Profiles and Quantization of the Blow Up Mass for Critical Nonlinear Schrödinger Equation

We consider finite time blow up solutions to the critical nonlinear Schrödinger equation For a suitable class of initial data in the energy space H¹, we prove that the solution splits in two parts: the first part corresponds to the singular part and accumulates a quantized amount of L² mass at the blow up point, the second part corresponds to the regular part and has a strong L² limit at blow up time.Part of this work has been supported by grant DMS-0111298.     

Schrödinger Maps and Anti-Ferromagnetic Chains

We study the problem of restricting Schrödinger maps onto Lagrangian submanifolds. The restriction is imposed by an infinite constraining potential. We show that, in the limit, solutions of the Schrödinger map equations converge to solutions of generalized wave map equations. This result is applied to the anti-ferromagnetic systems where we prove rigorously that the dynamics is governed by wave map equations into .     

Mean-Field- and Classical Limit of Many-Body Schrödinger Dynamics for Bosons

We present a new proof of the convergence of the N ?particle Schrödinger dynamics for bosons towards the dynamics generated by the Hartree equation in the mean-field limit. For a restricted class of two-body interactions, we obtain convergence estimates uniform in \({\hbar}\) , up to an exponentially small remainder. For \({\hbar = 0}\) , the classical dynamics in the mean-field limit is given by the Vlasov equation.     

Quantum Yang-Mills-Weyl Dynamics in the Schrödinger paradigm

Inspired by F. Wilczek’s QCD Lite, quantum Yang-Mills-Weyl Dynamics (YMWD) describes quantum interaction between gauge bosons (associated with a simple gauge group \(\mathbb{G}\) ) and larks (massless chiral fields charged by an irreducible unitary representation of \(\mathbb{G}\) ). Schrödinger representation of this quantum Yang-Mills-Weyl theory is based on a sesqui-holomorphic operator calculus of infinite-dimensional operators with variational derivatives. The spectrum of quantum YMWD in a compact bag is a sequence of eigenvalues convergent to +∞. The eigenvalues have finite multiplicities with respect to a von Neumann algebra with a regular trace. The spectrum is inversely proportional to the square of the running coupling constant. The rigorous mathematical theory is nonperturbative with a running coupling constant as the only ad hoc parameter. The application of the first mathematical principles is based on the properties of the compact simple Lie group \(\mathbb{G}\) .     

Pointwise bounds for Schrödinger eigenstates

In a different paper we constructed imaginary time Schrödinger operatorsH _q=–1/2+V acting onL ^q( ⁿ ,dx). The negative part of typical potential functionV was assumed to be inL +L ^q for somep>max{1,n/2}. Our proofs were based on the evaluation of Kac's averages over Brownian motion paths. The present paper continues this study: using probabilistic techniques we prove pointwise upper bounds forL ^q-Schrödinger eigenstates and pointwise lower bounds for the corresponding groundstate. The potential functionsV are assumed to be neither smooth nor bounded below. Consequently, our results generalize Schnol's and Simon's ones. Moreover probabilistic proofs seem to be shorter and more informative than existing ones.Laboratoire de Mathématiques de Marseille associé au C.N.R.S. L.A.225     

Time Decay of Scaling Critical Electromagnetic Schrödinger Flows

We obtain a representation formula for solutions to Schrödinger equations with a class of homogeneous, scaling-critical electromagnetic potentials. As a consequence, we prove the sharp ${L^1 \to L^\infty}$ time decay estimate for the 3D-inverse square and the 2D-Aharonov–Bohm potentials.     

Lorentz Transformations for the Schrödinger Equation

Abstract

We show that the free Schrödinger equation admits Lorentz space-time transformations when corresponding transformations of the ψ-function are nonlocal. Some consequences of this symmetry are discussed.

Dedicated to Wilhelm Fushchych – Inspirer, Mentor, Friend and Pioneer in non–Lie symmetry methods – on the occasion of his sixtieth birthday     

Tunneling Estimates for Magnetic Schrödinger Operators

We study the behavior of eigenfunctions in the semiclassical limit for Schr?dinger operators with a simple well potential and a (non-zero) constant magnetic field. We prove an exponential decay estimate on the low-lying eigenfunctions, where the exponent depends explicitly on the magnetic field strength. Received: 30 March 1998 / Accepted: 1 May 1998     

B-spline solver for one-electron Schrödinger equation

A numerical algorithm for solving the one-electron Schrödinger equation is presented. The algorithm is based on the Finite Element method, and the basis functions are tensor products of univariate B-splines. The application of cubic or higher order B-splines guarantees that the searched solution belongs to a continuous and one time differentiable function space, which is a desirable property in the Kohn–Sham equation context from the Density Functional Theory with pseudopotential approximation. The theoretical background of the numerical algorithm is presented, and additionally, the implementation on parallel computers with distributed memory is described. The current implementation of the algorithm uses the MPI, HYPRE and ParMETIS libraries to distribute matrices on processing units. Additionally, the LOBPCG algorithm from HYPRE library is used to solve the algebraic generalized eigenvalue problem. The proposed algorithm works for any smooth interaction potential, where the domain of the problem is a finite subspace of the ?³ space. The accuracy of the algorithm is demonstrated for a selected interaction potential. In the current stage, the algorithm can be applied to solve the linearized Kohn–Sham equation for molecular systems.     

Inverse Spectral Problems for Schrödinger Operators

In this article we find some explicit formulas for the semi-classical wave invariants at the bottom of the well of a Schrödinger operator. As an application of these new formulas for the wave invariants, we improve the inverse spectral results proved by Guillemin and Uribe in [GU]. They proved that under some symmetry assumptions on the potential V(x), the Taylor expansion of V(x) near a non-degenerate global minimum can be recovered from the knowledge of the low-lying eigenvalues of the associated Schrödinger operator in \({\mathbb R^n}\) . We prove similar inverse spectral results using fewer symmetry assumptions. We also show that in dimension 1, no symmetry assumption is needed to recover the Taylor coefficients of V(x).     

A Liouville-type Theorem for Schrödinger Operators

In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a symmetric critical operator P ₁, such that a nonzero subsolution of a symmetric nonnegative operator P ₀ is a ground state. Particularly, if P _j : = ?Δ + V _j , for j = 0,1, are two nonnegative Schrödinger operators defined on \(\Omega\subseteq \mathbb{R}^d\) such that P ₁ is critical in Ω with a ground state φ, the function \(\psi\nleq 0\) is a subsolution of the equation P ₀ u = 0 in Ω and satisfies \(\psi_+\leq C\varphi\) in Ω, then P ₀ is critical in Ω and \(\psi\) is its ground state. In particular, \(\psi\) is (up to a multiplicative constant) the unique positive supersolution of the equation P ₀ u = 0 in Ω. Similar results hold for general symmetric operators, and also on Riemannian manifolds.     

Critical assessment of the Schrödinger picture of quantum mechanics

We provide an example in which the Heisenberg and the Schrödinger pictures of quantum mechanics give different results, thus confirming the statement of P.A.M. Dirac that the two pictures may lead to inequivalent results. We consider a one-dimensional nonrelativistic charged harmonic oscillator (frequency ω₀ and mass m), and take into account the action of the radiation reaction and the vacuum electromagnetic forces on the charged oscillator. We show that the Heisenberg picture gives the correct value, ℏω₀/2, for the ground state energy of the harmonic oscillator in both cases of classical and quantized vacuum fields. In the case of the Schrödinger picture, considering classical vacuum fields, and using a simple calculation for the classical radiation reaction force that is valid in the limit of large mass (mc²⪢ℏω₀), we obtain the value ℏω₀ for the ground state energy of the harmonic oscillator. We show that the vacuum electromagnetic forces play a very important role in the understanding of this discrepancy.     

The Scaling Limit of the Critical One-Dimensional Random Schrödinger Operator

We consider two models of one-dimensional discrete random Schrödinger operators

$(H_n\psi)_\ell =\psi_{\ell -1}+\psi_{\ell +1}+v_\ell \psi_\ell$

, \({\psi_0=\psi_{n+1}=0}\) in the cases \({ v_k=\sigma \omega_k/\sqrt{n}}\) and \({ v_k=\sigma \omega_k/ \sqrt{k}}\) . Here ω _k are independent random variables with mean 0 and variance 1.

We show that the eigenvectors are delocalized and the transfer matrix evolution has a scaling limit given by a stochastic differential equation. In both cases, eigenvalues near a fixed bulk energy E have a point process limit. We give bounds on the eigenvalue repulsion, large gap probability, identify the limiting intensity and provide a central limit theorem.In the second model, the limiting processes are the same as the point processes obtained as the bulk scaling limits of the β-ensembles of random matrix theory. In the first model, the eigenvalue repulsion is much stronger.     

Colliding Solitons for the Nonlinear Schrödinger Equation

We study the collision of two fast solitons for the nonlinear Schrödinger equation in the presence of a slowly varying external potential. For a high initial relative speed ||v|| of the solitons, we show that, up to times of order ||v|| after the collision, the solitons preserve their shape (in L ²-norm), and the dynamics of the centers of mass of the solitons is approximately determined by the external potential, plus error terms due to radiation damping and the extended nature of the solitons. We remark on how to obtain longer time scales under stronger assumptions on the initial condition and the external potential.