In the last decade, there has been an increasing interest in the p-Laplacian, which plays an important role in geometry and partial differential equations. The p-Laplacian is a natural generalization of the Laplacian. Although the Laplacian has been much studied, not much is known about the nonlinear case p >1. Motivated by these facts, the purpose of the present paper is to review recent developments in the spectral theory of a specific class of quantum waveguides modeled by the Dirichlet Laplacian, i.e. p = 2, in unbounded tubes of uniform cross-section rotating w.r.t. the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension. We discuss how the spectrum depends upon three geometric deformations: straightness, asymptotic straightness, and bending. Precisely, if the reference curve is straight or asymptotic straight, the essential spectrum is preserved. While dealing with bent tubes, such geometry produces a spectrum below the first eigenvalue. All the results confirm the literature for the Laplacian operator. The results are obtained via a very delicate analysis since the nonlinearity given by the p-Laplacian operator adds different types of difficulties with respect to the linear situation. These results are contained in a work written jointly with D. Krejčiřík.[-]

We study the existence of non-trivial lower bounds for positive powers of the discrete Dirichlet Laplacian on the half line. Unlike in the continuous setting where both $-\Delta$ and $(-\Delta)^2$ admit a Hardy-type inequality, their discrete analogues exhibit a different behaviour. While the discrete Laplacian is subcritical, its square is critical and the threshold where the criticality of $(-\Delta)^\alpha$ first appears turns out to be $\alpha=3 / 2$. We provide corresponding (non-optimal) Hardy-type inequalities in the subcritical regime. Moreover, for the critical exponent $\alpha=2$, we employ a remainder factorisation strategy to derive a discrete Rellich inequality on a suitable subspace (with a weight improving upon the classical Rellich weight). Based on joint work with D. Krejčiřík and F. Štampach.[-]

Magic angles are a topic of current interest in condensed matter physics and refer to a remarkable theoretical (Bistritzer–MacDonald, 2011) and experimental (Jarillo-Herrero et al, 2018) discovery: two sheets of graphene twisted by a certain (magic) angle display unusual electronic properties, such as superconductivity. In this talk, we shall discuss a simple periodic Hamiltonian describing the chiral limit of twisted bilayer graphene (Tarnopolsky-Kruchkov-Vishwanath, 2019), whose spectral properties are thought to determine which angles are magical. We show that the corresponding eigenfunctions decay exponentially in suitable geometrically determined regions as the angle of twisting decreases, which can be viewed as a form of semiclassical analytic hypoellipticity. This is joint work with Maciej Zworski.[-]

The main question that we will investigate in this talk is: what does the spectrum of a quantum channel typically looks like? We will see that a wide class of random quantum channels generically exhibit a large spectral gap between their first and second largest eigenvalues. This is in close analogy with what is observed classically, i.e. for the spectral gap of transition matrices associated to random graphs. In both the classical and quantum settings, results of this kind are interesting because they provide examples of so-called expanders, i.e. dynamics that are converging fast to equilibrium despite their low connectivity. We will also present implications in terms of typical decay of correlations in 1D many-body quantum systems. If time allows, we will say a few words about ongoing investigations of the full spectral distribution of random quantum channels. This talk will be based on: arXiv:1906.11682 (with D. Perez-Garcia), arXiv:2302.07772 (with P. Youssef) and arXiv:2311.12368 (with P. Oliveira Santos and P. Youssef).[-]

We consider an acoustic waveguide modeled as follows:

$ \left \{\begin {matrix}
\Delta u+k^2(1+V)u=0& in & \Omega= \mathbb{R} \times]0,1[\\
\frac{\partial u}{\partial y}=0& on & \partial \Omega
\end{matrix}\right.$

where $u$ denotes the complex valued pressure, k is the frequency and $V \in L^\infty(\Omega)$ is a compactly supported potential.
It is well-known that they may exist non trivial solutions $u$ in $L^2(\Omega)$, called trapped modes. Associated eigenvalues $\lambda = k^2$ are embedded in the essential spectrum $\mathbb{R}^+$. They can be computed as the real part of the complex spectrum of a non-self-adjoint eigenvalue problem, defined by using the so-called Perfectly Matched Layers (which consist in a complex dilation in the infinite direction) [1].
We show here that it is possible, by modifying in particular the parameters of the Perfectly Matched Layers, to define new complex spectra which include, in addition to trapped modes, frequencies where the potential $V$ is, in some sense, invisible to one incident wave.
Our approach allows to extend to higher dimension the results obtained in [2] on a 1D model problem.[-]

We consider the operator $\mathcal{A}_h = -h^2 \Delta + iV$ in the semi-classical limit $h \to 0$, where $V$ is a smooth real potential with no critical points. We obtain both the left margin of the spectrum, as well as resolvent estimates on the left side of this margin. We extend here previous results obtained for the Dirichlet realization of $\mathcal{A}_h$ by removing significant limitations that were formerly imposed on $V$. In addition, we apply our techniques to the more general Robin boundary condition and to a transmission problem which is of significant interest in physical applications.[-]

The Invariant Subspace Problem for (separable) Hilbert spaces is a long-standing open question that traces back to Jonhn Von Neumann's works in the fifties asking, in particular, if every bounded linear operator acting on an infinite dimensional separable Hilbert space has a non-trivial closed invariant subspace. Whereas there are well-known classes of bounded linear operators on Hilbert spaces that are known to have non-trivial, closed invariant subspaces (normal operators, compact operators, polynomially compact operators,...), the question of characterizing the lattice of the invariant subspaces of just a particular bounded linear operator is known to be extremely difficult and indeed, it may solve the Invariant Subspace Problem.

In this talk, we will focus on those concrete operators that may solve the Invariant Subspace Problem, presenting some of their main properties, exhibiting old and new examples and recent results about them obtained in collaboration with Prof. Carl Cowen (Indiana University-Purdue University).[-]

Originally arisen to understand characterizing properties connected with dispersive phenomena, in the last decades the method of multipliers has been recognized as a useful tool in Spectral Theory, in particular in connection with proof of absence of point spectrum for both self-adjoint and non self-adjoint operators. In this seminar we will see the developments of the method reviewing some recent results concerning self-adjoint and non self-adjoint Schrödinger operators in different settings, specifically both when the configuration space is the whole Euclidean space \R^d and when we restrict to domains with boundaries. We will show how this technique allows to detect physically natural repulsive and smallness conditions on the potentials which guarantee total absence of eigenvalues. Some very recent results concerning Pauli and Dirac operators will be also presented.
The talk is based on joint works with L. Fanelli and D. Krejcirik.[-]