Andrea Posilicano

Università degli Studi dell'Insubria, Como · Department of Science & High Technology

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...

Given a self-adjoint operator A: D(A)⊆→ and a continuous linear operator τ: D(A)→ with Range τ′∩′={0}, a Banach space, we explicitly construct a family AτΘ of self-adjoint operators such that any AτΘ coincides with the original A on the kernel of τ. Such a family is obtained by giving a Kreın-like formula where the role of the deficiency spaces is...

We provide an asymptotic completeness criterion and a representation formula for the scattering matrix of the scattering couple (A_{\mathsf B},A) , where both A and A_{\mathsf B} are self-adjoint operator and A_{\mathsf B} formally corresponds to adding to A two terms, one regular and the other singular. In particular, our abstract results apply to...

Given an open, bounded and connected set $$\Omega \subset \mathbb {R}^{3}$$ Ω ⊂ R 3 and its rescaling $$\Omega _{\varepsilon }$$ Ω ε of size $$\varepsilon \ll 1$$ ε ≪ 1 , we consider the solutions of the Cauchy problem for the inhomogeneous wave equation $$\begin{aligned} (\varepsilon ^{-2}\chi _{\Omega _{\varepsilon }}+\chi _{\mathbb {R}^{3}\backs...

We present a much shorter and streamlined proof of an improved version of the results previously given in [A. Posilicano: On the Self-Adjointness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemarg...

We provide a detailed study of the spectral properties of the linear operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(\varepsilon )=-(\varepsilon ^{2}\chi _{\...

We provide a detailed study of the spectral properties of the linear operator $H(\varepsilon)=-(\varepsilon^{2}\chi_{\Omega_{\varepsilon}}+\chi_{\Omega^{c}_{\varepsilon}})\Delta$ modeling, through the wave equation $(\partial_{tt}+H(\varepsilon))u=0$, the dynamics of acoustic waves in the presence of a small inhomogeneity of size $\varepsilon$ havi...

Given an open, bounded and connected set Ω ⊂ R 3 and its rescaling Ω ε of size ε 1, we consider the solutions of the Cauchy problem for the inhomogeneous wave equation (ε −2 χ Ωε + χ R 3 \Ωε)∂ tt u = ∆u + f with initial data and source supported outside Ω ε ; here, χ S denotes the characteristic function of a set S. We provide the first-order ε-cor...

We present a much shorter and streamlined proof of the results previously given in [A. Posilicano: On the Self-Adjointness of $H+A^{*}+A$, Math. Phys. Anal. Geom. (2020)] concerning the self-adjoint realizations of formal QFT-like Hamiltonians of the kind $H+A^{*}+A$, where $H$ and $A$ play the role of the free field Hamiltonian and of the annihila...

Let ${\mathsf D}$ and ${\mathsf H}$ be the self-adjoint, one-dimensional Dirac and Schr\"odinger operators in $L^{2}(\mathbb{R};\mathbb{C}^{2})$ and $L^{2}(\mathbb{R};\mathbb{C})$ respectively. It is well known that, in absence of an external potential, the two operators are related through the equality ${\mathsf D}^2 = ({\mathsf H} + \frac{1}{4}){...

Let Δα,Y be the bounded from above self-adjoint realization in L2(R3) of the Laplacian with n point scatterers placed at Y={y1,…,yn}⊂R3, the parameters (α1,…αn)≡α∈Rn being related to the scattering properties of the obstacles. Let ufϵα,Y and ufϵ∅ denote the solutions of the wave equations corresponding to Δα,Y and to the free Laplacian Δ respective...

We provide a representation formula for the scattering matrix of the scattering couple $(A_B,A)$, where both $A$ and $A_B$ are self-adjoint operator and $A_B$ formally corresponds to adding to $A$ two terms, one regular and the other singular. In particular, our abstract results apply to the couple $(\Delta,\Delta_B)$, where $\Delta$ is the free se...

It is well known that the presence, in a homogeneous acoustic medium, of a small inhomogeneity (of size ε), enjoying a high contrast of both its mass density and bulk modulus, amplifies the generated total fields. This amplification is more pronounced when the incident frequency is close to the Minnaert frequency ωM. Here we provide an interpretati...

We consider the quantum evolution [Formula: see text] of a Gaussian coherent state [Formula: see text] localized close to the classical state [Formula: see text], where [Formula: see text] denotes a self-adjoint realization of the formal Hamiltonian [Formula: see text], with [Formula: see text] the derivative of Dirac’s delta distribution at [Formu...

It is well known that the presence, in a homogeneous acoustic medium, of a small inhomogeneity (of size $\varepsilon$), enjoying a high contrast of both its mass density and bulk modulus, amplifies the generated total fields. This amplification is more pronounced when the incident frequency is close to the Minnaert frequency $\omega_{M}$. Here we e...

We consider the dynamics of a quantum particle of mass m on a n -edges star-graph with Hamiltonian $$H_K=-(2m)^{-1}\hbar ^2 \Delta $$ H K = - ( 2 m ) - 1 ħ 2 Δ and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state....

Let $\Delta_{\alpha,Y}$ be the bounded from above self-adjoint realization in $L^{2}({\mathbb R}^{3})$ of the Laplacian with $n$ point scatterers placed at $Y=\{y_{1},\dots,y_{n}\}\subset{\mathbb R}^{3}$, the parameters $(\alpha_{1},\dots\alpha_{n})\equiv\alpha\in {\mathbb R}^{n}$ being related to the scattering properties of the obstacles. Let $u^...

We consider the quantum evolution $e^{-i\frac{t}{\hbar}H_{\beta}} \psi_{\xi}^{\hbar}$ of a Gaussian coherent state $\psi_{\xi}^{\hbar}\in L^{2}(\mathbb{R})$ localized close to the classical state $\xi \equiv (q,p) \in \mathbb{R}^{2}$, where $H_{\beta}$ denotes a self-adjoint realization of the formal Hamiltonian $-\frac{\hbar^{2}}{2m}\,\frac{d^{2}\...

Let H:dom(H)⊆F→F be self-adjoint and let A:dom(H)→F (playing the role of the annihilation operator) be H-bounded. Assuming some additional hypotheses on A (so that the creation operator A∗ is a singular perturbation of H), by a twofold application of a resolvent Kreı̆n-type formula, we build self-adjoint realizations Ĥ of the formal Hamiltonian H +...

We consider the semi-classical limit of the quantum evolution of Gaussian coherent states whenever the Hamiltonian H is given, as sum of quadratic forms, by \( H= -\frac{{\hbar ^{2}}}{2m}\,\frac{d^{2}\,}{dx^{2}}\,\dot{+}\,\alpha \delta _{0}\), with \(\alpha \in \mathbb R\) and \(\delta _{0}\) the Dirac delta-distribution at \(x=0\). We show that th...

We consider the nonlinear Dirac equation with Soler-type nonlinearity concentrated at one point and present a detailed study of the spectrum of linearization at solitary waves. We then consider two different perturbations of the nonlinearity which break the $\mathbf{SU}(1,1)$-symmetry: the first preserving and the second breaking the parity symmetr...

We consider the dynamics of a quantum particle of mass $m$ on a $n$-edges star-graph with Hamiltonian $H_K=-(2m)^{-1}\hbar^2 \Delta$ and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state. We define the limiting cla...

Let $H:D(H)\subseteq{\mathscr F}\to{\mathscr F}$ be self-adjoint and let $A:D(H)\to{\mathscr F}$ (playing the role of the annihilator operator) be $H$-bounded. Assuming some additional hypotheses on $A$ (so that the creation operator $A^{*}$ is a singular perturbation of $H$), by a twofold application of a resolvent Krein-type formula, we build sel...

We provide a limiting absorption principle for self-adjoint realizations of Dirac operators with electrostatic and Lorentz scalar δ-shell interactions supported on regular compact surfaces. Then, we show completeness of the wave operators and give a representation formula for the scattering matrix.

Let Δ Λ ≤ λ Λ \Delta _{\Lambda }\le \lambda _{\Lambda } be a semi-bounded self-adjoint realization of the Laplace operator with boundary conditions (Dirichlet, Neumann, semi-transparent) assigned on the Lipschitz boundary of a bounded obstacle Ω \Omega . Let u f Λ u^{\Lambda }_{f} and u f 0 u^{0}_{f} denote the solutions of the wave equations corre...

We correct a minor mistake in the final formula for the scattering matrix.

We provide a limiting absorption principle for self-adjoint realizations of Dirac operators with electrostatic and Lorentz scalar $\delta$-shell interactions supported on regular compact surfaces. Then we show completeness of the wave operators and give a representation formula for the scattering matrix.

We consider the semi-classical limit of the quantum evolution of Gaussian coherent states whenever the Hamiltonian $\mathsf H$ is given, as sum of quadratic forms, by $\mathsf H= -\frac{\hbar^{2}}{2m}\,\frac{d^{2}\,}{dx^{2}}\,\dot{+}\,\alpha\delta_{0}$, with $\alpha\in\mathbb R$ and $\delta_{0}$ the Dirac delta-distribution at $x=0$. We show that t...

Let $\Delta_{\Lambda}\le \lambda_{\Lambda}$ be a semi-bounded self-adjoint realization of the Laplace operator with boundary conditions assigned on the Lipschitz boundary of a bounded obstacle $\Omega$. Let $u^{\Lambda}_{f}$ and $u^{0}_{f}$ be the solutions of the wave equations corresponding to $\Delta_{\Lambda}$ and to the free Laplacian $\Delta$...

We provide a general scheme, in the combined frameworks of Mathematical Scattering Theory and Factorization Method, for inverse scattering for the couple of self-adjoint operators $(\widetilde\Delta,\Delta)$, where $\Delta$ is the free Laplacian in $L^{2}({\mathbb R}^{3})$ and $\widetilde\Delta$ is one of its singular perturbations, i.e., such that...

We study scattering for the couple (AF,A0) of Schrödinger operators in L²(R³) formally defined as A0=−Δ+αδπ0 and AF=−Δ+αδπF, α>0, where δπF is the Dirac δ-distribution supported on the deformed plane given by the graph of the compactly supported, Lipschitz continuous function F:R²→R and π0 is the undeformed plane corresponding to the choice F≡0. We...

Let $A_{Q}$ be the self-adjoint operator defined by the $Q$-function $Q:z\mapsto Q_{z}$ through the Krein-like resolvent formula $$(-A_{Q}+z)^{-1}= (-A_{0}+z)^{-1}+G_{z}WQ_{z}^{-1}VG_{\bar z}^{*}\,,\quad z\in Z_{Q}\,,$$ where $V$ and $W$ are bounded operators and $$Z_{Q}:=\{z\in\rho(A_{0}):\text{$Q_{z}$ and $Q_{\bar z }$ have a bounded inverse}\}\,...

We study scattering for the couple $(A_{0},A_{F})$ of Schr\"odinger operators in $L^2(\mathbb{R}^3)$ formally defined as $A_0 = -\Delta + \alpha\, \delta_{\pi_0}$ and $A_F = -\Delta + \alpha\, \delta_{\pi_F}$, $\alpha >0$, where $\delta_{\pi_F}$ is the Dirac $\delta$-distribution supported on the deformed plane given by the graph of the compactly s...

We investigate one-dimensional (2p × 2p)-matrix Dirac operators DX,α and DX,β with point matrix interactions on a discrete set X. Several results of [4] are generalized to the case of (p × p)-matrix interactions with p > 1. It is shown that a number of properties of the operators DX,α and DX,β (self-adjointness, discreteness of the spectrum, etc.)...

We give a criterion of asymptotic completeness and provide a representation of the scattering matrix for the scattering couple $(A_{0},A)$, where $A_{0}$ and $A$ are semi-bounded self-adjoint operators in $L^{2}(M,{\mathscr B},m)$ such that the set $\{u\in D(A_{0})\cap D(A):A_{0}u=Au\}$ is dense. No sort of trace-class condition on resolvent differ...

We give a criterion of asymptotic completeness and provide a representation of the scattering matrix for the scattering couple $(A_{0},A)$, where $A_{0}$ and $A$ are semi-bounded self-adjoint operators in $L^{2}(M,{\mathscr B},m)$ such that the set $\{u\in D(A_{0})\cap D(A):A_{0}u=Au\}$ is dense. No sort of trace-class condition on resolvent differ...

We consider two families of realizations of the 2p×2p–Dirac differential expression with point interactions on a discrete set X = {xn}n=1∞ ⊂ ℝ on a half–line (line) and generalize certain results from [10] to the matrix case. We show that these realizations are always self-adjoint. We investigate the nonrelativistic limit as the velocity of light t...

We prove uniqueness in inverse acoustic scattering in the case the density of the medium has an unbounded gradient across $\Sigma\subseteq\Gamma=\partial\Omega$, where $\Omega$ is a bounded open subset of $\mathbb{R}^{3}$ with a Lipschitz boundary. Such a result follows from a uniqueness result in inverse scattering for Schr\"odinger operators with...

We study the relative zeta function for the couple of operators $A_0$ and $A_\alpha$, where $A_0$ is the free unconstrained Laplacian in $L^2(\mathbf{R}^d)$ ($d \geq 2$) and $A_\alpha$ is the singular perturbation of $A_0$ associated to the presence of a delta interaction supported by a hyperplane. In our setting the operatorial parameter $\alpha$,...

We prove uniqueness in inverse acoustic scattering in the case the density of the medium has an unbounded gradient across $\Sigma\subseteq\Gamma=\partial\Omega$, where $\Omega$ is a bounded open subset of $\mathbb{R}^{3}$ with a Lipschitz boundary. This follows from a uniqueness result in inverse scattering for Schr\"odinger operators with singular...

We study the relative zeta function for the couple of operators $A_0$ and $A_\alpha$, where $A_0$ is the free unconstrained Laplacian in $L^2(\mathbf{R}^d)$ ($d \geq 2$) and $A_\alpha$ is the singular perturbation of $A_0$ associated to the presence of a delta interaction supported by a hyperplane. In our setting the operatorial parameter $\alpha$,...

We define and study the Cauchy problem for a one-dimensional (1-D) nonlinear Dirac equation with nonlinearities concentrated at one point. Global well-posedness is provided and conservation laws for mass and energy are shown. Several examples, including nonlinear Gesztesy- Šeba models and the concentrated versions of the Bragg resonance and 1-D Sol...

We study the spectrum, resonances and scattering matrix of a quantum hamiltonian on a "hybrid surface" consisting of a half-line attached by its endpoint to the vertex of a concave planar wedge. At the boundary of the wedge, outside the vertex, Dirichlet boundary are imposed. The system is tunable by varying the measure of the angle at the vertex.

Let $\Omega_-$ and $\Omega_+$ be two bounded smooth domains in $\mathbb{R}^n$, $n\ge 2$, separated by a hypersurface $\Sigma$. For $\mu>0$, consider the function $h_\mu=1_{\Omega_-}-\mu 1_{\Omega_+}$. We discuss self-adjoint realizations of the operator $L_{\mu}=-\nabla\cdot h_\mu \nabla$ in $L^2(\Omega_-\cup\Omega_+)$ with the Dirichlet condition...

Let $\Omega_-$ and $\Omega_+$ be two bounded smooth domains in $\mathbb{R}^n$, $n\ge 2$, separated by a hypersurface $\Sigma$. For $\mu>0$, consider the function $h_\mu=1_{\Omega_-}-\mu 1_{\Omega_+}$. We discuss self-adjoint realizations of the operator $L_{\mu}=-\nabla\cdot h_\mu \nabla$ in $L^2(\Omega_-\cup\Omega_+)$ with the Dirichlet condition...

We define and study the Cauchy problem for a 1-D nonlinear Dirac equation with nonlinearities concentrated at one point. Global well-posedness is provided and conservation laws for mass and energy are shown. Several examples, including nonlinear Gesztesy-\v{S}eba models and the concentrated versions of the Bragg Resonance, Gross-Neveu, and Soler ty...

We provide a limiting absorption principle for the self-adjoint realizations of Laplace operators corresponding to boundary conditions on (relatively open parts $\Sigma$ of) compact hypersurfaces $\Gamma=\partial\Omega$, $\Omega\subset{\mathbb{R}}^{n}$. For any of such self-adjoint operators we also provide the generalized eigenfunctions and the sc...

We provide a limiting absorption principle for the self-adjoint realizations of Laplace operators corresponding to boundary conditions on (relatively open parts $\Sigma$ of) compact hypersurfaces $\Gamma=\partial\Omega$, $\Omega\subset{\mathbb{R}}^{n}$. For any of such self-adjoint operators we also provide the generalized eigenfunctions and the sc...

We review the main results of our recent work on singular perturbations supported on bounded hypersurfaces. Our approach consists in using the theory of self-adjoint extensions of restrictions to build self-adjoint realizations of the n-dimensional Laplacian with linear boundary conditions on (a relatively open part of) a compact hypersurface. This...

We solve the Cauchy problem for the Schr\"odinger equation corresponding to the family of Hamiltonians $H_{\gamma(t)}$ in $L^{2}(\mathbb{R})$ which describes $\delta'$ interactions with time-dependent strength $1/\gamma(t)$. We prove that the strong solution of such a Cauchy problem exits whenever the map $t\mapsto\gamma(t)$ belongs to the fraction...

We solve the Cauchy problem for the Schr\"odinger equation corresponding to the family of Hamiltonians $H_{\gamma(t)}$ in $L^{2}(\mathbb{R})$ which describes a $\delta'$-interaction with time-dependent strength $1/\gamma(t)$. We prove that the strong solution of such a Cauchy problem exits whenever the map $t\mapsto\gamma(t)$ belongs to the fractio...

The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic operator on $\mathbb{R}% ^{n}$ with linear boundary conditions on (a relatively open part of) a compact hypersurface. Our approach allows to obtain Krein-like resolvent formulas where the reference operator...

Let $\Omega\subset\RE^n$ be bounded with a smooth boundary $\Gamma$ and let $S$ be the symmetric operator in $L^2(\Omega)$ given by the minimal realization of a second order elliptic differential operator. We give a complete classification of the Markovian self-adjoint extensions of $S$ by providing an explicit one-to-one correspondence between su...

Given a linear semi-bounded symmetric operator $S\ge -\omega$, we explicitly define, and provide their nonlinear resolvents, nonlinear maximal monotone operators $A_\Theta$ of type $\lambda>\omega$ (i.e. generators of one-parameter continuous nonlinear semi-groups of contractions of type $\lambda$) which coincide with the Friedrichs extension of $S...

We give a simple criterion so that a countable infinite direct sum of trace (evaluation) maps is a trace map. An application to the theory of self-adjoint extensions of direct sums of symmetric operators is provided; this gives an alternative approach to results recently obtained by Malamud-Neidhardt and Kostenko-Malamud using regularized direct su...

Given two different self-adjoint extensions of the same symmetric operator, we analyse the intersection of their point spectra. Some simple examples are provided.

By Birman and Skvortsov it is known that if $\Omegasf$ is a planar curvilinear polygon with $n$ non-convex corners then the Laplace operator with domain $H^2(\Omegasf)\cap H^1_0(\Omegasf)$ is a closed symmetric operator with deficiency indices $(n,n)$. Here we provide a Kre\u\i n-type resolvent formula for any self-adjoint extensions of such an ope...

We investigate spectral properties of Gesztesy-\v{S}eba realizations D_{X,\alpha} and D_{X,\beta} of the 1-D Dirac differential expression D with point interactions on a discrete set $X=\{x_n\}_{n=1}^\infty\subset \mathbb{R}.$ Here $\alpha := \{\alpha_{n}\}_{n=1}^\infty$ and \beta :=\{\beta_{n}\}_{n=1}^\infty \subset\mathbb{R}. The Gesztesy-\v{S}eb...

We consider a one dimensional evolution problem modelling the dynamics of an acoustic field coupled with a set of mechanical oscillators. We analyse solutions of the system of ordinary and partial differential equations with time-dependent boundary conditions describing the evolution in the limit of a continuous distribution of oscillators.

For a general self-adjoint Hamiltonian operator $H_0$ on the Hilbert space $L^2(\RE^d)$, we determine the set of all self-adjoint Hamiltonians $H$ on $L^2(\RE^d)$ that dynamically confine the system to an open set $\Omega \subset \RE^d$ while reproducing the action of $ H_0$ on an appropriate operator domain. In the case $H_0=-\Delta +V$ we constru...

Given a symmetric, semi-bounded, second order elliptic differential operator on a bounded domain with $C^{1,1}$ boundary, we provide a Krein-type formula for the resolvent difference between its Friedrichs extension and an arbitrary self-adjoint one. Comment: Final version, to appear in J. Phys. A: Math. Theor

In the case of a single point interaction we improve, by different techniques, the existence theorem for the unitary evolution generated by a Schr\"odinger operator with moving point interactions obtained by Dell'Antonio, Figari and Teta.

This paper treats the dynamics and scattering of a model of coupled oscillating systems, a finite dimensional one and a wave field on the half line. The coupling is realized producing the family of selfadjoint extensions of the suitably restricted self-adjoint operator describing the uncoupled dynamics. The spectral theory of the family is studied...

A one dimensional system made up of a compressible fluid and several mechanical oscillators, coupled to the acoustic field in the fluid, is analyzed for different settings of the oscillators array. The dynamical models are formulated in terms of singular perturbations of the decoupled dynamics of the acoustic field and the mechanical oscillators. D...

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 Sega...

We show that the boundary conditions entering in the definition of the self-adjoint operator describing the Laplacian plus a finite number of point interactions are local if and only if the corresponding wave equation has finite speed of propagation

In this paper we address the problem of wave dynam-ics in presence of concentrated nonlinearities. Given a vector field V on an open subset of C n and a discrete set Y ⊂ R 3 with n elements, we define a nonlinear operator ∆ V,Y on L 2 (R 3) which coincides with the free Laplacian when restricted to regular func-tions vanishing at Y , and which redu...

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 $\ddot\phi-(C_1+C_2)\dot\phi=-(B^2+C_1C_2)\phi$. Giv...

In this paper, we study the distorted Ornstein-Uhlenbeck processes associated with given densities on an abstract Wiener space. We prove that the laws of distorted Ornstein-Uhlenbeck processes converge in total variation norm if the densities converge in Sobolev space \( D^{1}_{2} \).

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 v...

Given the symmetric operator $A_N$ obtained by restricting the self-adjoint operator $A$ to $N$, a linear dense set, closed with respect to the graph norm, we determine a convenient boundary triple for the adjoint $A_N^*$ and the corresponding Weyl function. These objects provide us with the self-adjoint extensions of $A_N$ and their resolvents.

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(Theta) of A(N) such that D(A(Theta)) boolean AND D(A) = N can be addi...

We show how the scattering-into-cones and ux-across-surfaces theorems in Quantum Mechanics have very intuitive pathwise probabilistic versions based on some results by Carlen about large time behaviour of paths of Nelson's diusions. The quantum mechanical results can be then recovered by taking expectations in our pathwise statements.

Let A: D(A) ⊆ H → H be an injective self-adjoint operator and let τ: D(A) → X, X a Banach space, be a surjective linear map such that ‖τφ‖X ≤ c ‖Aφ‖H. Supposing that Kernelτ is dense in H, we of self-adjoint operators which are extensions of the define a family Aτ Θ symmetric operator A |{τ=0}. Any φ in the operator domain D(Aτ Θ) is characterized...

Given the abstract wave equation $\ddot\phi-\Delta_\alpha\phi=0$, where $\Delta_\alpha$ is the Laplace operator with a point interaction of strength $\alpha$, we define and study $\bar W_\alpha$, the associated wave generator in the phase space of finite energy states. We prove the existence of the phase flow generated by $\bar W_\alpha$, and descr...

Let $A:D(A)\subseteq\H\to\H$ be an injective self-adjoint operator and let $\tau:D(A)\to\X$, X a Banach space, be a surjective linear map such that $\|\tau\phi\|_\X\le c \|A\phi\|_\H$. Supposing that \text{\rm Range}$ (\tau')\cap\H' =\{0\}$, we define a family $A^\tau_\Theta$ of self-adjoint operators which are extensions of the symmetric operator...

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 dynami...

. We continue in this paper the analysis, begun in [NP], of the classical dynamics of the point limit of the Maxwell--Lorentz system in dipole approximation ( the Pauli--Fierz model ). Here, as a first step towards considering the full nonlinear system, we study the case in which a nonlinear external field of force is present. We study the flow of...

. We study the point limit of the linearized Maxwell--Lorentz equations describing the interaction, in the dipole approximation, of an extended charged particle with the electromagnetic field. We find that this problem perfectly fits into the framework of singular perturbations of the Laplacian; indeed we prove that the solutions of the Maxwell--Lo...

. We give a complete characterization, including a L'evy--Ito decomposition, of Poincar'e--invariant Markov processes on H 1 + Theta M 2 , the relativistic phase space in 1+1 space--time dimensions. Then, by means of such processes, we construct Poincar'e-- invariant Gaussian random fields, and we prove a "no--go" theorem for the random fields corr...

We prove a convergence theorem for sequences of Diffusion Processes corresponding to Dirichlet Forms of the kind ef ( f,g ) = \tfrac12ò\mathbbRd Ñf ( x ) Ñg( x )f2 ( x )dx\varepsilon _\phi \left( {f,g} \right) = \tfrac{1}{2}\int_{\mathbb{R}^d } {\nabla f} \left( x \right) \cdot \nabla g\left( x \right)\phi ^2 \left( x \right)dx .We obtain converg...

Let us consider a Nelson Diffusion defined by the family of probability densities ρt and by the drift vector field b. Making use of the Liapunov function denotes the Stein regularization of : the distance function of , we prove the nonattainability of the nodal set of p under the conditions , where U is a nbh. of Zρ

Some recent results on the convergence of Nelson diffusions are extended to the case of Schro¨dinger operators with time-dependent electromagnetic potentials. It is proven that the sequence {P n}n≥1 of measures on the canonical space of physical trajectories associated to the solutions of Schro¨dinger equations in Nelson’s scheme, corresponding to...

Let {Pn}n≥1 be the sequence of probability measures (Nelson diffusions) cor- responding to the sequence of infinitesimal characteristics {(vn,ρn)}n≥1. We prove that if (vn, ρn) → (v, ρ) w.r.t. the Guerra metric then P n → P in total variation, where P is the measure corresponding to (v, ρ). We also give an application to the case of convergence of...

Let t, t n ,n1, be solutions of Schrdinger equations with potentials form-bounded by –1/2 and initial data inH 1( d ). LetP, P n ,n1, be the probability measures on the path space =C(+, d ) given by the corresponding Nelson diffusions. We show that if { t n } n1 converges to t inH 1( d ), uniformly int over compact intervals, then converges to in...

In this note we use a banal relation between flows of vector fields, a sort of time-dependent Campbell—Baker—Hausdorff formula, to obtain a non-linear version of the variation of constants formula. We employ this formula to calculate the tangent to the mapping which assigns to each time-dependent vector fieldV with compact support the diffeomorphis...

We consider the problem of finding a normal form for differential equations in the neighbourhood of an equilibrium point, and produce general explicit estimates for both the normal form at a finite order and the remainder, using the method of Lie transforms. With such technique, the classical Poincar-Dulac theorems are recovered, and the problem of...