arXiv subjects

Jan Dereziński

Publications and source records attributed to Jan Dereziński.

At least 19 recordsLinked to original sources

arXiv2026

Exactly solvable Schrödinger operators related to the hypergeometric equation

We study one-dimensional Schrödinger operators defined as closed operators that are exactly solvable in terms of the Gauss hypergeometric function. We allow the potentials to be complex. These operators fall into three groups. The first group can be reduced to the Gegenbauer equation, up to an affine transformation, a special case of the hypergeometric equation. The two other groups, which we call {\em hypergeometric of the first}, resp. {\em second kind}, can be reduced to the general Gauss hypergeometric equation. Each of the group is subdivided in three families, acting to on the Hilbert space $L^2]-1,1[,$ $L^2(\rr_+)$ resp. $L^2(\rr)$. Motivated by geometric applications of these families, we call them {\em spherical}, {\em hyperbolic}, resp. {\em deSitterian}. All these families are known from applications in Quantum Mechanics: e.g. spherical hypergeometric Schrödinger operators of the first kind are often called {\em trigonometric Pöschl-Teller Hamiltonians}. For operators belonging to each family we compute their spectrum and determine their Green function (the integral kernel of their resolvent). We also describe transmutation identities that relate these Green functions. These identities interchange spectral parameters with coupling constants across different operator families. Finally, we describe how these operators arise from separation of variables of (pseudo-)Laplacians on symmetric manifolds. Our paper can be viewed as a sequel to \cite{DL}, where closed realizations of one-dimensional Schrödinger operators solvable in terms Kummer's confluent equation were studied.

math-ph↗

arXiv2026

Damping of phonons in Bose gas at low temperatures

We consider homogeneous Bose gas in a large cubic box with periodic boundary conditions interacting with a small potential with a positive Fourier transform. We compute the imaginary part of the phononic excitation spectrum in the lowest order of perturbation theory in thermodynamic limit at low temperatures and low momentum. Our analysis is based on perturbation theory of the standard Liouvillean. We use two approaches: the first, motivated by the standard representation of operator algebras, examines resonances near zero; the second analyzes the 2-point correlation function in the energy-momentum space.

math-ph↗

arXiv2025

Confluent functions, Laguerre polynomials and their (generalized) bilinear integrals

We review properties of confluent functions and the closely related Laguerre polynomials, and determine their bilinear integrals. As is well-known, these integrals are convergent only for a limited range of parameters. However, when one uses the generalized integral they can be computed essentially without restricting the parameters. This gives the (generalized) Gram matrix of Laguerre polynomials. If the parameters are not negative integers, then Laguerre polynomials are orthogonal, or at least pseudo-orthogonal in the case of generalized integrals. For negative integer parameters, the orthogonality relations are more complicated.

math.CA↗

arXiv2025

Propagators in curved spacetimes from operator theory

We discuss two distinct operator-theoretic settings useful for describing (or defining) propagators associated with a scalar Klein-Gordon field on a Lorentzian manifold $M$. Typically, we assume that $M$ is globally hyperbolic. The term propagator here refers to any Green function or bisolution of the Klein-Gordon equation pertinent to Quantum Field Theory. The off-shell setting is based on the Hilbert space $L^2(M)$. It leads to the definition of the operator-theoretic Feynman and anti-Feynman propagators, which often coincide with the so-called in-out Feynman and out-in anti-Feynman propagator. On some special spacetimes, the sum of the operator-theoretic Feynman and anti-Feynman propagator equals the sum of the forward and backward propagator. This is always true on static stable spacetimes and, curiously, in some other cases as well. The on-shell setting is based on the Krein space $\mathcal{W}_{\rm KG}$ of solutions of the Klein-Gordon equation. It allows us to define 2-point functions associated to two, possibly distinct, Fock states as the Klein-Gordon kernels of projectors onto maximal uniformly positive subspaces of $\mathcal{W}_{\rm KG}$. After a general discussion, we review a number of examples. We start with static and asymptotically static spacetimes, which are especially well-suited for Quantum Field Theory. Then we discuss FLRW spacetimes, reducible by a mode decomposition to 1-dimensional Schrödinger operators. We compare various approaches to de Sitter space where, curiously, the off-shell approach gives non-physical propagators. Finally, we discuss the universal cover of anti-de Sitter spaces, where the on-shell approach may require boundary conditions, unlike the off-shell approach.

math-ph↗

arXiv2025

A unified approach to hypergeometric class functions

Hypergeometric class equations are given by second order differential operators in one variable whose coefficient at the second derivative is a polynomial of degree $\leq2$, at the first derivative of degree $\leq1$ and the free term is a number. Their solutions, called hypergeometric class functions, include the Gauss hypergeometric function and its various limiting cases. The paper presents a unified approach to these functions. The main structure behind this approach is a family of complex 4-dimensional Lie algebras, originally due to Willard Miller. Hypergeometric class functions can be interpreted as eigenfunctions of the quadratic Casimir operator in a representation of Miller's Lie algebra given by differential operators in three complex variables. One obtains a unified treatment of various properties of hypergeometric class functions such as recurrence relations, discrete symmetries, power series expansions, integral representations, generating functions and orthogonality of polynomial solutions.

math.CA↗

arXiv2025

Exactly solvable Schrödinger operators related to the confluent equation

Our paper investigates one-dimensional Schrödinger operators defined as closed operators on $L^2(\mathbb{R})$ or $L^2(\mathbb{R}_+)$ that are exactly solvable in terms of confluent functions (or, equivalently, Whittaker functions). We allow the potentials to be complex. They fall into three families: Whittaker operators (or radial Coulomb Hamiltonians), Schrödinger operators with Morse potentials and isotonic oscillators. For each of them, we discuss the corresponding basic holomorphic family of closed operators and the integral kernel of their resolvents. We also describe transmutation identities that relate these resolvents. These identities interchange spectral parameters with coupling constants across different operator families. A similar analysis is performed for one-dimensional Schrödinger operators solvable in terms of Bessel functions (which are reducible to special cases of Whittaker functions). They fall into two families: Bessel operators and Schrödinger operators with exponential potentials. To make our presentation self-contained, we include a short summary of the theory of closed one-dimensional Schrödinger operators with singular boundary conditions. We also provide a concise review of special functions that we use.

math-ph↗

arXiv2025

Axial anomaly in the presence of arbitrary spinor interactions

We consider N Dirac fermions on a 4-dimensional Euclidean space with a quadratic interaction given by arbitrary external Clifford-valued fields. The divergence of the axial current satisfies on the classical level a relation that is violated after quantization. Using the Pauli-Villars method to regularize the fields, we find the conditions that guarantee the finiteness of the anomaly. We also find this anomaly. Our result generalizes the well-known computation of axial anomaly of Dirac fermions interacting with an external Yang-Mills field.

hep-th↗

arXiv2024

Bessel potentials and Green functions on pseudo-Euclidean spaces

We review properties of Bessel potentials, that is, inverse Fourier transforms of (regularizations of) $\frac{1}{(m^2+p^2)^{\fracμ{2}}}$ on a pseudoEuclidean space with signature $(q,d-q)$. We are mostly interested in the Lorentzian signature $(1,d-1)$, and the case $μ=2$, related to the Klein-Gordon equation $(-\Box+m^2)f=0$. We analyze properties of various ``two-point functions'', which play an important role in Quantum Field Theory, such as the retarded/advanced propagators or Feynman/antiFeynman propagators. We consistently use hypergeometric functions instead of Bessel functions, which makes most formulas much more transparent. We pay attention to distributional properties of various Bessel potentials. We include in our analysis the ``tachyonic case'', corresponding to the ``wrong'' sign in the Klein-Gordon equation.

math-ph↗

arXiv2024

Point potentials on Euclidean space, hyperbolic space and sphere in any dimension

In dimensions d= 1, 2, 3 the Laplacian can be perturbed by a point potential. In higher dimensions the Laplacian with a point potential cannot be defined as a self-adjoint operator. However, for any dimension there exists a natural family of functions that can be interpreted as Green's functions of the Laplacian with a spherically symmetric point potential. In dimensions 1, 2, 3 they are the integral kernels of the resolvent of well-defined self-adjoint operators. In higher dimensions they are not even integral kernels of bounded operators. Their construction uses the so-called generalized integral, a concept going back to Riesz and Hadamard. We consider the Laplace(-Beltrami) operator on the Euclidean space, the hyperbolic space and the sphere in any dimension. We describe the corresponding Green's functions, also perturbed by a point potential. We describe their limit as the scaled hyperbolic space and the scaled sphere approach the Euclidean space. Especially interesting is the behavior of positive eigenvalues of the spherical Laplacian, which undergo a shift proportional to a negative power of the radius of the sphere. We expect that in any dimension our constructions yield possible behaviors of the integral kernel of the resolvent of a perturbed Laplacian far from the support of the perturbation. Besides, they can be viewed as toy models illustrating various aspects of renormalization in Quantum Field Theory, especially the point-splitting method and dimensional regularization.

math-ph↗

arXiv2024

Beliaev damping in Bose gas

According to the Bogoliubov theory the low energy behaviour of the Bose gas at zero temperature can be described by non-interacting bosonic quasiparticles called phonons. In this work the damping rate of phonons at low momenta, the so-called Beliaev damping, is explained and computed with simple arguments involving the Fermi Golden Rule and Bogoliubov's quasiparticles.

math-ph↗

arXiv2023

Generalized integrals and point interactions

First we recall a method of computing scalar products of eigenfunctions of a Sturm-Liouville operator. This method is then applied to Macdonald and Gegenbauer functions, which are eigenfunctions of the Bessel, resp. Gegenbauer operators. The computed scalar products are well defined only for a limited range of parameters. To extend the obtained formulas to a much larger range of parameters, we introduce the concept of a generalized integral. The (standard as well as generalized) integrals of Macdonald and Gegenbauer functions have important applications to operator theory. Macdonald functions can be used to express the integral kernels of the resolvent (Green functions) of the Laplacian on the Euclidean space in any dimension. Similarly, Gegenbauer functions appear in Green functions of the Laplacian on the sphere and the hyperbolic space. In dimensions 1,2,3 one can perturb these Laplacians with a point potential, obtaining a well defined self-adjoint operator. Standard integrals of Macdonald and Gegenbauer functions appear in the formulas for the corresponding Green functions. In higher dimensions the Laplacian perturbed by point potentials does not exist. However, the corresponding Green function can be generalized to any dimension by using generalized integrals.

math-ph↗

arXiv2023

An Evolution Equation Approach to Linear Quantum Field Theory

In the first part of our paper we analyze bisolutions and inverses of (non-autonomous) evolution equations. We are mostly interested in pseudo-unitary evolutions on Krein spaces, which naturally arise in linear Quantum Field Theory. We prove that with boundary conditions given by a maximal positive and maximal negative space we can associate an inverse, which can be viewed as a generalization of the usual Feynman propagator. In the context of globally hyperbolic manifolds, the Feynman propagator turns out to be a distinguished inverse of the Klein-Gordon operator. Within the formalism of Quantum Field Theory on curved spacetimes, the Feynman propagator yields the expectation values of time-ordered products of fields between the in and out vacuum --the basic ingredient for Feynman diagrams.

math-ph↗

arXiv2023

Generalized integrals of Macdonald and Gegenbauer functions

We compute bilinear integrals involving Macdonald and Gegenbauer functions. These integrals are convergent only for a limited range of parameters. However, when one uses generalized integrals they can be computed essentially without restricting the parameters. The generalized integral is a linear functional extending the standard integral to a certain class of functions involving finitely many homogeneous non-integrable terms at the edpoints of the interval. For generic values of parameters, generalized bilinear integrals of Macdonald and Gegenbauer functions can be obtained by analytic continuation from the region in which the integrals are convergent. In the case of integer parameters we obtain expressions with explicit additional terms related to an anomaly, namely the failure of the generalized integral to be scaling invariant.

math.CA↗

arXiv2022

Holomorphic family of Dirac-Coulomb Hamiltonians in arbitrary dimension

We study massless 1-dimensional Dirac-Coulomb Hamiltonians, that is, operators on the half-line of the form $D_{ω,λ}:=\begin{bmatrix}-\frac{λ+ω}{x}&-\partial_x \\ \partial_x & -\frac{λ-ω}{x}\end{bmatrix}$. We describe their closed realizations in the sense of the Hilbert space $L^2(\mathbb R_+,\mathbb C^2)$, allowing for complex values of the parameters $λ,ω$. In physical situations, $λ$ is proportional to the electric charge and $ω$ is related to the angular momentum. We focus on realizations of $D_{ω,λ}$ homogeneous of degree $-1$. They can be organized in a single holomorphic family of closed operators parametrized by a certain 2-dimensional complex manifold. We describe the spectrum and the numerical range of these realizations. We give an explicit formula for the integral kernel of their resolvent in terms of Whittaker functions. We also describe their stationary scattering theory, providing formulas for a natural pair of diagonalizing operators and for the scattering operator. It is well-known that $D_{ω,λ}$ arise after separation of variables of the Dirac-Coulomb operator in dimension 3. We give a simple argument why this is still true in any dimension. Furthermore, we explain the relationship of spherically symmetric Dirac operators with the Dirac operator on the sphere and its eigenproblem. Our work is mainly motivated by a large literature devoted to distinguished self-adjoint realizations of Dirac-Coulomb Hamiltonians. We show that these realizations arise naturally if the holomorphy is taken as the guiding principle. Furthermore, they are infrared attractive fixed points of the scaling action. Beside applications in relativistic quantum mechanics, Dirac-Coulomb Hamiltonians are argued to provide a natural setting for the study of Whittaker (or, equivalently, confluent hypergeometric) functions.

math-ph↗

arXiv2022

Perturbed Bessel operators

We study perturbed Bessel operators $L_{m^2}=- \partial^2_x + ( m^2 - \frac14 )\frac{1}{x^2} + Q(x)$ on $L^2]0,\infty[$, where $m\in\mathbb{C}$ and $Q$ is a complex locally integrable potential. Assuming that $Q$ is integrable near $\infty$ and $x\mapsto x^{1-\varepsilon}Q(x)$ is integrable near $0$, with $\varepsilon\ge0$, we construct solutions to $L_{m^2} f = - k^2 f$ with prescribed behaviors near $0$. The special cases $m=0$ and $k=0$ are included in our analysis. Our proof relies on mapping properties of various Green's operators of the unperturbed Bessel operator. Then we determine all closed realizations of $L_{m^2}$ and show that they can be organized as holomorphic families of closed operators.

math.FA↗

arXiv2021

Momentum approach to the $1/r^2$ potential as a toy model of the Wilsonian renormalization

The Bessel operator, that is, the Schrödinger operator on the half-line with a potential proportional to $1/x^2$, is analyzed in the momentum representation. Many features of this analysis are parallel to the approach à la K. Wilson to Quantum Field Theory: one needs to impose a cutoff, add counterterms, study the renormalization group flow with its fixed points and limit cycles.

math-ph↗

arXiv2021

From Heun Class Equations to Painlevé Equations

In the first part of our paper we discuss linear 2nd order differential equations in the complex domain, especially Heun class equations, that is, the Heun equation and its confluent cases. The second part of our paper is devoted to Painlevé I-VI equations. Our philosophy is to treat these families of equations in a unified way. This philosophy works especially well for Heun class equations. We discuss its classification into 5 supertypes, subdivided into 10 types (not counting trivial cases). We also introduce in a unified way deformed Heun class equations, which contain an additional nonlogarithmic singularity. We show that there is a direct relationship between deformed Heun class equations and all Painlevé equations. In particular, Painlevé equations can be also divided into 5 supertypes, and subdivided into 10 types. This relationship is not so easy to describe in a completely unified way, because the choice of the ''time variable'' may depend on the type. We describe unified treatments for several possible ''time variables''.

math.CA↗

arXiv2021

On the domains of Bessel operators

We consider the Schrödinger operator on the halfline with the potential $(m^2-\frac14)\frac1{x^2}$, often called the Bessel operator. We assume that $m$ is complex. We study the domains of various closed homogeneous realizations of the Bessel operator. In particular, we prove that the domain of its minimal realization for $|\Re(m)|<1$ and of its unique closed realization for $\Re(m)>1$ coincide with the minimal second order Sobolev space. On the other hand, if $\Re(m)=1$ the minimal second order Sobolev space is a subspace of infinite codimension of the domain of the unique closed Bessel operator. The properties of Bessel operators are compared with the properties of the corresponding bilinear forms.

math-ph↗