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Andrea Posilicano

Publications and source records attributed to Andrea Posilicano.

At least 37 records · Page 2Linked to original sources
arXiv2019

The semi-classical limit with delta potentials

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 $\mathsf H= -\frac{\hbar^{2}}{2m}\,\frac{d^{2}\,}{dx^{2}}\,\dot{+}\,αδ_{0}$, with $α\in\mathbb R$ and $δ_{0}$ the Dirac delta-distribution at $x=0$. We show that the quantum evolution can be approximated, uniformly for any time away from the collision time and with an error of order $\hbar^{3/2-λ}$, $0\!<\!λ\!<\!3/2$, by the quasi-classical evolution generated by a self-adjoint extension of the restriction to $\mathcal C^{\infty}_{c}({\mathscr M}_{0})$, ${\mathscr M}_{0}:=\{(q,p)\!\in\!\mathbb R^{2}\,|\,q\!\not=\!0\}$, of ($-i$ times) the generator of the free classical dynamics; such a self-adjoint extension does not correspond to the classical dynamics describing the complete reflection due to the infinite barrier. Similar approximation results are also provided for the wave and scattering operators.

math-ph↗
arXiv2019

Asymptotic Completeness and S-Matrix for Singular Perturbations

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 differences is required. Applications to the case in which $A_{0}$ corresponds to the free Laplacian in $L^{2}({\mathbb R}^{n})$ and $A$ describes the Laplacian with self-adjoint boundary conditions on rough compact hypersurfaces are given.

math-ph↗
arXiv2018

On inverses of Krein's Q-functions

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 $$Z_{Q}:=\{z\inρ(A_{0}):\text{$Q_{z}$ and $Q_{\bar z }$ have a bounded inverse}\}\,.$$ We show that $$Z_{Q}\not=\emptyset\quad\Longrightarrow\quad Z_{Q}=ρ(A_{0})\cap ρ(A_{Q})\,.$$ We do not suppose that $Q$ is represented in terms of a uniformly strict, operator-valued Nevanlinna function (equivalently, we do not assume that $Q$ is associated to an ordinary boundary triplet), thus our result extends previously known ones. The proof relies on simple algebraic computations stemming from the first resolvent identity.

math.SP↗
arXiv2018

Uniqueness in inverse acoustic scattering with unbounded gradient across Lipschitz surfaces

We prove uniqueness in inverse acoustic scattering in the case the density of the medium has an unbounded gradient across $Σ\subseteqΓ=\partialΩ$, where $Ω$ is a bounded open subset of $\mathbb{R}^{3}$ with a Lipschitz boundary. This follows from a uniqueness result in inverse scattering for Schrödinger operators with singular $δ$-type potential supported on the surface $Γ$ and of strength $α\in L^{p}(Γ)$, $p>2$.

math.AP↗
arXiv2017

Relative-Zeta and Casimir energy for a semitransparent hyperplane selecting transverse modes

We study the relative zeta function for the couple of operators $A_0$ and $A_α$, where $A_0$ is the free unconstrained Laplacian in $L^2(\mathbf{R}^d)$ ($d \geq 2$) and $A_α$ 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 $α$, which is related to the strength of the perturbation, is of the kind $α=α(-Δ_{\parallel})$, where $-Δ_{\parallel}$ is the free Laplacian in $L^2(\mathbf{R}^{d-1})$. Thus $α$ may depend on the components of the wave vector parallel to hyperplane; in this sense $A_α$ describes a semitransparent hyperplane selecting transverse modes. As an application we give an expression for the associated thermal Casimir energy. Whenever $α=χ_{I}(-Δ_{\parallel})$, where $χ_{I}$ is the characteristic function of an interval $I$, the thermal Casimir energy can be explicitly computed.

math-ph↗
arXiv2017

Limiting Absorption Principle, Generalized Eigenfunctions and Scattering Matrix for Laplace Operators with Boundary conditions on Hypersurfaces

We provide a limiting absorption principle for the self-adjoint realizations of Laplace operators corresponding to boundary conditions on (relatively open parts $Σ$ of) compact hypersurfaces $Γ=\partialΩ$, $Ω\subset{\mathbb{R}}^{n}$. For any of such self-adjoint operators we also provide the generalized eigenfunctions and the scattering matrix; both these objects are written in terms of operator-valued Weyl functions. We make use of a Krein-type formula which provides the resolvent difference between the operator corresponding to self-adjoint boundary conditions on the hypersurface and the free Laplacian on the whole space ${\mathbb{R}}^{n}$. Our results apply to all standard examples of boundary conditions, like Dirichlet, Neumann, Robin, $δ$ and $δ'$-type, either assigned on $Γ$ or on $Σ\subsetΓ$.

math-ph↗
arXiv2016

Self-adjoint indefinite Laplacians

Let $Ω_-$ and $Ω_+$ be two bounded smooth domains in $\mathbb{R}^n$, $n\ge 2$, separated by a hypersurface $Σ$. For $μ>0$, consider the function $h_μ=1_{Ω_-}-μ1_{Ω_+}$. We discuss self-adjoint realizations of the operator $L_μ=-\nabla\cdot h_μ\nabla$ in $L^2(Ω_-\cupΩ_+)$ with the Dirichlet condition at the exterior boundary. We show that $L_μ$ is always essentially self-adjoint on the natural domain (corresponding to transmission-type boundary conditions at the interface $Σ$) and study some properties of its unique self-adjoint extension $\mathcal{L}_μ:=\overline{L_μ}$. If $μ\ne 1$, then $\mathcal{L}_μ$ simply coincides with $L_μ$ and has compact resolvent. If $n=2$, then $\mathcal{L}_1$ has a non-empty essential spectrum, $σ_\mathrm{ess}(\mathcal{L}_{1})=\{0\}$. If $n\ge 3$, the spectral properties of $\mathcal{L}_1$ depend on the geometry of $Σ$. In particular, it has compact resolvent if $Σ$ is the union of disjoint strictly convex hypersurfaces, but can have a non-empty essential spectrum if a part of $Σ$ is flat. Our construction features the method of boundary triplets, and the problem is reduced to finding the self-adjoint extensions of a pseudodifferential operator on $Σ$. We discuss some links between the resulting self-adjoint operator $\mathcal{L}_μ$ and some effects observed in negative-index materials.

math.SP↗
arXiv2016

Time dependent delta-prime interactions in dimension one

We solve the Cauchy problem for the Schrödinger equation corresponding to the family of Hamiltonians $H_{γ(t)}$ in $L^{2}(\mathbb{R})$ which describes a $δ'$-interaction with time-dependent strength $1/γ(t)$. We prove that the strong solution of such a Cauchy problem exits whenever the map $t\mapstoγ(t)$ belongs to the fractional Sobolev space $H^{3/4}(\mathbb{R})$, thus weakening the hypotheses which would be required by the known general abstract results. The solution is expressed in terms of the free evolution and the solution of a Volterra integral equation.

math-ph↗
arXiv2016

The 1-D Dirac equation with concentrated nonlinearity

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-Šeba models and the concentrated versions of the Bragg Resonance, Gross-Neveu, and Soler type models, all within the scope of the present paper, are given. The key point of the proof consists in the reduction of the original equation to a nonlinear integral equation for an auxiliary, space-independent variable (the "charge").

math-ph↗
arXiv2015

Nonlinear Maximal Monotone Extensions of Symmetric Operators

Given a linear semi-bounded symmetric operator $S\ge -ω$, we explicitly define, and provide their nonlinear resolvents, nonlinear maximal monotone operators $A_Θ$ of type $λ>ω$ (i.e. generators of one-parameter continuous nonlinear semi-groups of contractions of type $λ$) which coincide with the Friedrichs extension of $S$ on a convex set containing ${\mathscr D}(S)$. The extension parameter $Θ\subset{\mathfrak h}\times{\mathfrak h}$ ranges over the set of nonlinear maximal monotone relations on an auxiliary Hilbert space $\mathfrak h$ isomorphic to the deficiency subspace of $S$. Moreover $A_Θ+λ$ is a sub-potential operator (i.e. is the sub-differential of a lower semicontinuos convex function) whenever $Θ$ is sub-potential. Examples describing Laplacians with nonlinear singular perturbations supported on null sets and Laplacians with nonlinear boundary conditions on a bounded set are given.

math.FA↗
arXiv2014

Direct sums of trace maps and self-adjoint extensions

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 sums of boundary triplets.

math-ph↗
arXiv2014

Effective equation for a system of mechanical oscillators in an acoustic field

We consider a one dimensional evolution problem modeling the dynamics of an acoustic field coupled with a set of mechanical oscillators. We analyze 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.

math-ph↗
arXiv2014

Markovian Extensions of Symmetric Second Order Elliptic Differential Operators

Let $Ω\subset{\mathbb R}^n$ be bounded with a smooth boundary $Γ$ and let $S$ be the symmetric operator in $L^2(Ω)$ 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 such extensions and the class of Dirichlet forms in $L^2(Γ)$ which are additively decomposable by the bilinear form of the Dirichlet-to-Neumann operator plus a Markovian form. By such a result two further equivalent classifications are provided: the first one is expressed in terms of an additive decomposition of the bilinear forms associated to the extensions, the second one uses the additive decomposition of the resolvents provided by Krein's formula. The Markovian part of the decomposition allows to characterize the operator domain of the corresponding extension in terms of Wentzell-type boundary conditions. Some properties of the extensions, and of the corresponding Dirichlet forms, semigroups and heat kernels, like locality, regularity, irreducibility, recurrence, transience, ultracontractivity and Gaussian bounds are also discussed.

math.AP↗
arXiv2013

On the many Dirichlet Laplacians on a non-convex polygon and their approximations by point interactions

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\uın-type resolvent formula for any self-adjoint extensions of such an operator, i.e. for the set of self-adjoint non-Friedrichs Dirichlet Laplacians on $\Omegasf$, and show that any element in this set is the norm resolvent limit of a suitable sequence of Friedrichs-Dirichlet Laplacians with $n$ point interactions.

math-ph↗
arXiv2013

On the spectral theory of Gesztesy-Šeba realizations of 1-D Dirac operators with point interactions on a discrete set

We investigate spectral properties of Gesztesy-Šeba realizations D_{X,α} and D_{X,β} 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 $α:= \{α_{n}\}_{n=1}^\infty$ and β:=\{β_{n}\}_{n=1}^\infty \subset\mathbb{R}. The Gesztesy-Šeba realizations $D_{X,α}$ and $D_{X,β}$ are the relativistic counterparts of the corresponding Schrödinger operators $H_{X,α}$ and $H_{X,β}$ with $δ$- and $δ'$-interactions, respectively. We define the minimal operator D_X as the direct sum of the minimal Dirac operators on the intervals $(x_{n-1}, x_n)$. Then using the regularization procedure for direct sum of boundary triplets we construct an appropriate boundary triplet for the maximal operator $D_X^*$ in the case $d_*(X):=\inf\{|x_i-x_j| \,, i\not=j\} = 0$. It turns out that the boundary operators $B_{X,α}$ and $B_{X,β}$ parameterizing the realizations D_{X,α} and D_{X,β} are Jacobi matrices. These matrices substantially differ from the ones appearing in spectral theory of Schrödinger operators with point interactions. We show that certain spectral properties of the operators $D_{X,α}$ and $D_{X,β}$ correlate with the corresponding spectral properties of the Jacobi matrices $B_{X,α}$ and $B_{X,β}$, respectively. Using this connection we investigate spectral properties (self-adjointness, discreteness, absolutely continuous and singular spectra) of Gesztesy--{\vS}eba realizations. Moreover, we investigate the non-relativistic limit as the velocity of light $c\to\infty$. Most of our results are new even in the case $d_*(X)> 0.$

math-ph↗
arXiv2012

Self-adjoint, globally defined Hamiltonian operators for systems with boundaries

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 $Ω\subset \RE^d$ while reproducing the action of $ H_0$ on an appropriate operator domain. In the case $H_0=-Δ+V$ we construct these Hamiltonians explicitly showing that they can be written in the form $H=H_0+ B$, where $B$ is a singular boundary potential and $H$ is self-adjoint on its maximal domain. An application to the deformation quantization of one-dimensional systems with boundaries is also presented.

math-ph↗