Searcharxiv⌕ Search

arXiv subjects

Gheorghe Nenciu

Publications and source records attributed to Gheorghe Nenciu.

13 recordsLinked to original sources
arXiv2024

Effective Dynamics of Translationally Invariant Magnetic Schrödinger Equations in the High Field Limit

We study the large field limit in Schrödinger equations with magnetic vector potentials describing translationally invariant $B$-fields with respect to the $z$-axis. In a first step, using regular perturbation theory, we derive an approximate description of the solution, provided the initial data is compactly supported in the Fourier-variable dual to $z\in \mathbb R$. The effective dynamics is thereby seen to produce high-frequency oscillations and large magnetic drifts. In a second step we show, by using the theory of almost invariant subspaces, that this asymptotic description is stable under polynomially bounded perturbations that vanish in the vicinity of the origin.

math-ph↗
arXiv2022

Weak anisotropic Hardy inequality: essential self-adjointness of drift-diffusion operators on domains in $\mathbb{R}^d$, revisited

We consider the problem of essential self-adjointness of the drift-diffusion operator $H=-\frac{1}ρ\nabla\cdot ρ\mathbb D\nabla +V$ on domains $Ω\subset \mathbb{R}^d$ with $\mathcal{C}^2$-boundary $\partial Ω$ and for large classes of coefficients $ρ,\; \mathbb{D}$ and $V$. We give criteria showing how the behavior as $x \rightarrow \partial Ω$ of these coefficients balances to ensure essential self-adjointness of $H$. On the way we prove a weak anisotropic Hardy inequality which is of independent interest.

math-ph↗
arXiv2021

Essential self-adjointness of symmetric first-order differential systems and confinement of Dirac particles on bounded domains in $\mathbb{R}^d$

We prove essential self-adjointness of Dirac operators with Lorentz scalar potentials which grow sufficiently fast near the boundary $\partialΩ$ of the spatial domain $Ω\subset\mathbb R^d$. On the way, we first consider general symmetric first order differential systems, for which we identify a new, large class of potentials, called scalar potentials, ensuring essential self-adjointness. Furthermore, using the supersymmetric structure of the Dirac operator in the two dimensional case, we prove confinement of Dirac particles, i.e. essential self-adjointness of the operator, solely by magnetic fields $\mathcal{B}$ assumed to grow, near $\partialΩ$, faster than $1/\big(2\text{dist} (x, \partialΩ)^2\big)$.

math-ph↗
arXiv2018

On essential self-adjointness for first order differential operators on domains in $\mathbb{R}^d$

We consider general symmetric systems of first order linear partial differential operators on domains $Ω\subset \mathbb{R}^d$, and we seek sufficient conditions on the coefficients which ensure essential self-adjointness. The coefficients of the first order terms are only required to belong to $C^1(Ω)$ and there is no ellipticity condition. Our criterion writes as the completeness of an associated Riemannian structure which encodes the propagation velocities of the system. As an application we obtain sufficient conditions for confinement of energy for some wave propagation problems of classical physics.

math-ph↗
arXiv2017

Drift-diffusion equations on domains in $\mathbb{R}^d$: essential self-adjointness and stochastic completeness

We consider the problem of quantum and stochastic confinement for drift-diffusion equations on domains $ Ω\subset \mathbb R^d$. We obtain various sufficient conditions on the behavior of the coefficients near the boundary of $Ω$ which ensure the essential self-adjointness or stochastic completeness of the symmetric form of the drift-diffusion operator, $-\frac{1}{ρ_\infty}\,\nabla\cdot ρ_\infty\mathbb D\nabla$. The proofs are based on the method developed in [29] for quantum confinement on bounded domains in $\mathbb R^d$. In particular for stochastic confinement we combine the Liouville property with Agmon type exponential estimates for weak solutions.

math-ph↗
arXiv2017

On the adiabatic theorem when eigenvalues dive into the continuum

We consider a reduced two-channel model of an atom consisting of a quantum dot coupled to an open scattering channel described by a three-dimensional Laplacian. We are interested in the survival probability of a bound state when the dot energy varies smoothly and adiabatically in time. The initial state corresponds to a discrete eigenvalue which dives into the continuous spectrum and re-emerges from it as the dot energy is varied in time and finally returns to its initial value. Our main result is that for a large class of couplings, the survival probability of this bound state vanishes in the adiabatic limit. At the end of the paper we present a short outlook on how our method may be extended to cover other classes of Hamiltonians; details will be given elsewhere.

math-ph↗
arXiv2016

On the construction of composite Wannier functions

We give a constructive proof for the existence of an $N$-dimensional Bloch basis which is both smooth (real analytic) and periodic with respect to its $d$-dimensional quasi-momenta, when $1\leq d\leq 2$ and $N\geq 1$. The constructed Bloch basis is conjugation symmetric when the underlying projection has this symmetry, hence the corresponding exponentially localized composite Wannier functions are real. In the second part of the paper we show that by adding a weak, globally bounded but not necessarily constant magnetic field, the existence of a localized basis is preserved.

math-ph↗
arXiv2014

Metastable states when the Fermi Golden Rule constant vanishes

Resonances appearing by perturbation of embedded non-degenerate eigenvalues are studied in the case when the Fermi Golden Rule constant vanishes. Under appropriate smoothness properties for the resolvent of the unperturbed Hamiltonian, it is proved that the first order Rayleigh-Schrödinger expansion exists. The corresponding metastable states are constructed using this truncated expansion. We show that their exponential decay law has both the decay rate and the error term of order $\varepsilon^4$, where $\varepsilon$ is the perturbation strength.

math-ph↗
arXiv2012

Memory effects in non-interacting mesoscopic transport

Consider a quantum dot coupled to two semi-infinite one-dimensional leads at thermal equilibrium. We turn on adiabatically a bias between the leads such that there exists exactly one discrete eigenvalue both at the beginning and at the end of the switching procedure. It is shown that the expectation on the final bound state strongly depends on the history of the switching procedure. On the contrary, the contribution to the final steady-state corresponding to the continuous spectrum has no memory, and only depends on the initial and final values of the bias.

math-ph↗
arXiv2010

Faraday effect revisited: sum rules and convergence issues

This is the third paper of a series revisiting the Faraday effect. The question of the absolute convergence of the sums over the band indices entering the Verdet constant is considered. In general, sum rules and traces per unit volume play an important role in solid state physics, and they give rise to certain convergence problems widely ignored by physicists. We give a complete answer in the case of smooth potentials and formulate an open problem related to less regular perturbations.

math-ph↗
arXiv2007

Adiabatically switched-on electrical bias in continuous systems, and the Landauer-Buttiker formula

Consider a three dimensional system which looks like a cross-connected pipe system, i.e. a small sample coupled to a finite number of leads. We investigate the current running through this system, in the linear response regime, when we adiabatically turn on an electrical bias between leads. The main technical tool is the use of a finite volume regularization, which allows us to define the current coming out of a lead as the time derivative of its charge. We finally prove that in virtually all physically interesting situations, the conductivity tensor is given by a Landauer-B{ü}ttiker type formula.

cond-mat.mes-hall↗
arXiv2007

Schrödinger operators on the half line: Resolvent expansions and the Fermi golden rule at thresholds

We consider Schrödinger operators $H=- \d^2/\d r^2+V$ on $L^2([0,\infty))$ with the Dirichlet boundary condition. The potential $V$ may be local or non-local, with polynomial decay at infinity. The point zero in the spectrum of $H$ is classified, and asymptotic expansions of the resolvent around zero are obtained, with explicit expressions for the leading coefficients. These results are applied to the perturbation of an eigenvalue embedded at zero, and the corresponding modified form of the Fermi golden rule.

math-ph↗