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• Talk 01 — Oleksandr Konovalenko (O.Ya. Usikov Institute for Radiophysics and Electronics, Ukraine) — [computing…]
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  Title: Iterated completely positive maps in measurement-controlled dissipative two-qubit dynamics
  Abstract

  We consider a discrete-time dynamical model for a controlled open two-qubit system in an operator-theoretic setting. The evolution is defined by iterates of parameter-dependent completely positive trace-preserving maps acting on the space $\mathcal{T}(\mathcal{H})$ of trace-class operators, where $\mathcal{H} \cong \mathbb{C}^{4}$. The dissipative part of the dynamics is given by a quantum channel describing relaxation and dephasing induced by coupling to independent reservoirs, whereas repeated joint measurements in an entangled basis are encoded by additional completely positive maps. The full evolution is thus represented by compositions of bounded linear operators on $\mathcal{T}(\mathcal{H})$, whose restriction to the convex set of density operators generates a discrete noncommutative dynamical system. Our analysis is concerned with explicit representation formulas for the iterates, small-time asymptotics, and parameter-dependent stability properties of the resulting channel family. We derive closed expressions for the evolution of the matrix entries of the density operator under the combined dissipative and measurement-induced action and obtain asymptotic expansions for entanglement-related functionals in the short-time regime. In the weak-measurement case, we further derive analytic formulas for efficiency functionals measuring decoherence suppression and study their dependence on the measurement strength and on the number of iterations. This makes it possible to identify parameter regions in which the measurement component changes the effective action of the dissipative channel, in particular its contraction characteristics with respect to entanglement-sensitive quantities. We also examine a finite-time configuration motivated by quantum-information processing and compare the analytic asymptotics with numerical simulations. The agreement confirms that repeated measurement maps can substantially modify the effective dissipative evolution and improve finite-time preservation of entanglement. In this way, the model provides an explicit finite-dimensional setting for studying iterated completely positive maps, parameter-dependent perturbations, and stability properties of controlled open-system dynamics.

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• Talk 02 — Jan Dereziński (University of Warsaw, Poland) — [computing…]
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  Title: One-dimensional Schrödinger operators solvable in terms of the confluent and hypergeometric functions
  Abstract

  I will describe several holomorphic families of one-dimensional Schrödinger operators whose Green functions (the integral kernel of their resolvents) have simple expressions in terms of Bessel, or more generally confluent functions, and Gegenbauer, or more generally hypergeometric functions. Most of them were discovered by physicists in the early years of Quantum Mechanics. They have various surprising properties, including unexpected singularities and "transmutation identities", which link Green functions of distinct families. Some of these operators have nice geometric interpretations: e.g. Gegenbauer Hamiltonians arise when we separate the (pseudo-)Laplacian on the sphere, hyperbolic space and deSitter space. The talk is based on joint work with Jinyeop Lee and Pedram Karimi.

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• Talk 03 — Mateusz Piorkowski (KTH, Stockholm, Sweden) — [computing…]
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  Title: The regularity index of Sturm--Liouville operators with trace class resolvents
  Abstract

  In this talk I introduce the notion of the \emph{regularization index} $\ell \in\mathbb N \cup \{ \infty \}$ associated to the endpoints of nonoscillatory Sturm–Liouville differential expressions with trace class resolvents. This index extends the limit circle/limit point dichotomy in the sense that $\ell = 0$ at some endpoint if and only if the expression is in the limit circle case. In the limit point case $\ell > 0$, we present a natural interpretation of $\ell$ in terms of iterated Darboux transforms. We also show stability of the regularization index for a suitable class of perturbations, extending earlier work on perturbations of spherical Schr\"odinger operators to the case of general three terms Sturm--Liouville operators. If time permits, some connections to generalized Herglotz--Nevanlinna functions will also be presented. This is joint work with Jonathan Stanfill from Ohio State University.

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• Talk 04 — Monika Winklmeier (Universidad de los Andes, Colombia) — [computing…]
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  Title: On the eigenvalues of the Dirac operator
  Abstract

  Abstract will be posted when available.

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• Talk 05 — Andrés Felipe Patiño López (Universidad de los Andes, Colombia) — [computing…]
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  Title: Complete nonselfadjointness of Schrödinger-type operators on the bounded interval
  Abstract

  Let S be the minimal Laplacian defined in L²(0,1). In this talk, we present maximally dissipative extensions of the dissipative operator S + iV, where V is a bounded nonnegative operator. Our main focus is the analysis of the selfadjoint and symmetric subspaces associated with these maximally dissipative extensions. In particular, we describe how the selfadjoint and the symmetric subspaces can be describe in terms of the operator and the boundary conditions that define the extensions. This characterization allows us to establish criteria that determine whether or not a given maximally dissipative extension is completely nonselfadjoint. The case where S is the minimal Laplacian provides a concrete and important setting in which these ideas can be developed explicitly, illustrating the relationship between the operator V, the boundary conditions, and the structure of the corresponding dissipative extensions and their complete nonselfadjointness. Moreover, some of these arguments can be adapted to more general settings.

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• Talk 06 — Fritz Gesztesy (Baylor University, USA) — [computing…]
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  Title: Essential self-adjointness of fractional powers of the Laplacian perturbed by strongly singular homogeneous potential coefficients
  Abstract

  Authors: Fritz Gesztesy, Markus Hunziker, and Dorina Mitrea

  Abstract. For ∅ ≠ X ⊆ ℝⁿ discrete, n ∈ ℕ, we characterize the distributions in 𝒟′(ℝⁿ) with support contained in X and subsequently use this result to prove that C₀^∞(ℝⁿ \ X) is dense in the fractional Sobolev space H^s(ℝⁿ) if and only if s ∈ (−∞, n/2]. This fact is then used to show that

  (−Δ)^s|_{C0^∞(ℝⁿ \ X)} is essentially self-adjoint in L²(ℝⁿ)
  if and only if s ∈ (0, n/4] (i.e., if and only if n ≥ 4s),

  and analogously for the operator (−Δ + m²I)^s|_{C0^∞(ℝⁿ \ X)}, m ∈ (0, ∞).

  In addition, for n ∈ ℕ, s ∈ (0, n/4), applying the fractional Birman–Hardy–Rellich-type inequality combined with Kato–Rellich–Wüst perturbation theory, we prove essential self-adjointness (resp., self-adjointness) of

  [ (−Δ)^s + c | cot |^−2s ] |_H^2s(ℝⁿ)

  in L²(ℝⁿ) if c ∈ [−C_n,s, C_n,s] (resp., c ∈ (−C_n,s, C_n,s)). Here C_n,s is given by

  C_n,s = 2^2s Γ(n/4 + s) / Γ(n/4 − s),   s ∈ (0, n/4).
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• Talk 07 — Cody Stockdale (Clemson University, USA) — [computing…]
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  Title: Compact pseudodifferential and Fourier integral operators via localization
  Abstract

  We present a general framework of localized operators, i.e., operators whose matrix coefficients with respect to the Gabor frame are concentrated on the diagonal. We show that localized operators are bounded between modulation spaces, and we deduce their compactness from an easily verifiable weak compactness condition. We apply this abstract formalism to unify and extend existing theorems for pseudodifferential and Fourier integral operators, and to obtain new results for three-parameter pseudodifferential operators. This talk is based on joint work with Cody Waters.

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• Talk 08 — Cesar Mateo Bahos Orjuela (Universidad de los Andes, Colombia) — [computing…]
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  Title: Rank one unbounded perturbations of selfadjoint operators.
  Abstract

  In this talk we study possibly unbounded rank perturbations of a selfadjoint operator $A$ with discrete simple spectrum. We will always assume that the distance between two consecutive eigenvalues of $A$ is larger than some constant. We will characterize the eigenvalues of the perturbed operator as the zeros of the so-called characterstic function. We will show that they remain close in the L_2-norm to the eigenvalues of $A$. We will give conditions under which the root vectors of the perturbed operator is a Riesz basis.

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• Talk 09 — Edison Jair Leguizamon Quinche (Universidad de los Andes, Colombia) — [computing…]
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  Title: 1-D Schrödinger operators with singular interactions on non-discrete sets
  Abstract

  We study one–dimensional Schrödinger operators with singular interactions modeled by Dirac delta (δ) and delta-prime (δ′) potentials. Such interactions arise naturally in the modeling of quantum systems with highly localized forces and appear, for example, in models of particles moving through crystalline structures. Classical results usually assume that the interaction points form a discrete set with a positive minimal distance. In this work, we consider a more general setting in which the interaction points form a countable set that may fail to be discrete. To address this problem, the Schrödinger equation is formulated within the framework of measure differential equations, where the singular potentials are represented by Borel measures. This approach, developed by Eckhardt and Teschl, allows δ and δ′ interactions to be incorporated through appropriate jump conditions on the wave function or its quasi-derivative. Using the theory of quasi-derivatives and measure-valued potentials, we analyze the associated differential expression and its operator realizations. Within this framework, we investigate conditions ensuring the existence of self-adjoint realizations and study spectral properties of the corresponding operators, including aspects related to the limit point/limit circle classification and the possible presence of eigenvalues embedded in the essential spectrum.

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• Talk 10 — Ian Wood (University of Kent, UK) — [computing…]
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  Title: Spectrum of the Maxwell Equations for a Flat Interface between Non-Homogeneous Dispersive Media
  Abstract

  This talk concerns the time-harmonic Maxwell equation in three-dimensional space, divided into two half-spaces by a flat interface. The two half-spaces are filled with media whose electric permittivity is frequency-dependent and varies as a function of the distance from the interface only; this dependence is assumed to satisfy a regularity condition in each half-space but may be discontinuous at the interface. No specific model for the frequency dependence is assumed. For the associated operator pencil, we characterise subsets of the resolvent set and separate subsets of the Weyl spectrum corresponding to radiation away from the interface and along the interface, respectively. If the media are periodic in the direction orthogonal to the interface, a more explicit description of these sets can be given in terms of Floquet theory of related Sturm-Liouville equations. This is joint work with Malcolm Brown (Cardiff), Tomas Dohnal (Halle), Karl Michael Schmidt (Cardiff) and Michael Plum (Karlsruhe). 

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• Talk 11 — Matthias Hofmann (FernUniversität in Hagen, Germany) — [computing…]
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  Title: On the existence of Nehari ground states for the Nonlinear Schrödinger Equation on Discrete Graphs
  Abstract

  We present results on existence and study properties of standing wave solutions for the nonlinear Schrödinger equation on discrete graphs. We characterize for a self-adjoint realizations of Schrödinger operators conditions related with the geometry of the graph that guarantee discreteness of the spectrum and study ground states on the generalized Nehari manifold in order to prove the existence of standing wave solutions in the self-focusing and defocusing case. In this context, we show properties of the solutions, such as integrability. Furthermore, we discuss the bifurcation of solutions from the trivial solution in this context. The talk is based on a joint work with Setenay Akduman and Sedef Karakılıç.

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• Talk 12 — Alexander Kiselev (Independent Researcher; formerly affiliated with institutions in the UK and Europe) — [computing…]
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  Title: Effective properties of inhomogeneous media 
  Abstract

  I will discuss the deep connections between two different setups in homogenization theory for inhomogeneous periodic composites when the size of the periodic cell vanishes and the contrast between the material properties of the components constituting the composite increases. This increase can be either thought to be inversely proportionate to the squared cell size, leading to the so-called double porosity model, or be independent of the cell size. The two setups mentioned above are the static (e.g., electrostatic) and dynamic (e.g., convergence of resolvents in uniform operator topology), respectively. It is known that both formulations yield the same homogenized, or effective, medium in the case where the contrast is fixed (uniformly strongly elliptic case); I will discuss this and further generalisations to the case of increasing contrast.

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• Talk 13 — Yuliia Yershova (Michigan State University, USA) — [computing…]
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  Title: Effective static and quasi-static properties of periodic inhomogeneous media 
  Abstract

  In a wide class of periodic inhomogeneous media as the period vanishes it is possible to obtain an asymptotically equivalent description in the form of a homogeneous one. In particularly we mention the so-called effective conductivity problem as well as a number of other setups in mechanics and elasticity theory. We will discuss the relationship of these static and quasi-static results with those in the “dynamic” homogenisation. The latter involves norm-resolvent asymptotics for spectral parameter in a compact uniformly separated from the spectrum of the underlying linear operator. Using the machinery of Dirichlet-to-Neumann maps, I will show how to obtain new explicit characterizations for the effective properties of the homogenized medium both in the cases moderate and high contrast.

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