Group-theoretical origin of symmetries of hypergeometric class equations and functions

We show that properties of hypergeometric class equations and functions become transparent if we derive them from appropriate 2nd order differential equations with constant coefficients. More precisely, properties of the hypergeometric and Gegenbauer equation can be derived from generalized symmetries of the Laplace equation in 4, resp. 3 dimension. Properties of the confluent, resp. Hermite equation can be derived from generalized symmetries of the heat equation in 2, resp. 1 dimension. Finally, the theory of the ${}_1F_1$ equation (equivalent to the Bessel equation) follows from the symmetries of the Helmholtz equation in 2 dimensions. All these symmetries become very simple when viewed on the level of the 6- or 5-dimensional ambient space. Crucial role is played by the Lie algebra of generalized symmetries of these 2nd order PDE's, its Cartan algebra, the set of roots and the Weyl group. Standard hypergeometric class functions are special solutions of these PDE's diagonalizing the Cartan algebra. Recurrence relations of these functions correspond to the roots. Their discrete symmetries correspond to the elements of the Weyl group.Comment: Prepared for the Summer School "Complex Differential and Difference Equations" 02.09.2018 - 15.09.2018 in B\k{e}dlew

Some recent developments in the transmutation operator approach

This is a brief overviewof some recent developments in the transmutation operator approach to practical solution of mathematical physics problems. It introduces basic notions and results of transmutation theory, and gives a brief historical survey with some important references. Mainly applications to linear ordinary and partial differential equations and to related boundary value and spectral problems are discusse

Quantum Distillation of Hilbert Spaces, Semi-classics and Anomaly Matching

A symmetry-twisted boundary condition of the path integral provides a suitable framework for the semi-classical analysis of nonperturbative quantum field theories (QFTs), and we reinterpret it from the viewpoint of the Hilbert space. An appropriate twist with the unbroken symmetry can potentially produce huge cancellations among excited states in the state-sum, without affecting the ground states; we call this effect "quantum distillation". Quantum distillation can provide the underlying mechanism for adiabatic continuity, by preventing a phase transition under $S^1$ compactification. We revisit this point via the 't Hooft anomaly matching condition when it constrains the vacuum structure of the theory on $\mathbb{R}^d$ and upon compactification. We show that there is a precise relation between the persistence of the anomaly upon compactification, the Hilbert space quantum distillation, and the semi-classical analysis of the corresponding symmetry-twisted path integrals. We motivate quantum distillation in quantum mechanical examples, and then study its non-trivial action in QFT, with the example of the 2D Grassmannian sigma model $\mathrm{Gr}(N,M)$. We also discuss the connection of quantum distillation with large-$N$ volume independence and flavor-momentum transmutation.Comment: 29 pages, 2 figure

One-Dimensional Semi-Relativistic Hamiltonian with Multiple Dirac Delta Potentials

Física Teórica, Atómica y Óptic

Features and stability analysis of non-Schwarzschild black hole in quadratic gravity

Black holes are found to exist in gravitational theories with the presence of quadratic curvature terms and behave differently from the Schwarzschild solution. We present an exhaustive analysis for determining the quasinormal modes of a test scalar field propagating in a new class of black hole backgrounds in the case of pure Einstein-Weyl gravity. Our result shows that the field decay of quasinormal modes in such a non-Schwarzschild black hole behaves similarly to the Schwarzschild one, but the decay slope becomes much smoother due to the appearance of the Weyl tensor square in the background theory. We also analyze the frequencies of the quasinormal modes in order to characterize the properties of new back holes, and thus, if these modes can be the source of gravitational waves, the underlying theories may be testable in future gravitational wave experiments. We briefly comment on the issue of quantum (in)stability in this theory at linear order.Comment: 18 pages, 4 figures, 1 table, several references added, version published on JHE

Hadron models and related New Energy issues

The present book covers a wide-range of issues from alternative hadron models to their likely implications in New Energy research, including alternative interpretation of lowenergy reaction (coldfusion) phenomena. The authors explored some new approaches to describe novel phenomena in particle physics. M Pitkanen introduces his nuclear string hypothesis derived from his Topological Geometrodynamics theory, while E. Goldfain discusses a number of nonlinear dynamics methods, including bifurcation, pattern formation (complex GinzburgLandau equation) to describe elementary particle masses. Fu Yuhua discusses a plausible method for prediction of phenomena related to New Energy development. F. Smarandache discusses his unmatter hypothesis, and A. Yefremov et al. discuss Yang-Mills field from Quaternion Space Geometry. Diego Rapoport discusses theoretical link between Torsion fields and Hadronic Mechanic. A.H. Phillips discusses semiconductor nanodevices, while V. and A. Boju discuss Digital Discrete and Combinatorial methods and their likely implications in New Energy research. Pavel Pintr et al. describe planetary orbit distance from modified Schrödinger equation, and M. Pereira discusses his new Hypergeometrical description of Standard Model of elementary particles. The present volume will be suitable for researchers interested in New Energy issues, in particular their link with alternative hadron models and interpretation

On uniform controllability of 1D transport equations in the vanishing viscosity limit

We consider a one dimensional transport equation with varying vector field and a small viscosity coefficient, controlled by one endpoint of the interval. We give upper and lower bounds on the minimal time needed to control to zero, uniformly in the vanishing viscosity limit. We assume that the vector field varies on the whole interval except at one point. The upper/lower estimates we obtain depend on geometric quantities such as an Agmon distance and the spectral gap of an associated semiclassical Schr{\"o}dinger operator. They improve, in this particular situation, the results obtained in the companion paper [LL21]. The proofs rely on a reformulation of the problem as a uniform observability question for the semiclassical heat equation together with a fine analysis of localization of eigenfunctions both in the semiclassically allowed and forbidden regions [LL22], together with estimates on the spectral gap [HS84, All98]. Along the proofs, we provide with a construction of biorthogonal families with fine explicit bounds, which we believe is of independent interest

LECTURES ON NONLINEAR DISPERSIVE EQUATIONS I

CONTENTS J. Bona Derivation and some fundamental properties of nonlinear dispersive waves equations F. Planchon Schr\"odinger equations with variable coecients P. Rapha\"el On the blow up phenomenon for the L^2 critical non linear Schrodinger Equatio