Geometric Methods in Physics XXXIX: Workshop, Białystok, Poland, 2022 (Trends in Mathematics)

Preface
Contents
In Memoriam: Anatol Odzijewicz 1947–2022
In Memoriam: Emma Previato 1952–2022
Part I Session in Memory of Bogdan Mielnik
Incomplete Visions…
Memories About Bogdan Mielnik
	1 Bogdan as Seen by an Admiring Student
	2 Bogdan in Action
		2.1 Presenting Science to Broad Public
		2.2 Improvised Speeches
		2.3 Articles on Topics Beyond Physics
		2.4 A Movie Appearance
		2.5 Other Public Appearances
		2.6 Other Recollections
	3 Summary
	4 Conclusion
Scattering States of the Inverted Oscillator and Its Supersymmetric Partners
	1 Introduction
	2 Harmonic, Inverted, and Complex Oscillators
	3 Scattering States of the Inverted Oscillator
	4 Supersymmetric Partners of the Inverted Oscillator
	5 Scattering States of the Supersymmetric Partners of the Inverted Oscillator
	6 Conclusions
	References
The Klein Paradox in the Phase Space Quantum Mechanics
	1 Motivation
	2 The Klein Paradox
	3 Eigenfunctions of the System
	4 The Wigner Function for a 1D Scattering System
	5 The Current Density
	References
An Asymmetric Harmonic Oscillator
	1 Introduction
	2 Classical and Quantum Case
		2.1 Symmetric Harmonic Oscillator
		2.2 Asymmetric Harmonic Oscillator
		2.3 Properties of Solutions and Positions of Asymptotes
		2.4 Eigenvectors and Eigenvalues
	3 Wave Functions and Fock Representation
	4 Examples
	References
On the Construction of Position-Dependent Mass Models with Quadratic Spectra
	1 Introduction
	2 Position-Dependent Mass Hamiltonian Hierarchies with Quadratic Spectra
	3 The Solution of the Eigenvalue Problem
	4 Some Particular Examples
		4.1 Deformed Oscillator-Type Potential with Quadratic Spectrum
		4.2 Deformed Scarf I-Type Potentials
		4.3 Error Function-Like Potentials
	5 Summary
	References
Bogdan Mielnik\'s Contributions to the Factorization Method
	1 Introduction
	2 Harmonic Oscillator Factorization
	3 Mielnik\'s Factorization
	4 Second-Order Technique
		4.1 Real Case with c>0
		4.2 Confluent Case with c=0 mnr00,fs03
		4.3 Complex Case with c<0 fmr03
	5 Periodic Potentials
	6 State of the Art Up to 2004
	References
The 1-D Dirac Equation in the Phase Space Quantum Mechanics
	1 Motivation
	2 The 1-D Dirac Equation
	3 Phase Space for a System with Internal Degrees of Freedom
	4 Representation of States on Phase Space R R (s+1)
	5 Phase Space Model of the 1-D Dirac Equation
	References
Canonical Photon Position Operator with Commuting Components
	1 Introduction
	2 Construction of the Photon Position Operator
	3 Vector Fiber Bundle Approach to Quantum Mechanics of the Photon
	References
Quantum Version of Euler\'s Problem
	1 Introduction
	2 Separable and Maximally Entangled States of a Bipartite d d Quantum System
	3 Non-displaceable Manifolds and Mutually Entangled States
	4 Multipartite Systems and Absolutely Maximally Entangled States
	5 Classical Orthogonal Latin Squares
	6 Quantum Orthogonal Latin Squares
	7 Solution of the Quantum Euler Problem for d=6 and Its Geometric Consequences
	8 Concluding Remarks
	Appendix: Glossary of Key Mathematical Terms Used
		Algebraic
		Combinatorial
		Geometric
	References
Part II Contributions to the XXXIX Workshop
Representations of Solvable Lie Groups
	1 Introduction
	2 Preliminaries on Operator Algebras
		2.1 Very Basic Notions
		2.2 Finiteness Properties of von Neumann Algebras
		2.3 Group von Neumann Algebras
	3 Preliminaries on the Pukánszky Correspondence
	4 The Main Results
		4.1 Method of Proof of Theorem 19
	5 Examples of Groups with Factor Regular Representations
	6 Perspectives and Open Problems
	References
Pedal Coordinates and Orbits Inside Magnetic Dipole Field
	1 Introduction
	2 Magnetic Dipole Field Problem
		2.1 Bounded and Unbounded Components
	Appendix: Gallery
	References
Hidden Symmetries of the Type D Metrics
	1 Introduction
	2 Type D Metric and Its First Integrals
	3 Killing Tensors
	4 Test Particle Trajectories in the Type D Metric
		4.1 Associated Killing Vectors
		4.2 Associated Conformal Killing Tensors
		4.3 Trajectories of a Charged Particle
	5 Conclusions
	References
On Some Structures of Lie Algebroids
	1 Introduction
	2 Structures of Lie Algebroids Determined by Vector Fields
	3 Example
	4 Conclusions
	References
Predicting ``Anyons\'\': Implications of History for Science
	1 Anyons and Non-Abelian Anyons: The Easiest of Ideas
	2 Why So Long? Epistemological and Cognitive Obstacles
	3 Antecedent Ideas
	4 Three Independent Predictions
	5 Issues of Acknowledgment and Their Wider Consequences
	References
Quadratic Algebra and Spectrum of Superintegrable System
	1 Introduction
	2 Separation of Variables
	3 Integrals of Motion and Symmetry Algebra
	4 First Algebraic Derivation
	5 Second Algebraic Derivation
	6 Conclusion
	References
Unifying Classical and Quantum Physics + Quantum Fields andGravity
	1 Proper Quantization Rules for Two Different Kinds of Problems
		1.1 Proper Rules for Canonical Quantization
		1.2 Proper Rules for Affine Quantization
	2 The Unification of Classical and Quantum Realms
	3 The Quantization of Field Theories
		3.1 Examining the Territory
		3.2 An Affine Quantization of Field Theories
	4 The Quantization of Einstein\'s Gravity
		4.1 Examining the Territory
	5 Summary
	References
Algebraic Quantum Field Theory and Causal Symmetric Spaces
	1 Introduction
	2 Causal Symmetric Spaces and Euler Elements
	3 Wedge Regions in Causal Symmetric Spaces
	4 Unitary and Anti-unitary Representations
		4.1 Smooth Vectors and Distribution Vectors
		4.2 Anti-unitary Representations
	5 Standard Subspaces
	6 Cayley Type Spaces and Causal Compactifications
	References
The Parametrically Extended Kardar–Parisi–Zhang Equation, Its Dark-Type Generalization, and Integrability
	1 Introduction
	2 Dynamical Systems with Hidden Symmetries: Mathematical Preliminaries
		2.1 Generalized Eigenvalue Problem
		2.2 The Periodic Spectral Problem: Floquet Theory Aspects
		2.3 Generative Function of Conserved Quantities
		2.4 Gradient-Holonomic Integrability Analysis
		2.5 Conservation Laws and the Related Involutive Properties
	3 Nonlinear Dynamical Systems and Integrability Testing Algorithm
		3.1 The Noether–Lax Determining Equation
		3.2 An Optimal Control Problem Aspect
	4 Hidden Symmetry Analysis of the Parametrically Dependent Nonlinear Kardar–Parisi–Zhang Equation
		4.1 The Noether–Lax Equation and Its Asymptotic Solutions
		4.2 Conserved Quantities and Dark-Type Parametric Extensions of the Kardar–Parisi–Zhang Equation
	5 Conclusion
	References
Banach Poisson–Lie Group Structure on U(H)
	1 Introduction
	2 Definition of Banach Poisson–Lie groups
	3 Some Subspaces of u*(H) in Duality with u(H)
		3.1 Duality Pairing Between u(H) and b+1(H)
		3.2 A Subspace of u*(H) on which u(H) Acts Continuously by Coadjoint Action
	4 The Unitary Group U(H) as a Banach Poisson–Lie Group
	References
Symplectic Realizations of e(3)*
	1 Introduction
	2 Gyrostat Hamiltonian System
	3 Penrose Twistor Space as a Symplectic Realization of e(3)*
	4 Symplectic Realizations of e(3)* Defined by Symplectic Reduction
	5 Examples of Integrable Hamiltonian Systems on the Symplectic Realizations of e(3)*
	References
On Some Developments of the Stokes Phenomenon
	1 Introduction
	2 Meromorphic Linear Systems of Poincaré Rank 1 and Stokes Matrices
		2.1 The Unique Formal Fundamental Solution
		2.2 The Canonical Solutions with a Prescribed Asymptotics
		2.3 Stokes Matrices
		2.4 Isomonodromic Deformation
	3 The Analytic Aspect of the Stokes Matrices
		3.1 Expression of the Stokes Matrices
		3.2 Leading Asymptotics of the Solutions of Isomonodromy Equations
		3.3 Explicit Expression of the Stokes Matrices
		3.4 A Connection Formula of the Isomonodromy Equation
		3.5 Poisson Brackets in r-matrix Forms
	4 The Algebraic Aspect of the Stokes Matrices
		4.1 Formal Power Series Solutions and Yangians
		4.2 Stokes Matrices and Representation of Quantum Groups
		4.3 WKB Approximation and Crystal Basis
		4.4 The Asymptotics of Stokes Matrices and Gelfand-Tsetlin Basis
		4.5 A Model of the WKB Approximation of the Stokes Matrices
		4.6 Soliton Solutions
	References
Comparison of Two Topologies on Weighted Szegő Space
	1 Preliminaries
	2 Two Topologies
	References
Part III Abstracts of the Lectures at ``XI School on Geometry and Physics\'\'
Lecture Notes on Quantization and Group C*-Algebras
	1 Group C*-Algebras
		1.1 Noncommutative Topology
		1.2 Multiplier Algebras
		1.3 Group C*-Algebras
	2 Examples of Solvable Lie Groups G1G2 with C*(G1)C*(G2)
	3 Representation Theory of Nilpotent Lie Groups
		3.1 Basic Definitions
		3.2 Heisenberg Groups and Their Schrödinger Representations
		3.3 The Method of Coadjoint Orbits for General Nilpotent Lie Groups
	4 C*-Rigidity
	References
Supersymmetric Quantum Mechanics and Painlevé IV Transcendents
	1 Introduction
	2 Supersymmetric Quantum Mechanics
	3 Polynomial Heisenberg Algebras
	4 Harmonic Oscillator SUSY Partners
	5 General Systems Ruled by Second-Degree PHA
	6 PIV Transcendents Through SUSY
	7 Conclusions
	References
On Some Concepts in the Theory of Lie Algebroids
	1 Groupoids and Algebroids
	2 Algebroids and Supergeometry
	3 Higher-Order Analogues of Lie Algebroids
	References