Related papers: Higher order Schrodinger and Hartree-Fock equation…
We consider the inverse coefficient problem of simultaneously determining the space dependent electric potential, the zero-th order coupling term and the first order coupling vector of a two-state Schr\"odinger equation in an infinite…
We study the possible vanishing order of solutions to Schrodinger equation in case the potential is a bounded function. We give an upper bound on compact manifold. We also show that this result is sharp.
We develop a modified semi-classical approach to the approximate solution of Schrodinger's equation for certain nonlinear quantum oscillations problems. At lowest order, the Hamilton-Jacobi equation of the conventional semi-classical…
Extensions of the Hartree-Fock-Bogoliubov theory are worked out which are tailored for, (i) the consistent evaluation of fluctuations and correlations and (ii) the restoration through projection of broken symmetries. For both purposes we…
There is studied problem on existence of solutions to non-homogeneous differential equation of higher even order. Similar problem arises while studying soliton and soliton-like solutions to partial differential equations of integrable type.…
We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schr\"odinger equation with initial data $u_{0}\in X$, where $X\in\{M_{2,q}^{s}(\mathbb R), H^{\sigma}(\mathbb T),…
We present a mathematically rigorous quantum-mechanical treatment of a two-dimensional nonrelativistic quantum dual theories (with oscillator and Coulomb like potentials) on a plane and compare their spectra and the sets of eigenfunctions.…
A procedure for constructing bound state potentials is given. We show that, under the natural conditions imposed on a radial eigenvalue problem, the only special cases of the general central potential, which are exactly solvable and have…
Some focusing coupled Schrodinger equations are investigated. First, existence of ground state is obtained. Second, global and non global existence of solutions are discussed via potential-well method. Finally, strong instability of…
We consider the existence of bound and ground states for a family of nonlinear elliptic systems in $\mathbb{R}^N$, which involves equations with critical power nonlinearities and Hardy-type singular potentials. The equations are coupled by…
In this article we give an overview on some recent development of Littlewood-Paley theory for Schr\"odinger operators. We extend the Littlewood-Paley theory for special potentials considered in the authors' previous work. We elaborate our…
The Hartree-Fock equation admits homogeneous states that model infinitely many particles at equilibrium. We prove their asymptotic stability in large dimensions, under assumptions on the linearised operator. Perturbations are moreover…
We establish uniqueness and radial symmetry of ground states for higher-order nonlinear Schr\"odinger and Hartree equations whose higher-order differentials have small coefficients. As an application, we obtain error estimates for…
The Legendre transform expresses dynamics of a classical system through first-order Hamiltonian equations. We consider coherent state transforms with a similar effect in quantum mechanics: they reduce certain quantum Hamiltonians to…
Using supervector fields and graded forms along a morphism, we study the geometry of ordinary differential superequations, extend the formalism of higher order Lagrangian mechanics to the graded context and prove a generalization of…
Linear electric circuits composed of inductors and capacitors can serve as analogues of tight-binding models that describe the electronic band structure of materials. This mapping provides a versatile approach for exploring topological…
Higher-order quantum theory is an extension of quantum theory where one introduces transformations whose input and output are transformations, thus generalizing the notion of channels and quantum operations. The generalization then goes…
The semi-classical regime of standing wave solutions of a Schr\"odinger equation in presence of non-constant electric and magnetic potentials is studied in the case of non-local nonlinearities of Hartree type. It is show that there exists a…
For slowly varying fields on the scale of the lightest mass the logarithm of the vacuum functional of a massive quantum field theory can be expanded in terms of local functionals satisfying a form of the Schrodinger equation, the principal…
New families of time-dependent potentials related to the parametric oscillator are introduced. This is achieved by introducing some general time-dependent operators that factorize the appropriate constant of motion (quantum invariant) of…