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We consider the cubic nonlinear Schr\"odinger equation with an exceptional potential. We obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions. We prove the existence of…
Two-, three-, and four-boson systems are studied close to the unitary limit using potential models constructed to reproduce the minimal information given by the two-body scattering length $a$ and the two-body binding energy or virtual state…
We consider a class of biharmonic nonlinear Schr\"odinger equations with a focusing inhomogeneous power-type nonlinearity \[ i\partial_t u -\Delta^2 u+\mu\Delta u +|x|^{-b} |u|^\alpha u=0, \quad \left. u\right|_{t=0}=u_0 \in…
An analytical solution of the perturbed equations is obtained, which exists in all ergodic models of collisionless spherical stellar systems with a single length parameter. This solution corresponds to variations of this parameter, i.e.,…
We prove that an $L^\infty$ potential in the Schr\"odinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace…
The quantum behavior of electrons in bilayer graphene with applied magnetic fields is addressed. By using second-order supersymmetric quantum mechanics the problem is transformed into two intertwined one dimensional stationary Schr\"odinger…
We examine the minimization of a quadratic cost functional composed of the output and the final state of abstract infinite-dimensional evolution equations in view of existence of solutions and optimality conditions. While the initial value…
These notes present preliminary results regarding two different approximations of linear infinite-horizon optimal control problems arising in model predictive control. Input and state trajectories are parametrized with basis functions and a…
In this paper we consider integrable dispersive chains associated with the so called Energy Dependent Schrodinger operator. In a general case multi component reductions of these dispersive chains are new integrable systems, which are…
We consider time-dependent perturbations which are relatively bounded with respect to the square root of an unperturbed Hamiltonian operator, and whose commutator with the latter is controlled by the full perturbed Hamiltonian. The…
We explore the quantum Coulomb problem for two-body bound states, in $D=3$ and $D=3-2\epsilon$ dimensions, in detail, and give an extensive list of expectation values that arise in the evaluation of QED corrections to bound state energies.…
We present a set of smooth infinite energy global solutions (without spatial symmetry) to the non-integrable, nonlinear Schr\"odinger equations on $\Bbb R$. These solutions are space-time quasi-periodic with two frequencies each. Previous…
A control problem with terminal overdetermination is considered for the higher order nonlinear Schr\"odinger equation on a bounded interval. The boundary condition on the space derivative is chosen as the control. Results on global…
We study the nonlinear Schrodinger equations with a linear potential. A change of variables makes it possible to deduce results concerning finite time blow up and scattering theory from the case with no potential.
We consider the inverse problem of determining the time independent scalar potential of the dynamic Schr\"odinger equation in an infinite cylindrical domain, from one Neumann boundary observation of the solution. Assuming that this…
The purpose of this Comment is to show that the solutions to the zero energy Schr\"odinger equations for monomial central potentials discussed in a recently published Letter, may also be obtained from the corresponding free particle…
Bound states of the generalized spiked harmonic oscillator potential are calculated accurately by using the generalized pseudospectral method. Energy eigenvalues, various expectation values, radial densities are obtained through a…
We consider the Schrodinger equation with a logarithmic nonlinearity and a repulsive harmonic potential. Depending on the parameters of the equation, the solution may or may not be dispersive. When dispersion occurs, it does with an…
We present a scalable tensor-based approach to computing input-normal/output-diagonal nonlinear balancing transformations for control-affine systems with polynomial nonlinearities. This transformation is necessary to determine the states…
We construct a generalized-gradient approximation for the exchange-energy density of finite two-dimensional systems. Guided by non-empirical principles, we include the proper small-gradient limit and the proper tail for the exchange-hole…