Extensions of standard one-dimensional supersymmetric quantum mechanics are discussed. Supercharges involving higher order derivatives are introduced leading to an algebra which incorporates a higher order polynomial in the Hamiltonian. We study scattering amplitudes for that problem. We also study the role of a dilatation of the spatial coordinate leading to a q-deformed supersymmetric algebra. An explicit model for the scattering amplitude is constructed in terms of a hypergeometric function which corresponds to a reflectionless potential with infinitely many bound states.

Several aspects of complex-valued potentials generating a real and positive spectrum are discussed. In particular, we construct complex-valued potentials whose corresponding Schrödinger eigenvalue problem can be solved analytically.

Two new methods for investigation of two-dimensional quantum systems, whose Hamiltonians are not amenable to separation of variables, are proposed. 1)The first one -SU SY − separation of variables -is based on the intertwining relations of Higher order SUSY Quantum Mechanics (HSUSY QM) with supercharges allowing for separation of variables.2)The second one is a generalization of shape invariance. While in one dimension shape invariance allows to solve algebraically a class of (exactly solvable) quantum problems, its generalization to higher dimensions has not been yet explored.Here we provide a formal framework in HSUSY QM for two-dimensional quantum mechanical systems for which shape invariance holds. Given the knowledge of one eigenvalue and eigenfunction, shape invariance allows to construct a chain of new eigenfunctions and eigenvalues. These methods are applied to a two-dimensional quantum system, and partial explicit solvability is achieved in the sense that only part of the spectrum is found analytically and a limited set of eigenfunctions is constructed explicitly.

An experimental indication of negative heat capacity in excited nuclear systems is inferred from the event by event study of energy fluctuations in Au quasi-projectile sources formed in Au + Au collisions at 35 A.MeV. The excited source configuration is reconstructed through a calorimetric analysis of its de-excitation products. Fragment partitions show signs of a critical behavior at about 5 A.MeV excitation energy. In the same energy range the heat capacity shows a negative branch providing a direct evidence of a first order liquid gas phase transition.Phase transitions are the prototype of a complex system behavior which goes beyond the simple sum of individual properties [1]. In macroscopic systems the thermostatistical potential presents non analytical behaviors which unambiguously marks a phase transition. Non analytical behaviors of infinite systems originate from anomalies of the thermostatistical potentials in finite systems [2,3]. Specifically in microcanonical finite systems, the entropy is known to present a convex intruder in 1-st order phase transitions associated to a negative heat capacity between two poles. A 2-nd order phase transition is characterized by the merging of the two poles.The experimental study of phase transitions in finite systems has recently attracted a strong interest from various communities. Bose condensates with a small number of particles [4], melting of solid atomic clusters [5], vaporization of atomic nuclei [6] are examples of attempts to study phase transitions in finite systems. The problem usually encountered with these small systems is how to control the equilibrium and how to extract the thermostatistical variables from observable quantities in order to identify the possible phase transition. This is for instance the case in heavy ion reactions in which excited nuclear systems are formed. Comparing the observed decay channels with statistical models [2,7] it seems that a certain degree of equilibration is reached [8,9] but up to now it has not been possible to unambiguously identify the presence of the expected liquid-gas phase transition.It has recently been shown [3] that for a given total energy the average partial energy stored in a part of the system is a good microcanonical thermometer while the associated fluctuations can be used to construct the heat capacity. In the case of a phase transition anomalously large fluctuations are expected as a consequence of the divergence and of the possible negative branch of the heat capacity. Let us consider an equilibrated system which can be decomposed into two independent components so that the energy is simply the sum of the two partial energies E t = E 1 + E 2 and that the total level density W t ≡ exp(S t ) is the folding product of the two partial level densities W i ≡ exp(S i ).An example of such a decomposition is given by the kinetic and the potential energies in the absence of velocity dependent interactions.The probability distribution of the partial energy where: , 2) are the heat capacities calculated for th...

We extend the standard intertwining relations used in supersymmetrical (SUSY) quantum mechanics which involve real superpotentials to complex superpotentials. This allows us to deal with a large class of non-Hermitian Hamiltonians and to study in general the isospectrality between complex potentials. In very specific cases we can construct in a natural way "quasicomplex" potentials which we define as complex potentials having a global property so 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.