Related papers: Recursive Schr\" odinger Equation Approach to Fast…
We introduce the notion of Schr\"odinger integral operators and prove sharp local and global regularity results for these (including propagators for the quantum mechanical harmonic oscillator). Furthermore we introduce general classes of…
With growing demand for time-domain simulations of correlated many-body systems, the development of efficient and stable integration schemes for the time-dependent Schr\"odinger equation is of keen interest in modern electronic structure…
In this paper, we obtain pointwise convergence of solutions to the Schrodinger equation along a class of curves in $\mathbb{R}^{2}$ by the polynomial partitioning.
By their very nature, field-theoretical Hamiltonians are derived in momentum representation. To solve the corresponding integro-differential equations is more difficult than to solve the simpler differential equations in configuration space…
We show trilinear Strichartz estimates in one and two dimensions on frequency-dependent time intervals. These improve on the corresponding linear estimates of periodic solutions to the Schr\"odinger equation. The proof combines decoupling…
The projection formalism for calculating effective Hamiltonians and resonances is generalized to the nonlocal and/or nonhermitian case, so that it is applicable to the reduction of relativistic systems (Bethe-Salpeter equations), and to…
Quantization techniques have been applied in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and efficient calibration of large…
The $O(N)$ stochastic propagation method, which relies on the numerical solution of the time-dependent Schr\"odinger equation using random initial states, is widely used in large-scale first-principles calculations. In this work, we…
We derive an effective equation for the dynamics of many identical bosons in dimension one in the presence of a tiny impurity. The interaction between every pair of bosons is mediated by the impurity through a positive three-body potential.…
The Schr\"odinger equation in the presence of an external electromagnetic field is an important problem in computational quantum mechanics. It also provides a nice example of a differential equation whose flow can be split with benefit into…
We introduce a simple and stable computational method for ill-posed partial differential equation (PDE) problems. The method is based on Schr\"odingerization, introduced in [S. Jin, N. Liu and Y. Yu, arXiv:2212.13969][S. Jin, N. Liu and Y.…
We establish sharp-in-time kernel and dispersive estimates for the Schr\"odinger equation on non-compact Riemannian symmetric spaces of any rank. Due to the particular geometry at infinity and the Kunze-Stein phenomenon, these properties…
We introduce an algorithm for the solution of a system of radial Schr\"odinger equations describing the inelastic scattering of particles with spin in a partial wave with definite total angular momentum. The system of differential equations…
We analyze the preservation properties of a family of reversible splitting methods when they are applied to the numerical time integration of linear differential equations defined in the unitary group. The schemes involve complex…
The quantum dynamics of an ensemble of interacting electrons in an array of random scatterers is treated using a new numerical approach for the calculation of average values of quantum operators and time correlation functions in the Wigner…
We present and discuss a detailed derivation of a new analytical method that systematically improves the convergence of path integrals of a generic $N$-fold discretized theory. We develop an explicit procedure for calculating a set of…
Background: Path integrals are a powerful tool for solving problems in quantum theory that are not amenable to a treatment by perturbation theory. Most path integral computations require an analytic continuation to imaginary time. While…
Integrable fractional equations such as the fractional Korteweg-deVries and nonlinear Schr\"odinger equations are key to the intersection of nonlinear dynamics and fractional calculus. In this manuscript, the first discrete/differential…
A solution $\psi $ to Schr\"odinger's equation needs some degree of regularity in order to allow the construction of a Bohmian mechanics from the integral curves of the velocity field $\hbar \Im \left( \bigtriangledown \psi /m\psi \right)…
Small-amplitude weakly coupled oscillators of the Klein-Gordon lattices are approximated by equations of the discrete nonlinear Schrodinger type. We show how to justify this approximation by two methods, which have been very popular in the…