arXiv subjects

Alessandro Michelangeli

Publications and source records attributed to Alessandro Michelangeli.

At least 19 recordsLinked to original sources

arXiv2026

The Dynamical Landscape of Beggar-My-Neighbour: Ultra-long Matches, Loops, and Infinite Matches

We present a rigorous mathematical and computational analysis of the deterministic card game \emph{Beggar-My-Neighbour}. By establishing a formal state-space framework, we investigate the game's dynamical landscape, focussing on the dichotomy between terminating and non-terminating matches. Extensive numerical simulations reveal that the distribution of finite match durations \emph{approximates} an exponential decay, with relevant deviations, confirming an emergent memory-less dynamics. This statistical behaviour is further analysed in the context of ultra-long matches, where we identify characteristic multi-scale oscillatory patterns and entropic regimes. Theoretically, we address the problem of backwards determinism, formalising the lack of injectivity of the trick function even within the set of reachable states. Crucially, we contribute to the recent resolution of the long-standing question regarding the existence of infinite games. We introduce an automated `Infinite Loop Factory' algorithm which, by implementing adaptive insertion strategies, proves effective in identifying non-terminating cycles with balanced initial deck configurations, thereby confirming the existence of non-terminating dynamics in standard and generalised settings of the game.

math.GM↗

arXiv2026

Complexity scaling and optimal policy degeneracy in quantum reinforcement learning via analytically solvable unitary-control-then-measure models

We propose and analyse a class of analytically solvable models of quantum reinforcement learning (QRL), formulated as finite-horizon Markov decision processes in finite-dimensional Hilbert spaces. The models are built around a `unitary-control-then-measure' protocol, in which a learning agent applies unitary transformations to a quantum state and interleaves each control step with a projective measurement onto a prescribed reference basis. Exact closed-form expressions for trajectory probabilities, rewards, and the expected return are derived for four concrete realisations: a closed-chain and an anti-periodic qubit implementation, a qutrit model with ladder coupling, and a four-level two-qubit system. Two structural features of these QRL protocols are then analysed. First, we identify and quantify the reduction in the computational complexity of the expected return, from the nominally exponential $O(e^N)$ scaling in the trajectory length~$N$ to an explicit power-law $O(N^{\mathcal{I}})$, driven by two rigorously established mechanisms, a trajectory equivalence and a sparsity of the transition graph, besides a third, conjectured one: a spectral concentration of the return, at the optimal policy, onto the polynomially populated trajectory classes. Second, we characterise the degeneracy of optimal policies. The low-dimensional models exhibit unique optima whose asymptotic behaviour with~$N$ is governed by the quantum Zeno effect, while the four-level system displays both plateau-type quasi-degeneracy at large horizons and genuine discrete degeneracy at critical energy parameters -- phenomena with no counterpart in the measurement-free quantum optimal control landscape.

math.GM↗

arXiv2025

Non-relativistic limit of Dirac Hamiltonians with Aharonov-Bohm fields

We characterise the families of self-adjoint Dirac and Schrödinger operators with Aharonov-Bohm magnetic field, and we exploit the non-relativistic limit of infinite light speed to connect the former to the latter. The limit consists of the customary removal of the rest energy and of a suitable scaling, with the light speed, of the short-scale boundary condition of self-adjointness. This ensures that the scattering length of the Aharonov-Bohm interaction is preserved along the limit. Noteworthy is the fact that the whole family of Dirac-AB operators is mapped, in the non-relativistic limit, into the physically relevant sub-family of $s$-wave, angular-momentum-commuting, Schrö\-dinger-AB Hamiltonians with relativistic Dirac approximants.

math-ph↗

arXiv2024

Open problems and perspectives on solving Friedrichs systems by Krylov approximation

We set up, at the abstract Hilbert space setting, the general question on when an inverse linear problem induced by an operator of Friedrichs type admits solutions belonging to (the closure of) the Krylov subspace associated to such operator. Such Krylov solvability of abstract Friedrichs systems allows to predict when, for concrete differential inverse problems, truncation algorithms can or cannot reproduce the exact solutions in terms of approximants from the Krylov subspace.

math.FA↗

arXiv2024

On creating new essential spectrum by self-adjoint extension of gapped operators

Given a densely defined and gapped symmetric operator with infinite deficiency index, it is shown how self-adjoint extensions admitting arbitrarily prescribed portions of the gap as essential spectrum are identified and constructed within a general extension scheme. The emergence of new spectrum in the gap by self-adjoint extension is a problem with a long history and recent deep understanding, and yet it remains topical in several recent applications. Whereas it is already an established fact that, in case of infinite deficiency index, any kind of spectrum inside the gap can be generated by a suitable self-adjoint extension, the present discussion has the virtue of showing the clean and simple operator-theoretic mechanism of emergence of such extensions.

math.SP↗

arXiv2023

Self-adjoint extension schemes and modern applications to quantum Hamiltonians

This monograph contains revised and enlarged materials from previous lecture notes of undergraduate and graduate courses and seminars delivered by both authors over the last years on a subject that is central both in abstract operator theory and in applications to quantum mechanics: to decide whether a given densely defined and symmetric operator on Hilbert space admits a unique self-adjoint realisation, namely its operator closure, or whether instead it admits an infinite multiplicity of distinct self-adjoint extensions, and in the latter case to classify them and characterise their main features (operator and quadratic form domains, spectrum, etc.) This is at the same time a very classical, well established field, corresponding to the first part of the monograph, and a territory of novel, modern applications, a selection of which, obviously subjective to some extent, but also driven by a pedagogical criterion, is presented in depth in the second part. A number of models are discussed, which are receiving today new or renewed interest in mathematical physics, in particular from the point of view of realising certain operators of interests self-adjointly, classifying their self-adjoint extensions as actual quantum Hamiltonians, studying their spectral and scattering properties, and the like, but also from the point of view of intermediate technical questions that have theoretical interest per se, such as characterising the corresponding operator closures and adjoints.

math-ph↗

arXiv2023

On a comparison between absolute and relative self-adjoint extension schemes

The problem of connecting the operator parameters that label the same self-adjoint extension of a given symmetric operator, respectively, within the 'absolute' von Neumann extension scheme and the 'relative' boundary-triplet-induced extension scheme (i.e., a la Kreĭn-Višik-Birman) is discussed, and quantitative connections between the two parameters are established in the limit of deficiency spaces at complex spectral points converging to the deficiency space at a real spectral point.

math.FA↗

arXiv2022

On the characterisation of fragmented Bose-Einstein condensation and its emergent effective evolution

Fragmented Bose-Einstein condensates are large systems of identical bosons displaying \emph{multiple} macroscopic occupations of one-body states, in a suitable sense. The quest for an effective dynamics of the fragmented condensate at the leading order in the number of particles, in analogy to the much more controlled scenario for complete condensation in one single state, is deceptive both because characterising fragmentation solely in terms of reduced density matrices is unsatisfactory and ambiguous, and because as soon as the time evolution starts the rank of the reduced marginals generically passes from finite to infinite, which is a signature of a transfer of occupations on infinitely many more one-body states. In this work we review these difficulties, we refine previous characterisations of fragmented condensates in terms of marginals, and we provide a quantitative rate of convergence to the leading effective dynamics in the double limit of infinitely many particles and infinite energy gap.

math-ph↗

arXiv2022

Quantum Geometric Confinement and Dynamical Transmission in Grushin Cylinder

We classify the self-adjoint realisations of the Laplace-Beltrami operator minimally defined on an infinite cylinder equipped with an incomplete Riemannian metric of Grushin type, in the class of metrics yielding an infinite deficiency index. Such realisations are naturally interpreted as Hamiltonians governing the geometric confinement of a Schrödinger quantum particle away from the singularity, or the dynamical transmission across the singularity. In particular, we characterise all physically meaningful extensions qualified by explicit local boundary conditions at the singularity. Within our general classification we retrieve those distinguished extensions previously identified in the recent literature, namely the most confining and the most transmitting one.

math.DG↗

arXiv2022

Standing waves and global well-posedness for the 2d Hartree equation with a point interaction

We study a class of two-dimensional non-linear Schrödinger equations with point-like singular perturbation and Hartree non-linearity. The point-like singular perturbation of the free Laplacian induces appropriate perturbed Sobolev spaces that are necessary for the study of ground states and evolution flow. We include in our treatment both mass sub-critical and mass critical Hartree non-linearities. Our analysis is two-fold: we establish existence, symmetry, and regularity of ground states, and we demonstrate the well-posedness of the associated Cauchy problem in the singular perturbed energy space. The first goal, unlike other treatments emerging in parallel with the present work, is achieved by a non-trivial adaptation of the standard properties of Schwartz symmetrisation for the modified Weinstein functional. This produces, among others, modified Gagliardo-Nirenberg type inequalities that allow to efficiently control the non-linearity and obtain well-posedness by energy methods. The evolution flow is proved to be global in time in the defocusing case, and in the focusing and mass sub-critical case. It is also global in the focusing and mass critical case, for initial data that are suitably small in terms of the best Gagliardo-Nirenberg constant.

math.AP↗

arXiv2022

Inverse linear problems on Hilbert space and their Krylov solvability

This monograph is centred at the intersection of three mathematical topics, that are theoretical in nature, yet with motivations and relevance deep rooted in applications: the linear inverse problems on abstract, in general infinite-dimensional Hilbert space; the notion of Krylov subspace associated to an inverse problem, i.e., the cyclic subspace built upon the datum of the inverse problem by repeated application of the linear operator; the possibility to solve the inverse problem by means of Krylov subspace methods, namely projection methods where the finite-dimensional truncation is made with respect to the Krylov subspace and the approximants converge to an exact solution to the inverse problem.

math.FA↗

arXiv2021

Lieb-Robinson bounds and growth of correlations in Bose mixtures

For a mixture of interacting Bose gases initially prepared in a regime of condensation (uncorrelation), it is proved that in the course of the the time evolution observables of disjoint sets of particles of each species have correlation functions that remain asymptotically small in the total number of particles and display a controlled growth in time. This is obtained by means of ad hoc estimates of Lieb-Robinson type on the propagation of the interaction, established here for the multi-component Bose mixture.

quant-ph↗

arXiv2021

Heat equation with inverse-square potential of bridging type across two half-lines

The heat equation with inverse square potential on both half-lines of $\mathbb{R}$ is discussed in the presence of \emph{bridging} boundary conditions at the origin. The problem is the lowest energy (zero-momentum) mode of the transmission of the heat flow across a Grushin-type cylinder, a generalisation of an almost Riemannian structure with compact singularity set. This and related models are reviewed, and the issue is posed of the analysis of the dispersive properties for the heat kernel generated by the underlying positive self-adjoint operator. Numerical integration is shown that provides a first insight and relevant qualitative features of the solution at later times.

math.AP↗

arXiv2021

Convergence of the conjugate gradient method with unbounded operators

In the framework of inverse linear problems on infinite-dimensional Hilbert space, we prove the convergence of the conjugate gradient iterates to an exact solution to the inverse problem in the most general case where the self-adjoint, non-negative operator is unbounded and with minimal, technically unavoidable assumptions on the initial guess of the iterative algorithm. The convergence is proved to always hold in the Hilbert space norm (error convergence), as well as at other levels of regularity (energy norm, residual, etc.) depending on the regularity of the iterates. We also discuss, both analytically and through a selection of numerical tests, the main features and differences of our convergence result as compared to the case, already available in the literature, where the operator is bounded.

math.NA↗

arXiv2021

Quantum particle across Grushin singularity

A class of models is considered for a quantum particle constrained on degenerate Riemannian manifolds known as Grushin cylinders, and moving freely subject only to the underlying geometry: the corresponding spectral analysis is developed in detail in view of the phenomenon of transmission across the singularity that separates the two half-cylinders. Whereas the classical counterpart always consists of a particle falling in finite time along the geodesics onto the metric's singularity locus, the quantum models may display geometric confinement, or on the opposite partial transmission and reflection. All the local realisations of the free (Laplace-Beltrami) quantum Hamiltonian are examined as non-equivalent protocols of transmission/reflection and the structure of their spectrum is characterised, including when applicable their ground state and positivity. Besides, the stationary scattering analysis is developed and transmission and reflection coefficients are calculated. This allows to comprehend the distinguished status of the so-called `bridging' transmission protocol previously identified in the literature, which we recover and study within our systematic analysis.

math.SP↗

arXiv2021

Self-adjointness in Quantum Mechanics: a pedagogical path

Observables in quantum mechanics are represented by self-adjoint operators on Hilbert space. Such ubiquitous, well-known, and very foundational fact, however, is traditionally subtle to be explained in typical first classes in quantum mechanics, as well as to senior physicists who have grown up with the lesson that self-adjointness is "just technical". The usual difficulties are to clarify the connection between the demand for certain physical features in the theory and the corresponding mathematical requirement of self-adjointness, and to distinguish between self-adjoint and hermitian operator not just at the level of the mathematical definition but most importantly from the perspective that mere hermiticity, without self-adjointness, does not ensure the desired physical requirements and leaves the theory inconsistent. In this work we organise an amount of standard facts on the physical role of self-adjointness into a coherent pedagogical path aimed at making quantum observables emerge as necessarily self-adjoint, and not merely hermitian operators. Next to the central core of our line of reasoning -- the necessity of a non-trivial declaration of a domain to associate with the formal action of an observable, and the emergence of self-adjointness as a consequence of fundamental physical requirements -- we include some complementary materials consisting of a few instructive mathematical proofs and a short retrospective, ranging from the past decades to the current research agenda, on the self-adjointness problem for quantum Hamiltonians of relevance in applications.

quant-ph↗

arXiv2021

Krylov solvability under perturbations of abstract inverse linear problems

When a solution to an abstract inverse linear problem on Hilbert space is approximable by finite linear combinations of vectors from the cyclic subspace associated with the datum and with the linear operator of the problem, the solution is said to be a Krylov solution, i.e., it belongs to the Krylov subspace of the problem. Krylov solvability of the inverse problem allows for solution approximations that, in applications, correspond to the very efficient and popular Krylov subspace methods. We study here the possible behaviours of persistence, gain, or loss of Krylov solvability under suitable small perturbations of the inverse problem -- the underlying motivations being the stability or instability of Krylov methods under small noise or uncertainties, as well as the possibility to decide a priori whether an inverse problem is Krylov solvable by investigating a potentially easier, perturbed problem. We present a whole scenario of occurrences in the first part of the work. In the second, we exploit the weak gap metric induced, in the sense of Hausdorff distance, by the Hilbert weak topology, in order to conveniently monitor the distance between perturbed and unperturbed Krylov subspaces.

math.FA↗

arXiv2021

On general convergence behaviours of finite-dimensional approximants for abstract linear inverse problems

In the framework of abstract linear inverse problems in infinitedimensional Hilbert space we discuss generic convergence behaviours of approximate solutions determined by means of general projection methods, namely outside the standard assumptions of Petrov-Galerkin truncation schemes. This includes a discussion of the mechanisms why the error or the residual generically fail to vanish in norm, and the identification of practically plausible sufficient conditions for such indicators to be small in some weaker sense. The presentation is based on theoretical results together with a series of model examples and numerical tests.

math.NA↗