arXiv subjects

Didier Robert

Publications and source records attributed to Didier Robert.

24 records · Page 2Linked to original sources

arXiv2009

Asymptotic expansion for nonlinear eigenvalue problems

In this paper we consider generalized eigenvalue problems for a family of operators with a quadratic dependence on a complex parameter. Our model is $L(λ)=-\triangle +(P(x)-λ)^2$ in $L^2(\R^d)$ where $P$ is a positive elliptic polynomial in $\R^d$ of degree $m\geq 2$. It is known that for $d$ even, or $d=1$, or $d=3$ and $m\geq 6$, there exist $λ\in\C$ and $u\in L^2(\R^d)$, $u\neq 0$, such that $L(λ)u=0$. In this paper, we give a method to prove existence of non trivial solutions for the equation $L(λ)u=0$, valid in every dimension. This is a partial answer to a conjecture in \cite{herowa}.

math-ph↗

arXiv2008

Supersymmetry vs ghosts

We consider the simplest nontrivial supersymmetric quantum mechanical system involving higher derivatives. We unravel the existence of additional bosonic and fermionic integrals of motion forming a nontrivial algebra. This allows one to obtain the exact solution both in the classical and quantum cases. The supercharges $Q, \bar Q$ are not anymore Hermitially conjugate to each other, which allows for the presence of negative energies in the spectrum. We show that the spectrum of the Hamiltonian is unbounded from below. It is discrete and infinitely degenerate in the free oscillator-like case and becomes continuous running from $-\infty$ to $\infty$ when interactions are added. Notwithstanding the absence of the ground state, there is no collapse, which suggests that a unitary evolution operator may be defined.

math-ph↗

arXiv2005

Quadratic Quantum Hamiltonians revisited

Time dependent quadratic Hamiltonians are well known as well in classical mechanics and in quantum mechanics. In particular for them the correspondance between classical and quantum mechanics is exact. But explicit formulas are non trivial (like the Mehler formula). Moreover, a good knowlege of quadratic Hamiltonians is very useful in the study of more general quantum Hamiltonians and associated Schrödinger equations in the semiclassical regime. Our goal here is to give our own presentation of this important subject. We put emphasis on computations with Gaussian coherent states. Our main motivation to do that is application concerning revivals and Loschmidt echo.

math-ph↗

arXiv2004

Non linear eigenvalue problems

In this paper we consider generalized eigenvalue problems for a family of operators with a polynomial dependence on a complex parameter. This problem is equivalent to a genuine non self-adjoint operator. We discuss here existence of non trivial eigenstates for models coming from analytic theory of smoothness for P.D.E. We shall review some old results and present recent improvements on this subject.

math-ph↗

arXiv2003

Semiclassical analysis of a nonlinear eigenvalue problem and non analytic hypoellipticity

We give a semiclassical analysis of a nonlinear eigenvalue problem arising from the study of the failure of analytic hypoellipticity and obtain a general family of hypoelliptic, but not analytic hypoelliptic operators.

math.AP↗

arXiv2002

Semiclassical results in the linear response theory

We consider a quantum system of non-interacting fermions at temperature T, in the framework of linear response theory. We show that semiclassical theory is an appropriate framework to describe some of their thermodynamic properties, in particular through asymptotic expansions in $\hbar$ (Planck constant) of the dynamical susceptibilities. We show how the closed orbits of the classical motion in phase space manifest themselves in these expansions, in the regime where T is of the order of $\hbar$.

math-ph↗