arXiv subjects

Ivan Beschastnyi

Publications and source records attributed to Ivan Beschastnyi.

At least 19 recordsLinked to original sources

arXiv2026

Hamiltonian engineering via pulses: beyond group averaging

We develop a geometric framework for Hamiltonian engineering in finite-dimensional quantum systems using ideal control pulses. Starting from a bilinear Schrödinger equation with unbounded control amplitudes, we construct the closed pulse group and use extensions of control systems and Filippov's relaxation theorem to obtain a family of effective Hamiltonians given by the convex hull of the drift's adjoint orbit plus the Lie algebra of the pulse group. The geometry of this orbitope describes possibilities beyond group averaging. Using the isotypic decomposition of the adjoint representation, we characterize its affine hull and show that the group average lies in its interior. This yields locally accessible families of effective Hamiltonians around the invariant part of the drift. We apply the framework to recover the necessary and sufficient condition for dynamical decoupling from arbitrary interactions with a finite-dimensional bath. For connected abelian pulse groups, we describe the relevant representation decomposition through restricted roots. Finally, for qubit networks with identical pairwise couplings, we give a qualitative characterization and quantitative estimation of effective Hamiltonians that can be generated using our framework.

quant-ph↗

arXiv2026

The $S$-resolvent estimates for the Spinor Dirac operator on manifolds with boundary conditions

The aim of this paper is to show that the spectral theory based on the S-spectrum is particularly well suited for the Dirac operator on manifolds, even in cases where the operator is not self adjoint. Traditionally, for non-self adjoint operators in the Clifford setting, the literature has often referred to the right spectrum. However, a more comprehensive approach is provided by the theory of the $S$-spectrum, which is the appropriate notion for general operators on Clifford modules. In this work, we show that this theory is particularly well suited for bisectorial Clifford operators. By using the $S$-spectrum, which naturally contains the right eigenvalues, we prove bisectorial estimates for the $S$-resolvent associated with the spinor Dirac operator under various boundary conditions.

math.SP↗

arXiv2026

Sub-Riemannian structures and non-transitive Cartan geometries via Lie groupoids

In this paper we discuss how to associate a suitable non-transitive version of a Cartan connection to sub-Riemannian manifolds of corank 1 (including contact and quasi-contact sub-Riemannian manifolds) with non-necessarily constant sub-Riemannian symbols. In particular, we recast the variation of the sub-Riemannian symbols into a suitable "type" map, which is constant if and only if the symbols are constant. We then consider the (non-transitive) groupoid of sub-Riemannian symmetries and investigate its smoothness, properness, regularity, and other properties in relation with the type map. Last, we describe how to build a "non-transitive" analogue of a Cartan connection on top of such (Lie) groupoid, obtained as the sum of a tautological form with a multiplicative Ehresmann connection. We conclude by illustrating our results on concrete examples in dimension 5.

math.DG↗

arXiv2025

Open qubit parameter identification with bounded pulses

We address the problem of parameter identification for a single open qubit subjected to relaxation and dephasing. Our approach is based on selecting a minimal set of carefully chosen qubit configurations that can be reliably prepared and measured in order to provide an interpretable methodology of parameter identification while potentially minimizing experimental overhead. The protocol relies on saturating control pulses to generate these configurations. In an idealized regime of infinite-amplitude pulses, we demonstrate that the parameters can be reconstructed analytically from the measured observables. We then consider large but finite pulses as a perturbation of this ideal regime and provide bounds on the estimation error introduced by the practical implementation. This framework allows us to separate the sources of uncertainty in the estimation procedure, distinguishing between statistical fluctuations arising from repeated measurements and modeling errors due to deviations from the ideal pulse regime.

math.OC↗

arXiv2025

Singular extremals of optimal control problems with $L^1$ cost

We study the optimal control problem for a control-affine system, where we want to minimize the $L^1$ norm of the control. First, we show how Pontryagin Maximum Principle (PMP) applies to this problem and we divide the extremal trajectories into two categories: regular and singular extremals. Then, we obtain a strong generalized Legendre-Clebsch condition for singular extremals and we show that this condition together with the absence of conjugate points is sufficient to ensure local strong optimality. We provide also some geometric examples where we apply our results. Finally, we prove that generalized Legendre-Clebsch condition is necessary for optimality.

math.OC↗

arXiv2025

The $S$-resolvent estimates for the Dirac operator on hyperbolic and spherical spaces

This seminal paper marks the beginning of our investigation into on the spectral theory based on $S$-spectrum applied to the Dirac operator on manifolds. Specifically, we examine in detail the cases of the Dirac operator $\mathcal{D}_H$ on hyperbolic space and the Dirac operator $\mathcal{D}_S$ on the spherical space, where these operators, and their squares $\mathcal{D}_H^2$ and $\mathcal{D}_S^2$, can be written in a very explicit form. This fact is very important for the application of the spectral theory on the $S$-spectrum. In fact, let $T$ denote a (right) linear Clifford operator, the $S$-spectrum is associated with a second-order polynomial in the operator $T$, specifically the operator defined as $ Q_s(T) := T^2 - 2s_0T + |s|^2. $ This allows us to associate to the Dirac operator boundary conditions that can be of Dirichlet type but also of Robin-like type. Moreover, our theory is not limited to Hilbert modules; it is applicable to Banach modules as well. The spectral theory based on the $S$-spectrum has gained increasing attention in recent years, particularly as it aims to provide quaternionic quantum mechanics with a solid mathematical foundation from the perspective of spectral theory. This theory was extended to Clifford operators, and more recently, the spectral theorem has been adapted to this broader context. The $S$-spectrum is crucial for defining the so-called $S$-functional calculus for quaternionic and Clifford operators in various forms. This includes bounded as well as unbounded operators, where suitable estimates of sectorial and bi-sectorial type for the $S$-resolvent operator are essential for the convergence of the Dunford integrals in this setting.

math.FA↗

arXiv2024

Schrödinger operators with non-integer power-law potentials and Lie-Rinehart algebras

We study Schrödinger operators $H:= -Δ+ V$ with potentials $V$ that have power-law growth (not necessarily polynomial) at 0 and at $\infty$ using methods of Lie theory (Lie-Rinehart algebras) and microlocal analysis. More precisely, we show that $H$ is ''generated'' in a certain sense by an explicit Lie-Rinehart algebra. This allows then to construct a suitable (microlocal) calculus of pseudodifferential operators that provides further properties of $H$. Classically, this microlocal analysis method was used to study $H$ when the power-laws describing the potential $V$ have integer exponents. Thus, the main point of this paper is that this integrality condition on the exponents is not really necessary for the microlocal analysis method to work. While we consider potentials following (possibly non-integer) power-laws both at the origin and at infinity, our results extend right away to potentials having power-law singularities at several points. The extension of the classical microlocal analysis results to potentials with non-integer power-laws is achieved by considering the setting of Lie-Rinehart algebras and of the continuous family groupoids integrating them. (The classical case relies instead on Lie algebroids and Lie groupoids.)

math.DG↗

arXiv2024

Self-adjointness criteria and self-adjoint extensions of the Laplace-Beltrami operator on $α$-Grushin manifolds

The Grushin plane serves as one of the simplest examples of a sub-Riemannian manifold whose distribution is of non-constant rank. Despite the fact that the singular set where this distribution drops rank is itself a smoothly embedded submanifold, many basic results in the spectral theory of differential operators associated to this geometry remain open, with the question of characterizing self-adjoint extensions being a recent question of interest both in sub-Riemannian geometry and mathematical physics. In order to systematically address these questions, we introduce an exotic calculus of pseudodifferential operators adapted to the geometry of the singularity, closely related to the 0-calculus of Mazzeo arising in asymptotically hyperbolic geometry. Extending results of arXiv:2011.03300, arXiv:1105.4687, arXiv:1609.01724, this calculus allows us to give criterion for essential self-adjointness of the Curvature Laplacian, $Δ-cS$ for $c>0$ (here $S$ is the scalar curvature). When this operator is not essentially self-adjoint, we determine several natural self-adjoint extensions. Our results generalize to a broad class of differential operators which are elliptic in this calculus.

math.DG↗

arXiv2023

Analysis on noncompact manifolds and Index Theory: Fredholm conditions and Pseudodifferential operators

We provide Fredholm conditions for compatible differential operators on certain Lie manifolds (that is, on certain possibly non-compact manifolds with nice ends). We discuss in more detail the case of manifolds with cylindrical, hyperbolic, and Euclidean ends, which are all covered by particular instances of our results. We also discuss applications to Schrödinger operators with singularities of the form r^{-2γ}$, $γ\in \RR_+$.

math.AP↗

arXiv2023

Embedding the Grushin Cylinder in ${\bf R}^3$and Schroedinger evolution

We consider the evolution of a free quantum particle on the Grushin cylinder, under different type of quantizations. In particular we are interested to understand if the particle can cross the singular set, i.e., the set where the structure is not Riemannian. We consider intrinsic and extrinsic quantizations, where the latter are obtained by embedding the Grushin structure isometrically in ${\bf R}^3$ (with singularities). As a byproduct we provide formulas to embed the Grushin cylinder in ${\bf R^3}$ that could be useful for other purposes. Such formulas are not global, but permit to study the embedding arbitrarily close to the singular set. We extend these results to the case of $α$-Grushin cylinders.

math.FA↗

arXiv2022

Morse index and determinant of block Jacobi matrices via optimal control

We describe the relation between block Jacobi matrices and minimization problems for discrete time optimal control problems. Using techniques developed for the continuous case, we provide new algorithms to compute spectral invariants of block Jacobi matrices. Some examples and applications are presented.

math.OC↗

arXiv2022

Index theorems for graph-parametrized optimal control problems

In this paper we prove Morse index theorems for a big class of constrained variational problems on graphs. Such theorems are useful in various physical and geometric applications. Our formulas compute the difference of Morse indices of two Hessians related to two different graphs or two different sets of boundary conditions. Several applications such as the iteration formulas or lower bounds for the index are proved.

math.OC↗

arXiv2021

Quantum confinement for the curvature Laplacian $-Δ+cK$ on 2D-almost-Riemannian manifolds

Two-dimension almost-Riemannian structures of step 2 are natural generalizations of the Grushin plane. They are generalized Riemannian structures for which the vectors of a local orthonormal frame can become parallel. Under the 2-step assumption the singular set $Z$, where the structure is not Riemannian, is a 1D embedded submanifold. While approaching the singular set, all Riemannian quantities diverge. A remarkable property of these structures is that the geodesics can cross the singular set without singularities, but the heat and the solution of the Schrödinger equation (with the Laplace-Beltrami operator $Δ$) cannot. This is due to the fact that (under a natural compactness hypothesis), the Laplace-Beltrami operator is essentially self-adjoint on a connected component of the manifold without the singular set. In the literature such phenomenon is called quantum confinement. In this paper we study the self-adjointness of the curvature Laplacian, namely $-Δ+cK$, for $c\in(0,1/2)$ (here $K$ is the Gaussian curvature), which originates in coordinate-free quantization procedures (as for instance in path-integral or covariant Weyl quantization). We prove that there is no quantum confinement for this type of operators.

math.FA↗

arXiv2021

Closure of the Laplace-Beltrami operator on 2D almost-Riemannian manifolds and semi-Fredholm properties of differential operators on Lie manifolds

The problem of determining the domain of the closure of the Laplace-Beltrami operator on a 2D almost-Riemannian manifold is considered. Using tools from theory of Lie groupoids natural domains of perturbations of the Laplace-Beltrami operator are found. The main novelty is that the presented method allows us to treat geometries with tangency points. This kind of singularity is difficult to treat since those points do not have a tubular neighbourhood compatible with the almost-Riemannian metric.

math.DG↗

arXiv2021

Jacobi Fields in Optimal Control I: Morse and Maslov Indices

In this paper we discuss a general framework based on symplectic geometry for the study of second order conditions in constrained variational problems on curves. Using the notion of L-derivatives we construct Jacobi curves, which represent a generalization of Jacobi fields from the classical calculus of variations, but which also works for non-smooth extremals. This construction includes in particular the previously known constructions for specific types of extremals. We state and prove Morse-type theorems that connect the negative inertia index of the Hessian of the problem to some symplectic invariants of Jacobi curves.

math.OC↗

arXiv2020

Cartan connections for stochastic developments on sub-Riemannian manifolds

Analogous to the characterisation of Brownian motion on a Riemannian manifold as the development of Brownian motion on a Euclidean space, we construct sub-Riemannian diffusions on equinilpotentisable sub-Riemannian manifolds by developing a canonical stochastic process arising as the lift of Brownian motion to an associated model space. The notion of stochastic development we introduce for equinilpotentisable sub-Riemannian manifolds uses Cartan connections, which take the place of the Levi-Civita connection in Riemannian geometry. We first derive a general expression for the generator of the stochastic process which is the stochastic development with respect to a Cartan connection of the lift of Brownian motion to the model space. We further provide a necessary and sufficient condition for the existence of a Cartan connection which develops the canonical stochastic process to the sub-Riemannian diffusion associated with the sub-Laplacian defined with respect to the Popp volume. We illustrate the construction of a suitable Cartan connection for free sub-Riemannian structures with two generators and we discuss an example where the condition is not satisfied.

math.DG↗

arXiv2019

An obstruction to small-time controllability of the bilinear Schr{ö}dinger equation

In this article we discuss which controllability properties of classical Hamiltonian systems are preserved after quantization. We discuss some necessary and some sufficient conditions for small-time controllability of classical systems and quantum systems using the WKB method. In particular, we investigate the conjecture that if the classical system is not small-time controllable, then the corresponding quantum system is not small-time controllable either.

math.OC↗

arXiv2019

Jacobi Fields in Optimal Control II: One-dimensional Variations

In this second paper of the series we specify general theory developed in the first paper. Here we study the structure of Jacobi fields in the case of an analytic system and piece-wise analytic control. Moreover, we consider only 1-dimensional control variations. Jacobi fields are piece-wise analytic in this case but may have much more singularities than the control. We derive ODEs that these fields satisfy on the intervals of regularity and study behavior of the fields in a neighborhood of a singularity where the ODE becomes singular and the Jacobi fields may have jumps.

math.OC↗