arXiv subjects

I. Nenciu

Publications and source records attributed to I. Nenciu.

4 recordsLinked to original sources

arXiv2011

On essential self-adjointness for magnetic Schroedinger and Pauli operators on the unit disc in R^2

We study the question of magnetic confinement of quantum particles on the unit disk $\ID$ in $\IR^2$, i.e. we wish to achieve confinement solely by means of the growth of the magnetic field $B(\vec x)$ near the boundary of the disk. In the spinless case we show that $B(\vec x)\ge \frac{\sqrt 3}{2}\cdot\frac{1}{(1-r)^2}-\frac{1}{\sqrt 3}\frac{1}{(1-r)^2\ln \frac{1}{1-r}}$, for $|\vec x|$ close to 1, insures the confinement provided we assume that the non-radially symmetric part of the magnetic field is not very singular near the boundary. Both constants $\frac{\sqrt 3}{2}$ and $-\frac{1}{\sqrt 3}$ are optimal. This answers, in this context, an open question from Y. Colin de Verdière and F. Truc. We also derive growth conditions for radially symmetric magnetic fields which lead to confinement of spin 1/2 particles.

math-ph↗

arXiv2008

On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in R^n

Let $Ω$ be a bounded domain in $R^n$ with $C^2$-smooth boundary of co-dimension 1, and let $H=-Δ+V(x)$ be a Schrödinger operator on $Ω$ with potential V locally bounded. We seek the weakest conditions we can find on the rate of growth of the potential V close to the boundary which guarantee essential self-adjointness of H on $C_0^\infty(Ω)$. As a special case of an abstract condition, we add optimal logarithmic type corrections to the known condition $V(x)\geq \frac{3}{4d(x)^2}$, where $d(x)=dist(x,\partialΩ)$. The constant 1 in front of each logarithmic term in Theorem 2 is optimal. The proof is based on a refined Agmon exponential estimate combined with a well known multidimensional Hardy inequality.

math-ph↗

arXiv2005

CMV: the unitary analogue of Jacobi matrices

We discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices recently introduced by Cantero, Moral, and Velazquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, we describe the analogues of well-known properties of Jacobi matrices: foliation by co-adjoint orbits, a natural symplectic structure, algorithmic reduction to this shape, Lax representation for an integrable lattice system (Ablowitz-Ladik), and the relation to orthogonal polynomials. As offshoots of our analysis, we will construct action/angle variables for the finite Ablowitz-Ladik hierarchy and describe the long-time behaviour of this system.

math.SG↗

arXiv2004

Matrix models for circular ensembles

We describe an ensemble of (sparse) random matrices whose eigenvalues follow the Gibbs distribution for n particles of the Coulomb gas on the unit circle at inverse temperature beta. Our approach combines elements from the theory of orthogonal polynomials on the unit circle with ideas from recent work of Dumitriu and Edelman. In particular, we resolve a question left open by them: find a tri-diagonal model for the Jacobi ensemble.

math.SP↗