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Davide Tramontana

Publications and source records attributed to Davide Tramontana.

6 recordsLinked to original sources
arXiv2026

The metaplectic semigroup and its applications to time-frequency analysis and evolution operators

We develop a systematic analysis of the metaplectic semigroup $\mathrm{Mp}_+(d,\mathbb{C})$ associated with positive complex symplectic matrices, a notion introduced almost simultaneously and independently by Hörmander, Brunet, Kramer, and Howe, thereby extending the classical metaplectic theory beyond the unitary setting. While the existing literature has largely focused on propagators of quadratic evolution equations, for which results are typically obtained via Mehler formulas, our approach is operator-theoretic and symplectic in spirit and adapts techniques from the standard metaplectic group $\mathrm{Mp}(d,\mathbb{R})$ to a substantially broader framework that is not driven by differential problems or particular propagators. This point of view provides deeper insight into the structure of the metaplectic semigroup, and allows us to investigate its generators, polar decomposition, and intertwining relations with complex conjugation and with the Wigner distribution. We then exploit these structural results to characterize, from a metaplectic perspective, classes of time-frequency representations satisfying prescribed structural properties. Finally, we discuss further implications for parabolic equations with complex quadratic Hamiltonians, we study the boundedness of their propagators on modulation spaces, we obtain estimates in time of their operator norms. Finally, we apply our theory to the study of propagation of Wigner singularities.

math.AP↗
arXiv2026

Schrödinger ultrahyperbolic equations with singular coefficients

In this paper we investigate the Cauchy problem for Schrödinger ultrahyperbolic equations with singular (less than continuous) coefficients. We prove $H^\infty$ well-posedness in the very weak sense under suitable assumptions of the distributional structure of the coefficients and decay on the lower order terms. Consistency is proven with the classical $H^\infty$-results when the equation coefficients are smooth.

math.AP↗
arXiv2026

On the microlocal phase-space concentration of Wigner distributions associated with Schrödinger evolutions

In this work, we investigate the microlocal properties of the evolutions of Schrödinger equations using metaplectic Wigner distributions. So far, only restricted classes of metaplectic Wigner distributions, satisfying particular structural properties, have allowed the analysis of microlocal properties. We first extend the microlocal results to all metaplectic Wigner distributions, including the well-known Kohn-Nirenberg quantization, and examine these findings in the framework of Fourier integral operators with quadratic phase. Finally, we analyze the phase space concentration of the (cross) Wigner distribution arising from the interaction of two states, with particular attention to interactions generated by certain Schrödinger evolutions. These contributions enable a more refined study of the so-called ghost frequencies.

math.AP↗
arXiv2025

Subelliptic Random Walks on Riemannian Manifolds and Their Convergence to Equilibrium

The aim of this work is to study the convergence to equilibrium of an $(h,ρ)$-subelliptic random walk on a closed, connected Riemannian manifold $(M,g)$ associated with a subelliptic second-order differential operator $A$ on $M$. In such a random walk, $h$ roughly represents the step size and $ρ$ the speed at which it is carried out. To construct the random walk and prove the convergence result, we employ a technique due to Fefferman and Phong, which reduces the problem to the study of a constant-coefficient operator $\tilde{A}$ that is locally equivalent to our second-order subelliptic operator $A$, in the sense that the diffusion generated by $\tilde{A}$ induces a local diffusion for $A$. By using the compactness of $M$ this local diffusion can be lifted to a global diffusion, and the convergence result is then obtained via the spectral theory of the associated Markov operator.

math.AP↗
arXiv2025

Smoothing effect for third order operators with variable coefficients

In this work we study the smoothing effect of some variable coefficient operators of the form $D_t-A$, where $A$ is a Weyl-quantized pseudo-differential operator of order $m=2,3$. The class under consideration includes, among others, KdV-type and ultrahyperbolic Schrödinger operators. We prove homogeneous and inhomogeneous smoothing estimates and use them to get well-posedness results for some NLIVPs with derivative nonlinearities. Finally, we investigate the so called non-trapping property of the bicharacteristic curves of the principal symbol of our operators.

math.AP↗