arXiv subjects
Publications and source records attributed to Biagio Cassano.
We study the Cauchy problem for biwave type equations with coefficients depending only on time. Assuming $\mathcal{C}^\kappa$ regularity of the principal symbol and suitable Levi conditions on the lower order terms, we establish well-posedness in Gevrey spaces.
In [DO99,KY99], the strong unique continuation property from the origin is established for $H_{loc}^1$-solutions to the massless Dirac differential inequality $|{D}_n u | \leq \frac{C}{|x|}|u|$, in dimension $n\geq 2$ and with $C<\frac12$. We show that $\frac12$ is the largest possibile constant in this result, providing an example in $\mathbb{R}^2$ of a (non-trivial) solution of the inequality. Also, we show properties of unique continuation from the origin for solutions to the inequality $|D_n u | \leq \frac{C}{ |x|^γ}|u|$, for $γ>1$, $C>0$. Finally, we establish the strong unique continuation property for the Dirac operator from the point at infinity.
We consider the massless Dirac operator with the MIT bag boundary conditions on an unbounded three-dimensional circular cone. For convex cones, we prove that this operator is self-adjoint defined on four-component $H^1$--functions satisfying the MIT bag boundary conditions. The proof of this result relies on separation of variables and spectral estimates for one-dimensional fiber Dirac-type operators. Furthermore, we provide a numerical evidence for the self-adjointness on the same domain also for non-convex cones. Moreover, we prove a Hardy-type inequality for such a Dirac operator on convex cones, which, in particular, yields stability of self-adjointness under perturbations by a class of unbounded potentials. Further extensions of our results to Dirac operators with quantum dot boundary conditions are also discussed.
In this paper, we investigate properties of unique continuation for hyperbolic Schrödinger equations with time-dependent complex-valued electric fields and time-independent real magnetic fields. We show that positive masses inside of a bounded region at a single time propagate outside the region and prove gaussian lower bounds for the solutions, provided a suitable average in space-time cylinders is taken.
We investigate the properties of self-adjointness of a two-dimensional Dirac operator on an infinite sector with infinite mass boundary conditions and in presence of a Coulomb-type potential with the singularity placed on the vertex. In the general case, we prove the appropriate Dirac-Hardy inequality and exploit the Kato-Rellich theory. In the explicit case of a Coulomb potential, we describe the self-adjoint extensions for all the intensities of the potential relying on a radial decomposition in partial wave subspaces adapted to the infinite-mass boundary conditions. Finally, we integrate our results giving a description of the spectrum of these operators.
We investigate Hardy-Rellich inequalities for perturbed Laplacians. In particular, we show that a non-trivial angular perturbation of the free operator typically improves the inequality, and may also provide an estimate which does not hold in the free case. The main examples are related to the introduction of a magnetic field: this is a manifestation of the diamagnetic phenomenon, which has been observed by Laptev and Weidl in \cite{LW1999} for the Hardy inequality, later by Evans and Lewis in \cite{EL2005} for the Rellich inequality; however, to the best of our knowledge, the so called Hardy-Rellich inequality has not yet been investigated in this regards. After showing the optimal inequality, we prove that the best constant is not attained by any function in the domain of the estimate.
In this paper we introduce a notion of magnetic field in the Heisenberg group and we study its influence on spectral properties of the corresponding magnetic (sub-elliptic) Laplacian. We show that uniform magnetic fields uplift the bottom of the spectrum. For magnetic fields vanishing at infinity, including Aharonov--Bohm potentials, we derive magnetic improvements to a variety of Hardy-type inequalities for the Heisenberg sub-Laplacian. In particular, we establish a sub-Riemannian analogue of Laptev and Weidl sub-criticality result for magnetic Laplacians in the plane. Instrumental for our argument is the validity of a Hardy-type inequality for the Folland--Stein operator, that we prove in this paper and has an interest on its own.
In this paper we investigate spectral properties of the damped elastic wave equation. Deducing a correspondence between the eigenvalue problem of this model and the one of Lamé operators with non self-adjoint perturbations, we provide quantitative bounds on the location of the point spectrum in terms of suitable norms of the damping coefficient.
We investigate the validity of gaussian lower bounds for solutions to an electromagnetic Schrödinger equation with a bounded time-dependent complex electric potential and a time-independent vector magnetic potential. We prove that, if a suitable geometric condition is satisfied by the vector potential, then positive masses inside of a bounded region at a single time propagate outside the region, provided a suitable average in space-time cylinders is taken.
In this work we consider the two-dimensional Dirac operator with general local singular interactions supported on a closed curve. A systematic study of the interaction is performed by decomposing it into a linear combination of four elementary interactions: electrostatic, Lorentz scalar, magnetic, and a fourth one which can be absorbed by using unitary transformations. We address the self-adjointness and the spectral description of the underlying Dirac operator, and moreover we describe its approximation by Dirac operators with regular potentials.
This paper is devoted to providing quantitative bounds on the location of eigenvalues, both discrete and embedded, of non self-adjoint Lamé operators of elasticity $-Δ^\ast + V$ in terms of suitable norms of the potential $V$. In particular, this allows to get sufficient conditions on the size of the potential such that the point spectrum of the perturbed operator remains empty. In three dimensions we show full spectral stability under suitable form-subordinated perturbations: we prove that the spectrum is purely continuous and coincides with the non negative semi-axis as in the free case.
We provide quantitative estimates on the location of eigenvalues of one-dimensional discrete Dirac operators with complex $\ell^p$-potentials for $1\leq p\leq\infty$. As a corollary, subsets of the essential spectrum free of embedded eigenvalues are determined for small $\ell^1$-potential. Further possible improvements and sharpness of the obtained spectral bounds are also discussed.
We consider the four-component two-valley Dirac operator on a wedge in $\mathbb{R}^2$ with infinite mass boundary conditions, which enjoy a flip at the vertex. We show that it has deficiency indices $(1,1)$ and we parametrize all its self-adjoint extensions, relying on the fact that the underlying two-component Dirac operator is symmetric with deficiency indices $(0,1)$. The respective defect element is computed explicitly. We observe that there exists no self-adjoint extension, which can be decomposed into an orthogonal sum of two two-component operators. In physics, this effect is called mixing the valleys.
We determine the largest non-trivial rate of exponential decay at infinity for solutions to the Dirac equation \begin{equation*} \mathcal{D}_n ψ+ \mathbb{V} ψ= 0 \quad \text{ in }\mathbb{R}^n, \end{equation*} being $\mathcal{D}_n$ the massless Dirac operator in dimension $n\geq 2$ and $\mathbb{V}$ a (possibly non-Hermitian) matrix-valued perturbation such that $|\mathbb{V}(x)| \sim |x|^{-ε}$ at infinity, for $-\infty < ε< 1$. Moreover, we show that our results are sharp for $n =2,3$, providing explicit examples of solutions that have the prescripted decay, in presence of a potential with the related behaviour at infinity.
We determine explicitly a boundary triple for the Dirac operator $H:=-iα\cdot \nabla + mβ+ \mathbb V(x)$ in $\mathbb R^3$, for $m\in\mathbb R$ and $\mathbb V(x)= |x|^{-1} ( ν\mathbb{I}_4 +μβ-i λα\cdot{x}/|x|\,β)$, with $ν,μ,λ\in \mathbb R$. Consequently we determine all the self-adjoint realizations of $H$ in terms of the behaviour of the functions of their domain in the origin. When $\sup_{x} |x||\mathbb V(x)| \leq 1$, we discuss the problem of selecting the distinguished extension requiring that its domain is included in the domain of the appropriate quadratic form.
We prove a sharp Hardy-type inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrix-valued potentials $\mathbf V$ of Coulomb type: we characterise its eigenvalues in terms of the Birman-Schwinger principle and we bound its discrete spectrum from below, showing that the \emph{ground-state energy} is reached if and only if $\mathbf V$ verifies some {rigidity} conditions. In the particular case of an electrostatic potential, these imply that $\mathbf V$ is the Coulomb potential.
We describe the self-adjoint realizations of the operator $H:=-iα\cdot \nabla + mβ+ \mathbb V(x)$, for $m\in\mathbb R $, and $\mathbb V(x)= |x|^{-1} ( ν\mathbb{I}_4 +μβ-i λα\cdot{x}/{|x|}\,β)$, for $ν,μ,λ\in \mathbb R$. We characterize the self-adjointness in terms of the behaviour of the functions of the domain in the origin, exploiting Hardy-type estimates and trace lemmas. Finally, we describe the distinguished extension.
We prove that the decay of the eigenfunctions of harmonic oscillators, uniform electric or magnetic fields is not stable under 0-order complex perturbations, even if bounded, of these Hamiltonians, in the sense that we can produce solutions to the evolutionary Schrödinger flows associated to the Hamiltonians, with a stronger Gaussian decay at two distinct times. We then characterize, in a quantitative way, the sharpest possible Gaussian decay of solutions as a function of the oscillation frequency or the strength of the field, depending on the Hamiltonian which is considered. This is connected to the Hardy's Uncertainty Principle for free Schrödinger evolutions.