Abstract:
The evolution problem for a quantum particle confined in a 1D box and interacting with one fixed point through a time dependent point interaction is considered. Under suitable assumptions of regularity for the time profile of the Hamiltonian, we prove the existence of strict solutions to the corresponding Schr\"odinger equation. The result is used to discuss the stability and the steady-state local controllability of the wavefunction when the strenght of the interaction is used as a control parameter.

Abstract:
We present a simple recipe to construct the Green's function associated with a Hamiltonian of the form H=H_0+V, where H_0 is a Hamiltonian for which the associated Green's function is known and V is a delta-function potential. We apply this result to the case in which H_0 is the Hamiltonian of a free particle in D dimensions. Field theoretic concepts such as regularization, renormalization, dimensional transmutation and triviality are introduced naturally in order to deal with an infinity which shows up in the formal expression of the Green's function for D>1.

Abstract:
1d Bose gas interacting through delta, delta' and double-delta function potentials is shown to be equivalent to a delta anyon gas allowing exact Bethe ansatz solution. In the noninteracting limit it describes an ideal gas with generalized exclusion statistics and solves some recent controversies.

Abstract:
Periodic nanostructures can display the dynamics of arrays of atoms while enabling the tuning of interactions in ways not normally possible in Nature. We examine one dimensional arrays of a ``synthetic atom,'' a one dimensional ring with a nearest neighbor Coulomb interaction. We consider the classical limit first, finding that the singly charged rings possess antiferroelectric order at low temperatures when the charge is discrete, but that they do not order when the charge is treated as a continuous classical fluid. In the quantum limit Monte Carlo simulation suggests that the system undergoes a quantum phase transition as the interaction strength is increased. This is supported by mapping the system to the 1D transverse field Ising model. Finally we examine the effect of magnetic fields. We find that a magnetic field can alter the electrostatic phase transition producing a ferroelectric groundstate, solely through its effect of shifting the eigenenergies of the quantum problem.

Abstract:
A vortex lattice ratchet effect has been investigated in Nb films grown on arrays of nanometric Ni triangles, which induce periodic asymmetric pinning potentials. The vortex lattice motion yields a net dc-voltage when an ac driving current is applied to the sample and the vortex lattice moves through the field of asymmetric potentials. This ratchet effect is studied taking into account the array geometry, the temperature, the number of vortices per unit cell of the array and the applied ac currents.

Abstract:
The problem of one-dimensional quantum wire along which a moving particle interacts with a linear array of N delta-function potentials is studied. Using a quantum waveguide approach, the transfer matrix is calculated to obtain the transmission probability of the particle. Results for arbitrary N and for specific regular arrays are presented. Some particular symmetries and invariances of the delta-function potential array for the N = 2 case are analyzed in detail. It is shown that perfect transmission can take place in a variety of situations.

Abstract:
We investigate the localization observed recently for locally non-hermitian Hamiltonians by studying the effect of the amplification on the scaling behavior of the transmission and reflection phases in 1D periodic chains of $\delta$-potentials. The amplification here is represented by an imaginary term added to the on-site potential. It is found that both phases of the transmission and reflection amplitudes are strongly affected by the amplification term. In particular, the phases in the region of amplification become independent of the length scale while they oscillate strongly near the maximum transmission (or reflection). The interference effects on the phase in passive systems are used to interpret those observed in the presence of amplification. The phases of the transmission and reflection are found to oscillate in passive systems whith increasing periods in the allowed band for the transmission phase while for the reflection phase, its initial value is always less than $\pi /2$ in this band.

Abstract:
Consider the discrete 1D Schr\"odinger operator on $\Z$ with an odd $2k$ periodic potential $q$. For small potentials we show that the mapping: $q\to $ heights of vertical slits on the quasi-momentum domain (similar to the Marchenko-Ostrovski maping for the Hill operator) is a local isomorphism and the isospectral set consists of $2^k$ distinct potentials. Finally, the asymptotics of the spectrum are determined as $q\to 0$.

Abstract:
For 1D Dirac operators Ly= i J y' + v y, where J is a diagonal 2x2 matrix with entrees 1,-1 and v(x) is an off-diagonal matrix with L^2 [0,\pi]-entrees P(x), Q(x) we characterize the class X of pi-periodic potentials v such that: (i) the smoothness of potentials v is determined only by the rate of decay of related spectral gaps gamma (n) = | \lambda (n,+) - \lambda (n,-)|, where \lambda (..) are the eigenvalues of L=L(v) considered on [0,\pi] with periodic (for even n) or antiperiodic (for odd n) boundary conditions (bc); (ii) there is a Riesz basis which consists of periodic (or antiperiodic) eigenfunctions and (at most finitely many) associated functions. In particular, X contains symmetric potentials X_{sym} (\overline{Q} =P), skew-symmetric potentials X_{skew-sym} (\overline{Q} =-P), or more generally the families X_t defined for real nonzero t by \overline{Q} =t P. Finite-zone potentials belonging to X_t are dense in X_t. Another example: if P(x)=a exp(2ix)+b exp(-2ix), Q(x)=Aexp(2ix)+Bexp(-2ix) with complex a, b, A, B \neq 0, then the system of root functions of L consists eventually of eigenfunctions. For antiperiodic bc this system is a Riesz basis if |aA|=|bB| (then v \in X), and it is not a basis if |aA| \neq |bB|. For periodic bc the system of root functions is a Riesz basis (and v \in X) always.

Abstract:
We present a systematic treatment of the bound state structure of a short-range attractive interatomic potential in one, two, and three dimensions as its range approaches zero. This allows the evaluation of the utility of delta function potentials in the modeling of few-body systems such as nuclei, atoms, and clusters. The relation to scattering by delta function potentials is also discussed.