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Andrea Posilicano

Publications and source records attributed to Andrea Posilicano.

51 records · Page 3Linked to original sources
arXiv2008

Self-adjoint Extensions of Restrictions

We provide a simple recipe for obtaining all self-adjoint extensions, together with their resolvent, of the symmetric operator $S$ obtained by restricting the self-adjoint operator $A:\D(A)\subseteq\H\to\H$ to the dense, closed with respect to the graph norm, subspace $\N\subset \D(A)$. Neither the knowledge of $S^*$ nor of the deficiency spaces of $S$ is required. Typically $A$ is a differential operator and $\N$ is the kernel of some trace (restriction) operator along a null subset. We parametrise the extensions by the bundle $π:\E(\fh)\to¶(\fh)$, where $¶(\fh)$ denotes the set of orthogonal projections in the Hilbert space $\fh\simeq \D(A)/\N$ and $π^{-1}(Π)$ is the set of self-adjoint operators in the range of $Π$. The set of self-adjoint operators in $\fh$, i.e. $π^{-1}(1)$, parametrises the relatively prime extensions. Any $(Π,Θ)\in \E(\fh)$ determines a boundary condition in the domain of the corresponding extension $A_{Π,Θ}$ and explicitly appears in the formula for the resolvent $(-A_{Π,Θ}+z)^{-1}$. The connection with both von Neumann's and Boundary Triples theories of self-adjoint extensions is explained. Some examples related to quantum graphs, to Schrödinger operators with point interactions and to elliptic boundary value problems are given.

math-ph↗
arXiv2005

Rigorous Dynamics and Radiation Theory for a Pauli-Fierz Model in the Ultraviolet Limit

The present paper is devoted to the detailed study of quantization and evolution of the point limit of the Pauli-Fierz model for a charged oscillator interacting with the electromagnetic field in dipole approximation. In particular, a well defined dynamics is constructed for the classical model, which is subsequently quantized according to the Segal scheme. To this end, the classical model in the point limit is reformulated as a second order abstract wave equation, and a consistent quantum evolution is given. This allows a study of the behaviour of the survival and transition amplitudes for the process of decay of the excited states of the charged particle, and the emission of photons in the decay process. In particular, for the survival amplitude the exact time behaviour is found. This is completely determined by the resonances of the systems plus a tail term prevailing in the asymptotic, long time regime. Moreover, the survival amplitude exhibites in a fairly clear way the Lamb shift correction to the unperturbed frequencies of the oscillator.

math-ph↗
arXiv2005

Singular Perturbations of Abstract Wave equations

Given, on the Hilbert space $\H_0$, the self-adjoint operator $B$ and the skew-adjoint operators $C_1$ and $C_2$, we consider, on the Hilbert space $\H\simeq D(B)\oplus\H_0$, the skew-adjoint operator $$W=[\begin{matrix} C_2&\uno -B^2&C_1\end{matrix}]$$ corresponding to the abstract wave equation $\ddotϕ-(C_1+C_2)\dotϕ=-(B^2+C_1C_2)ϕ$. Given then an auxiliary Hilbert space $\fh$ and a linear map $τ:D(B^2)\to\fh$ with a kernel $\K$ dense in $\H_0$, we explicitly construct skew-adjoint operators $W_Θ$ on a Hilbert space $\H_Θ\simeq D(B)\oplus\H_0\oplus \fh$ which coincide with $W$ on $\N\simeq\K\oplus D(B)$. The extension parameter $Θ$ ranges over the set of positive, bounded and injective self-adjoint operators on $\fh$. In the case $C_1=C_2=0$ our construction allows a natural definition of negative (strongly) singular perturbations $A_Θ$ of $A:=-B^2$ such that the diagram $$ \begin{CD} W @>>> W_Θ@AAA @VVV A@>>> A_Θ\end{CD} $$ is commutative.

math.FA↗
arXiv2005

Wave equation with concentrated nonlinearities

In this paper we address the problem of wave dynamics in presence of concentrated nonlinearities. Given a vector field $V$ on an open subset of $\CO^n$ and a discrete set $Y\subset\RE^3$ with $n$ elements, we define a nonlinear operator $Δ_{V,Y}$ on $L^2(\RE^3)$ which coincides with the free Laplacian when restricted to regular functions vanishing at $Y$, and which reduces to the usual Laplacian with point interactions placed at $Y$ when $V$ is linear and is represented by an Hermitean matrix. We then consider the nonlinear wave equation $\ddot ϕ=Δ_{V,Y}ϕ$ and study the corresponding Cauchy problem, giving an existence and uniqueness result in the case $V$ is Lipschitz. The solution of such a problem is explicitly expressed in terms of the solutions of two Cauchy problem: one relative to a free wave equation and the other relative to an inhomogeneous ordinary differential equation with delay and principal part $\dotζ+V(ζ)$. Main properties of the solution are given and, when $Y$ is a singleton, the mechanism and details of blow-up are studied.

math-ph↗
arXiv2004

Asymptotic flux across hypersurfaces for diffusion processes

We suggest a rigorous definition of the pathwise flux across the boundary of a bounded open set for transient finite energy diffusion processes. The expectation of such a flux has the property of depending only on the current velocity $v$, the nonsymmetric (as regards time reversibility) part of the drift. In the case the diffusion has a limiting velocity we then define the asymptotic (as $R\uparrow\infty$) flux across subsets of the sphere of radius $R$ and compute its expectation, again in terms of $v$.

math.PR↗
arXiv2003

Convergence of symmetric diffusions on Wiener spaces

We prove convergence of symmetric diffusions on Wiener spaces by using stopping times arguments and capacity techniques. The drifts of the diffusions can be singular, we require the densities of the processes to be neither bounded from above nor away from zero.

math.PR↗
arXiv2003

Self-Adjoint Extensions by Additive Perturbations

Let $A_\N$ be the symmetric operator given by the restriction of $A$ to $\N$, where $A$ is a self-adjoint operator on the Hilbert space $\H$ and $\N$ is a linear dense set which is closed with respect to the graph norm on $D(A)$, the operator domain of $A$. We show that any self-adjoint extension $A_Θ$ of $A_\N$ such that $D(A_Θ)\cap D(A)=\N$ can be additively decomposed by the sum $A_Θ=\A+T_Θ$, where both the operators $\A$ and $T_Θ$ take values in the strong dual of $D(A)$. The operator $\A$ is the closed extension of $A$ to the whole $\H$ whereas $T_Θ$ is explicitly written in terms of a (abstract) boundary condition depending on $\N$ and on the extension parameter $Θ$, a self-adjoint operator on an auxiliary Hilbert space isomorphic (as a set) to the deficiency spaces of $A_\N$. The explicit connection with both Kre\uın's resolvent formula and von Neumann's theory of self-adjoint extensions is given.

math.FA↗
arXiv2002

Boundary Conditions for Singular Perturbations of Self-Adjoint Operators

Let $A:D(A)\subseteq\H\to\H$ be an injective self-adjoint operator and let $τ:D(A)\to\X$, X a Banach space, be a surjective linear map such that $\|τϕ\|_\X\le c \|Aϕ\|_\H$. Supposing that \text{\rm Range}$ (τ')\cap\H' =\{0\}$, we define a family $A^τ_Θ$ of self-adjoint operators which are extensions of the symmetric operator $A_{|\{τ=0\}.}$. Any $ϕ$ in the operator domain $D(A^τ_Θ)$ is characterized by a sort of boundary conditions on its univocally defined regular component $\phireg$, which belongs to the completion of D(A) w.r.t. the norm $\|Aϕ\|_\H$. These boundary conditions are written in terms of the map $τ$, playing the role of a trace (restriction) operator, as $τ\phireg=ΘQ_ϕ$, the extension parameter $Θ$ being a self-adjoint operator from X' to X. The self-adjoint extension is then simply defined by $A^τ_Θϕ:=A \phireg$. The case in which $Aϕ=T*ϕ$ is a convolution operator on LD, T a distribution with compact support, is studied in detail.

math.FA↗
arXiv2001

Wave Equations with Point Interactions in Finite Energy Space

Given the abstract wave equation $\ddotϕ-Δ_αϕ=0$, where $Δ_α$ is the Laplace operator with a point interaction of strength $α$, we define and study $\bar W_α$, the associated wave generator in the phase space of finite energy states. We prove the existence of the phase flow generated by $\bar W_α$, and describe its most relevant properties with particular emphasis on the associated symplectic structure and scattering theory

math-ph↗
arXiv2000

A Krein-Like Formula for Singular Perturbations of Self-Adjoint Operators and Applications

Given a self-adjoint operator $A:D(A)\subseteq\calH\to\calH$ and a continuous linear operator $τ:D(A)\to\X$ with Range$ τ'\cap\calH' ={0}$, $\X$ a Banach space, we explicitly construct a family $A^τ_Θ$ of self-adjoint operators such that any $A^τ_Θ$ coincides with the original $A$ on the kernel of $τ$. Such a family is obtained by giving a Kre\uın-like formula where the role of the deficiency spaces is played by the dual pair $(\X,\X')$; the parameter $Θ$ belongs to the space of symmetric operators from $\X'$ to $\X$. When $\X=\C$ one recovers the ``$\calH_{-2}$ -construction'' of Kiselev and Simon and so, to some extent, our results can be regarded as an extension of it to the infinite rank case. Considering the situation in which $\calH=L^2(\RE^n)$ and $τ$ is the trace (restriction) operator along some null subset, we give various applications to singular perturbations of non necessarily elliptic pseudo-differential operators, thus unifying and extending previously known results.

math.FA↗
arXiv1999

Delta Interactions and Electrodynamics of Point Particles

We report on some recent work of the authors showing the relations between singular (point) perturbation of the Laplacian and the dynamical system describing a charged point particle interacting with the self-generated radiation field (the Maxwell-Lorentz system) in the dipole approximation. We show that in the limit of a point particle, the dynamics of the system is described by an abstract wave equation containing a selfadjoint operator $H_m$ of the class of point interactions; the classical Abraham-Lorentz-Dirac third order equation, or better its integrated second order version, emerges as the evolution equation of the singular part of the field and is related to the boundary conditions entering in the definition of the operator domain of $H_m$. We also give the Hamiltonian structure of the limit model and, in the case of no external force, we study the reduced dynamics on the linear stable manifold.

math-ph↗