--- title: Bessel-type Potentials in Analysis and Quantum Models url: https://www.emergentmind.com/topics/bessel-type-potential type: topic --- # Bessel-type Potentials in Analysis and Quantum Models A Bessel-type potential is a potential-theoretic or spectral object whose defining mechanism is governed by Bessel or modified Bessel functions, by Bessel convolution, or by the Fourier multiplier \((1+4\pi^2|\xi|^2)^{-s/2}\). In the cited literature, the expression is used in several distinct but related senses: as an explicit interaction law in Schrödinger operators, as an exactly solvable potential whose eigenfunctions are Bessel functions, as a kernel-generated potential \(G_{a,\nu} *_a \mu\) in weighted analysis, and as the operator underlying Bessel potential spaces \(H^{s,p}\) [1810.02008], [1611.02467], [2409.18445], [2503.04310]. A common misconception is that the term names a single standardized potential. The literature instead uses it for a family of constructions connected by Bessel kernels, Bessel translations, or Bessel-function solvability. ## 1. Terminological range and core definitions In quantum-mechanical usage, a canonical example is the two-dimensional attractive Bessel-Macdonald potential \[ H = -\frac{\hbar^2}{2\mu}\Delta(\mathbf{x}) - \alpha K_0(\beta \|\mathbf{x}\|), \] where \(K_0\) is the modified Bessel function of the second kind. In this setting, \(K_0\) decays exponentially at infinity and diverges logarithmically at the origin, so the interaction is attractive and finite-range in the sense used in planar \({\rm QED}_3\) models [1810.02008]. In harmonic analysis, a Bessel-type potential is often the Bessel convolution of a Radon measure with a Bessel kernel, \[ G_{a,\nu} *_a \mu(x), \] defined on \(\mathbb{R}_+^n\) with respect to the weighted measure \(d\mu_a(x)=x^a dx\) and the generalized Bessel translation \(T_t\) [2409.18445]. A nonlinear variant is the Wolff-type Bessel potential \(W_{a,v,p}\mu\), central to capacity and removability questions [2303.13437]. In function-space theory, the basic Bessel potential operator is \[ A^s f = \mathcal{F}^{-1}\left[(1+4\pi^2|\xi|^2)^{-s/2}\hat f(\xi)\right], \] which generates the scale \(H^{s,p}(\mathbb{R}^n)\) and interpolates between Lebesgue and Sobolev spaces [2503.04310]. | Usage | Representative form | Typical setting | |---|---|---| | Explicit interaction potential | \(-\alpha K_0(\beta\|x\|)\) | planar Schrödinger operators | | Exactly solvable quantum potential | \(-g^2 e^{-|x|}\), \(g^2 e^{2|x|}\), \(-\frac{\hbar^2}{2m}U_0(1+|x|/d)^{-2}\) | one-dimensional spectral problems | | Kernel-generated potential | \(G_{a,\nu} *_a \mu\) | Bessel convolution and trace inequalities | | Potential operator | \(A^s\) | fractional function spaces | ## 2. Exactly solvable quantum models One major meaning of Bessel-type potential is a potential for which the stationary Schrödinger equation reduces to Bessel’s or the modified Bessel equation. For the non-analytic attractive exponential potential \[ V(x)= -g^2 e^{-|x|}, \] the substitution \(\rho(x)=2g e^{-|x|/2}\) converts the bound-state equation into the Bessel equation of order \(2\kappa\). Physical bound states are proportional to \(J_{2\kappa}(\rho(x))\); odd states satisfy \(J_{2\kappa}(2g)=0\), and even states satisfy \(J'_{2\kappa}(2g)=0\). The same model admits closed-form scattering amplitudes obtained by matching Bessel solutions at the non-analytic point \(x=0\) [1605.07310]. A complementary confining model is \[ V(x)= g^2 e^{2|x|}. \] Here the Schrödinger equation reduces to a modified Bessel equation of pure imaginary order. Normalizability selects \(K_{i\nu}(g e^{|x|})\), because \(K_{i\nu}\) decays for large argument whereas \(I_{i\nu}\) grows. Even eigenvalues are specified by the zeros of \(K'_{i\nu}(g)\), and odd eigenvalues by the zeros of \(K_{i\nu}(g)\), both viewed as functions of the order \(\nu\) at fixed \(g\) [1611.02467]. This shifts quantization from zeros in the argument to zeros in the order, which is one of the distinctive features of this solvable class. Another explicitly solvable single-well example is \[ V(x) = -\frac{\hbar^2}{2m} U_0 \frac{1}{(1 + |x|/d)^2}. \] With \(\xi=\kappa d(1+|x|/d)\) and \(\alpha=\sqrt{\frac14-U_0d^2}\), the reduced equation becomes \[ \xi^2 f''(\xi) + \xi f'(\xi) - (\xi^2 + \alpha^2) f(\xi)=0, \] and the square-integrable solution is \(\psi(x)=\sqrt d\,\xi^{1/2}K_\alpha(\xi)\). Odd levels satisfy \(K_\alpha(\kappa d)=0\), while even levels satisfy \[ \frac{K_{\alpha+1}(\kappa d)}{K_\alpha(\kappa d)}=\frac{\alpha+\frac12}{\kappa d} \] [1307.5441]. The solvable class also extends beyond these explicit models. A six-parameter second-order Bessel-type differential equation with irregular singularity at the origin was solved by the Tridiagonal Representation Approach using series in Bessel polynomials, with expansion coefficients given by orthogonal polynomials in parameter space. The same framework was used to generate solutions of Schrödinger equations for novel potentials [2006.13821]. ## 3. The Bessel-Macdonald potential in two-dimensional Schrödinger theory The paper “On the two-dimensional Schrödinger operator with an attractive potential of the Bessel-Macdonald type” studies \[ H = -\frac{\hbar^2}{2\mu}\Delta(\mathbf{x}) - \alpha K_0(\beta \|\mathbf{x}\|), \] motivated by the non-relativistic approximation of parity-preserving \({\rm QED}_3\) and by two-quasiparticle scattering in planar condensed-matter models [1810.02008]. After separation of variables, the radial operator for angular momentum \(m\) is \[ H_m = -\frac{\hbar^2}{2\mu}\left(\frac{d^2}{dr^2} + \frac{1}{r}\frac{d}{dr}\right) + \frac{\hbar^2 m^2}{2\mu r^2} - \alpha K_0(\beta r), \] and the dimensionless effective coupling is \[ C = \frac{2\mu \alpha}{\hbar^2 \beta^2}. \] The operator-theoretic structure is central. The free Hamiltonian is self-adjoint on its natural Sobolev domain, and the potential is \(H_0\)-bounded with arbitrarily small bound. By the Kato-Rellich theorem, \(H=H_0+V\) is self-adjoint on the domain of \(H_0\). The potential is real-valued, belongs to \(L^2(\mathbb{R}^2)\), vanishes at infinity, and the operator is bounded from below by \(-C\alpha/(4\pi)\) [1810.02008]. Its spectral picture is correspondingly explicit. The essential spectrum is \[ \sigma_{\mathrm{ess}}(H)=[0,+\infty), \] and all discrete eigenvalues lie in \((-C\alpha/(4\pi),0)\). Using Setô-type estimates, the number of bound states satisfies \[ N_2^0 < 1+\frac{C}{2} \quad \text{for } m=0, \qquad N_2^m < \frac{C}{2m} \quad \text{for } m\ge 1. \] For zero angular momentum, the analysis yields an isolated two-quasiparticle bound state when \(C\in(0,2)\), and the lowest energy gap is estimated by \[ \Delta=\frac{C\alpha}{4\pi}. \] The same paper connects these results to high-\(T_c\) \(s\)-wave Cooper-type superconductors and to \(s\)-wave electron-polaron–electron-polaron bound states in mass-gap graphene systems [1810.02008]. ## 4. Bessel kernels, Bessel convolution, and nonlinear potentials In weighted harmonic analysis, Bessel-type potentials are defined on \(\mathbb{R}_+^n\) using Bessel translation and Bessel convolution. For a Radon measure \(\mu\), the basic potential is \[ G_{a,\nu} *_a \mu(x), \] and the trace inequality \[ \|G_{a,\nu} *_a f\|_{L^p(\mu)} \leq C \|f\|_{L^p(d\mu_a)} \] admits a Kerman-Sawyer type characterization. Chegaar and Horváth proved that, for doubling \(\mu\), this inequality is equivalent to a cube testing condition and to a nonlinear potential inequality involving \(G_{a,\nu} *_a \mu\); the semigroup property \[ G_{a,\nu} *_a G_{a,u} = G_{a,\nu+u} \] is a key structural ingredient [2409.18445]. The same framework yields spectral information for the Schrödinger-Bessel operator \[ H_a=-\Delta_a-V, \qquad \Delta_a u = \sum_{i=1}^n \left[\frac{\partial^2 u}{\partial x_i^2} + \frac{a_i}{x_i}\frac{\partial u}{\partial x_i}\right]. \] Under a doubling hypothesis on the density \(v(x)\), two-sided estimates are obtained for the least eigenvalue \(\lambda_1(H_a)\) in terms of cube-localized integrals of the Bessel potential \(G_{a,2} *_a v|_Q\) [2409.18445]. A nonlinear potential theory based on the same convolution structure was developed for the Bessel kernel \(G_{a,v}\). The Wolff-type potential is \[ W_{a,v,p} \mu(x) = \int_0^\infty \left( \frac{T_x(X_{B^+(0,r)} *_a \mu)(x)}{r^{n+|a|-v}} \right)^{p'-1} \frac{dr}{r}, \] and the corresponding \(B\)-\(p\) capacity is \[ C_{p,a,v}(E) = \inf \left\{ \|f\|_{L^p_a}^p : G_{a,v} *_a f(x)\ge 1 \ \forall x\in E \right\}. \] Within this theory, compact sets are removable for the Laplace-Bessel equation if and only if the relevant capacity is zero, a Wolff-type inequality controls \(\|G_{a,v} *_a \mu\|_{L^{p',a}}\) by \(\|W_{a,v,p}\mu\|_{L^1_a}\), and explicit capacity criteria are obtained for Cantor-type sets [2303.13437]. ## 5. Bessel potential operators and potential spaces A second large branch of the subject treats Bessel potentials as regularizing operators and the associated spaces as fractional Sobolev scales. For \(s\in\mathbb{R}\) and \(1n\), then \(H^{s,p}\hookrightarrow C^{0,\alpha}\) with \(\alpha=s-\frac{n}{p}\). Compact embeddings on bounded Lipschitz domains were proved from three viewpoints: abstract complex interpolation, translation estimates derived from the Riesz fractional gradient, and comparison with Gagliardo spaces [2503.04310], [2506.01677]. This function-space meaning of Bessel potential extends well beyond \(\mathbb{R}^n\). On Ahlfors regular metric spaces, Bessel-type potentials are built from a Coifman-type approximation of the identity: \[ J_a f(x)=\int_X k_a(x,y)f(y)\,dm(y). \] These operators improve Lipschitz, Besov, and Hajłasz–Sobolev regularity, define Sobolev potential spaces \(\mathcal{L}_a^p(X)=J_a(L^p(X))\), and for small orders are invertible through a fractional derivative \(D_a\). In the Euclidean case, the resulting spaces coincide with classical Bessel potential spaces [1506.08182]. On the unit square, bivariate Bessel-Potential spaces admit tensor-product quarklet characterizations derived from univariate boundary-adapted quarklets and the Hansen–Sickel intersection representation [2403.14388]. ## 6. Extensions: Fourier-Bessel, pseudo-Euclidean, periodic, and multilinear theories Potential operators associated with Fourier-Bessel expansions provide a classical analogue of Hardy-Littlewood-Sobolev theory. For \(\nu>-1\), the Bessel potential operator \[ \mathcal{B}_\sigma^\nu=(\operatorname{Id}+\mathcal{L}^\nu)^{-\sigma} \] has a kernel represented through the subordinated Poisson kernel, and the corresponding operators satisfy sharp strong type, weak type, and restricted weak type \((p,q)\) bounds in the Fourier-Bessel setting [1212.6342]. This is one route by which Bessel-type potentials enter special-function harmonic analysis. In pseudo-Euclidean geometry, Bessel potentials are inverse Fourier transforms of regularizations of \((m^2+p^2)^{-\mu/2}\). For Lorentzian signature \((1,d-1)\), they are closely tied to Green functions for the Klein-Gordon equation, including retarded, advanced, Feynman, and anti-Feynman propagators, and the analysis can be formulated systematically in terms of hypergeometric functions rather than Bessel functions. The same framework also covers the tachyonic case [2406.08327]. On the \(n\)-torus, the periodic Bessel potential operator is \[ J_{s,\pi}(u)=\sum_{k\in\mathbb{Z}^n}(1+|k|^2)^{s/2}c_k(u)f_k, \] and the periodic spaces \(H_p^s(\mathbb{T}^n)\) are defined by \(J_{s,\pi}(u)\in L^p(\mathbb{T}^n)\). Under explicit hypotheses on the smoothness indices, the multiplier space between \(H_p^s(\mathbb{T}^n)\) and \(H_q^t(\mathbb{T}^n)\) is exactly \[ H_q^t(\mathbb{T}^n)\cap H_{p'}^{-s}(\mathbb{T}^n) \] [2212.05681]. Operator theory on Bessel potential spaces has also been developed for Mellin convolution equations. A basic noncommutation identity is \[ \mathbf{\Lambda}^s_{-\gamma}\mathbf{K}_c^1 = c^{-s}\mathbf{K}_c^1\mathbf{\Lambda}^s_{-c\gamma}, \] which underlies lifting Mellin convolution operators from Bessel potential spaces to Lebesgue spaces and leads to Fredholm and index formulae for the resulting operator algebra [1502.02756]. Recent work extends the theory from linear to bilinear operators. Bilinear Bessel potentials are defined by \[ \mathcal J_s(f,g)(x)=\int_{\mathbb R^n} G_s(y)\,f(x-y)\,g(x+y)\,dy, \] and their boundedness from \(L^p\times L^q\) into Lebesgue and Lorentz spaces is characterized in the region \[ \frac1p+\frac1q-\frac{s}{n}\le \frac1r \le \frac1p+\frac1q. \] In several cases the optimal Lorentz indices are identified by explicit counterexamples [2603.15962]. A broader generalization replaces the classical Bessel kernel by a generalized Bessel-McDonald kernel \(G(x)=\varphi(|x|)\), allowing non-power singularities at the origin. In that setting, embeddings of generalized Bessel potentials into Calderón spaces are characterized by order-sharp moduli of smoothness and by the embedding of an associated cone \(\mathcal K\) into the target function space [2007.08286]. This suggests that the notion of Bessel-type potential is best understood not as a single formula, but as a recurring analytic mechanism spanning solvable quantum models, spectral theory, nonlinear capacities, and fractional function spaces. Source: https://www.emergentmind.com/topics/bessel-type-potential