous monitor system was installed. The system is a robotic station that every two hours measures 20 benchmarks \ufb01xed in the rock blocks and columns in the landslide area. Today, tens centimetre of displacements are recorded. The data show an increase of the deformations rate with the distance from the main scarp. At the moment, the evolution of the slope is controlled by continuous creeping processes and by period with accelerates deformations. These last correlate with wet and snow melt periods, which are responsible of the variation in the groundwater regime and of the changes in the material behaviour within the sliding zon

Large-scale slope instability affecting SS63 near the Cerreto Pass (northern Apennines, Italy)

National Road SS63 is endangered by large scale slope instability phenomena few km north of the Cerreto Pass (northern Apennines, Italy). At the Col Piagneto slope, SS63 was closed two times in the past 50 years after landslide events. In 1974, the collapse of portion of a rock slide triggered a debris slide that buried more than 50 m of the road track. In 2008, the reactivation of an earth slide in eastern portion of the slope damaged about 400 m of the “new” track built to by-pass the 1974 risk area. In order to evaluate if it is worthwhile keeping the old track open to traffic, or it is maybe better to repair the new track a comprehensive study was financed by Emilia Romagna Region. Research methods included: high resolution DEM, landslides mapping, continuous displacement and groundwater monitoring. Results allowed to assess different semi-quantitative risk scenario for the road structures or vehicles