Singular Potentials Research Articles

On the Absence of Weak Solutions of Nonlinear Nonnegative Higher Order Parabolic Inequalities with a Nonlocal Source

The paper proves the absence of solutions of semilinear parabolic inequalities and higher order systems with a singular potential and nonlocal sources. The proofs are based on the test function method developed by E. Mitidieri and S.I. Pokhozhaev.

Propagation of Moments for Large Data and Semiclassical Limit to the Relativistic Vlasov Equation

We investigate the semiclassical limit from the semirelativistic Hartree–Fock equation describing the time evolution of a system of fermions in the mean-field regime with a relativistic dispersion law and interacting through a singular potential of the form , , , and , with the convention if . For mixed states, we show convergence in the Schatten norms with an explicit rate towards the Weyl transform of a solution to the relativistic Vlasov equation with singular potentials, thus generalizing [E. Dietler, S. Rademacher, and B. Schlein, J. Stat. Phys., 172 (2018), pp. 398–433], where the case of smooth potentials has been treated. Moreover, we provide new results on the well-posedness theory of the relativistic Vlasov equations with singular interactions.

A Time-Fractional Schrödinger Equation with Singular Potentials on the Boundary

A Schrödinger equation with a time-fractional derivative, posed in (0,∞)×I, where I=]a,b], is investigated in this paper. The equation involves a singular Hardy potential of the form λ(x−a)2, where the parameter λ belongs to a certain range, and a nonlinearity of the form μ(x−a)−ρ|u|p, where ρ≥0. Using some a priori estimates, necessary conditions for the existence of weak solutions are obtained.

Isolated singularities for semilinear elliptic systems with Hardy potential

In this paper, we analyze the asymptotic behavior near zero of positive solutions for a class of semilinear elliptic systems with Hardy-type singular potential. In addition, we address the questions of existence and uniqueness for the solutions.

A One-Dimensional Time-Fractional Damped Wave Equation with a Convection Term

We investigate a semilinear time-fractional damped wave equation in one dimension, posed in a bounded interval. The considered equation involves a convection term and singular potentials on one extremity of the interval. A Dirichlet boundary condition depending on the time-variable is imposed. Using nonlinear capacity estimates, we establish sufficient conditions for the nonexistence of weak solutions to the considered problem. In particular, when the boundary condition is independent of time, we show the existence of a Fujita-type critical exponent.

Nonlocalization of singular potentials in quantum dynamics

Nonlocal modeling has drawn more and more attention and becomes steadily more powerful in scientific computing. In this paper, we demonstrate the superiority of a first-principle nonlocal model—Wigner function—in treating singular potentials which are often used to model the interaction between point charges in quantum science. The nonlocal nature of the Wigner equation is fully exploited to convert the singular potential into the Wigner kernel with weak or even no singularity, and thus highly accurate numerical approximations are achievable, which are hardly designed when the singular potential is taken into account in the local Schrödinger equation. The Dirac delta function, the logarithmic, and the inverse power potentials are considered. Numerically converged Wigner functions under all these singular potentials are obtained with a fourth-order accurate operator splitting spectral method, and display many interesting quantum behaviors as well.

Eigenvalue spacing for 1D singular Schrödinger operators

The aim of this paper is to provide uniform estimates for the eigenvalue spacing of one-dimensional semiclassical Schrödinger operators with singular potentials on the half-line. We introduce a new development of semiclassical measures related to families of Schrödinger operators that provides a means of establishing uniform non-concentration estimates within that class of operators. This dramatically simplifies analysis that would typically require detailed WKB expansions near the turning point, near the singular point and several gluing type results to connect various regions in the domain.

Periodic orbit theory of Bethe-integrable quantum systems: an N-particle Berry–Tabor trace formula

One of the fundamental results of semiclassical theory is the existence of trace formulae showing how spectra of quantum mechanical systems emerge from massive interference among amplitudes related with time-periodic structures of the corresponding classical limit. If it displays the properties of Hamiltonian integrability, this connection is given by the celebrated Berry–Tabor trace formula, and the periodic structures it is built on are tori supporting closed trajectories in phase space. Here we show how to extend this connection into the domain of quantum many-body systems displaying integrability in the sense of the Bethe ansatz, where a classical limit cannot be rigorously defined due to the presence of singular potentials. Formally following the original derivation of Berry and Tabor (1976 Proc. R. Soc. A 349 101), but applied to the Bethe equations without underlying classical structure, we obtain a many-particle trace formula for the density of states of N interacting bosons on a ring, the Lieb–Liniger model. Our semiclassical expressions are in excellent agreement with quantum mechanical results for and 4 particles. For N = 2 we relate our results to the quantization of billiards with mixed boundary conditions. Our work paves the way towards the treatment of the important class of integrable many-body systems by means of semiclassical trace formulae pioneered by Michael Berry in the single-particle context.

Bound and ground states of coupled “NLS-KdV” equations with Hardy potential and critical power

We consider the existence of bound and ground states for a family of nonlinear elliptic systems in RN, which involves equations with critical power nonlinearities and Hardy-type singular potentials. The equations are coupled by what we call “Schrödinger-Korteweg-de Vries” non-symmetric terms, which arise in some phenomena of fluid mechanics. By means of variational methods, ground states are derived for several ranges of the positive coupling parameter ν. Moreover, by using min-max arguments, we seek bound states under some energy assumptions.

Sharp Weyl laws with singular potentials

We consider the Laplace--Beltrami operator on a three-dimensional Riemannian manifold perturbed by a potential from the Kato class and study whether various forms of Weyl's law remain valid under this perturbation. We show that a pointwise Weyl law holds, modified by an additional term, for any Kato class potential with the standard sharp remainder term. The additional term is always of lower order than the leading term, but it may or may not be of lower order than the sharp remainder term. In particular, we provide examples of singular potentials for which this additional term violates the sharp pointwise Weyl law of the standard Laplace-Beltrami operator. For the proof we extend the method of Avakumovi\'c to the case of Schr\"odinger operators with singular potentials.

Relativistic equations with singular potentials

The first part of this paper concern with the study of the Lorentz force equation q′1-|q′|2′=E→(t,q)+q′×B→(t,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left( \\frac{q'}{\\sqrt{1-|q'|^2}}\\right) '= \\overrightarrow{E}(t,q)+q'\ imes \\overrightarrow{B}(t,q) \\end{aligned}$$\\end{document}in the relevant physical configuration where the electric field overrightarrow{E} has a singularity in zero. By using Szulkin’s critical point theory, we prove the existence of T-periodic solutions provided that T and the electric and magnetic fields interact properly. In the last part, we employ both a variational and a topological argument to prove that the scalar relativistic pendulum-type equation q′1-(q′)2′+q=G′(q)+h(t),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left( \\frac{q'}{\\sqrt{1-(q')^2}}\\right) ' +q = G^{\\prime }(q) +h(t), \\end{aligned}$$\\end{document}admits at least a periodic solution when hin L^1 (0, T) and G is singular at zero.

Optimal Distributed Control of Two-Dimensional Navier–Stokes–Cahn–Hilliard System with Chemotaxis and Singular Potential

Optimal Distributed Control of Two-Dimensional Navier–Stokes–Cahn–Hilliard System with Chemotaxis and Singular Potential

Existence and asymptotic behavior of solutions for Choquard equations with singular potential under Berestycki–Lions type conditions

We prove the existence and asymptotic behavior of solutions to the following problem: − Δ u + V ( x ) u − g ( x ) u = ( I α ∗ F ( u ) ) f ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , where g ( x ) : = μ | x | is called the Coulomb potential, g ( x ) : = β | x | 2 is called the Hardy potential (the inverse-square potential). μ , β > 0 are parameters, I α : R N ⟶ R is the Riesz potential. Moreover, the nonlinearity f satisfies Berestycki–Lions type conditions which are introduced by Moroz and Van Schaftingen (Trans. Amer. Math. Soc. 367 (2015) 6557–6579). When μ ∈ ( 0 , α ( N − 2 ) / 2 ( α + 1 ) ) and β ∈ ( 0 , α ( N − 2 ) 2 / 4 ( 2 + α ) ), under some mild assumptions on V, we establish the existence and asymptotic behavior of solutions. Particularly, our results extend some relate ones in the literature.

Multiple solution for a fourth-order nonlinear eigenvalue problem with singular and sublinear potential

"Let $(M,g)$ be a Cartan-Hadamard manifold. For certain positive numbers $\mu$ and $\lambda$, we establish the multiplicity of solutions to the problem $$\Delta_g^2 u-\Delta_g u+u=\mu \frac{u}{d_g(x_0,x)^4}+\lambda \alpha(x)f(u),\ \mbox{ in } M,$$ where $x_0\in M$, while $f:\R\to\R$ is continuous function, superlinear at zero and sublinear at infinity."

“Renormalization-group-invariant effective field theory” for few-nucleon systems is cutoff dependent

We consider nucleon-nucleon scattering using the formulation of chiral effective field theory which is claimed to be renormalization group invariant. The cornerstone of this framework is the existence of a well-defined infinite-cutoff limit for the scattering amplitude at each order of the expansion, which should not depend on a particular regulator form. Focusing on the ${}^{3}{P}_{0}$ partial wave as a representative example, we show that this requirement can in general not be fulfilled beyond the leading order, in spite of the perturbative treatment of subleading contributions to the amplitude. Several previous studies along these lines, including the next-to-leading order calculation by B. Long and C. J. Yang [Phys. Rev. C 84, 057001 (2011)] and a toy model example with singular long-range potentials by B. Long and U. van Kolck [Ann. Phys. 323, 1304 (2008)], are critically reviewed and scrutinized in detail.

Existence of ground‐state solutions for p‐Choquard equations with singular potential and doubly critical exponents

AbstractWe are concerned with the following Choquard equation: where , , , , is the p‐Laplacian, is the Riesz potential, and F is the primitive of f which is of critical growth due to the Hardy–Littlewood–Sobolev inequality. Under different range of θ and almost necessary conditions on the nonlinearity f in the spirit of Berestycki–Lions‐type conditions, we divide this paper into three parts. By applying the refined Sobolev inequality with Morrey norm and the generalized version of the Lions‐type theorem, some existence results are established. It is worth noting that our method is not involving the concentration–compactness principle.

Well-posedness of a Navier–Stokes–Cahn–Hilliard system for incompressible two-phase flows with surfactant

We investigate a diffuse-interface model that describes the dynamics of viscous incompressible two-phase flows with surfactant. The resulting system of partial differential equations consists of a sixth-order Cahn–Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth-order Cahn–Hilliard equation for the local concentration of the surfactant. The former has a smooth potential, while the latter has a singular potential. Both equations are coupled with a Navier–Stokes system for the (volume averaged) fluid velocity. The evolution system is endowed with suitable initial conditions, a no-slip boundary condition for the velocity field and homogeneous Neumann boundary conditions for the phase functions as well as for the chemical potentials. We first prove the existence of a global weak solution, which turns out to be unique in two dimensions. Stronger regularity assumptions on the initial data allow us to prove the existence of a unique global (respectively, local) strong solution in two (respectively, three) dimensions. In the two-dimensional case, we derive a continuous dependence estimate with respect to the norms controlled by the total energy. Moreover, we establish instantaneous regularization properties of global weak solutions for [Formula: see text]. In particular, we show that the surfactant concentration stays uniformly away from the pure states 0 and 1 after some positive time.

Dyonic black holes in the theory of two electromagnetic potentials. II

The results obtained in our previous paper are now extended to the case of stationary axially symmetric dyonic black boles within the theory of two electromagnetic potentials. We slightly enlarge the classical Ernst formalism by introducing, with the aid of the $t$- and $\varphi$-components of the dual potential $B_\mu$, the magnetic potential $\Phi_m$ which, similar to the known electric potential $\Phi_e$, also takes constant value on the black hole horizon. We analyze in detail the case of the dyonic Kerr-Newman black hole and show how the Komar mass must be evaluated correctly in this stationary dyonic model. In particular, we rigorously prove the validity of the standard Tomimatsu mass formula and point out that attempts to "improve" it made in recent years are explained by misunderstanding of the auxiliary role that singular potentials play in the description of magnetic charges. Our approach is symmetrical with respect to electric and magnetic charges and, like in the static case considered earlier, Dirac strings of all kind are excluded from the physical picture of the stationary black hole dyonic spacetimes.

Multimorbidity research in Sub-Saharan Africa: Proceedings of an interdisciplinary workshop

As life expectancies rise globally, the number of people living with multiple chronic health conditions – commonly referred to as ‘multimorbidity’ – is rising. Multimorbidity has been recognised as especially challenging to respond to in countries whose health systems are under-funded, fragmented, and designed primarily for acute care, including in sub-Saharan Africa. A growing body of research in sub-Saharan Africa has sought to better understand the particular challenges multimorbidity poses in the region and to develop context-sensitive responses. However, with multimorbidity still crystallising as a subject of enquiry, there remains considerable heterogeneity in conceptualising multimorbidity across disciplines and fields, hindering coordinated action. In June 2022, 60 researchers, practitioners, and stakeholders with regional expertise from nine sub-Saharan African countries gathered in Blantyre, Malawi to discuss ongoing multimorbidity research across the region. Drawing on insights from disciplines including epidemiology, public health, clinical medicine, anthropology, history, and sociology, participants critically considered the meaning, singular potential, and limitations of the concept of multimorbidity in sub-Saharan Africa. The workshop emphasised the need to move beyond a disease-centred concept of multimorbidity to one foregrounding patients’ values, needs, and social context; the importance of foregrounding structures and systems rather than behaviour and lifestyles; the value of a flexible (rather than standard) definition of multimorbidity; and the need to leverage local knowledge, expertise, resources, and infrastructure. The workshop further served as a platform for exploring opportunities for training, writing, and ongoing collaboration.

The Gell-Mann–Brueckner Formula for the Correlation Energy of the Electron Gas: A Rigorous Upper Bound in the Mean-Field Regime

We prove a rigorous upper bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of interaction potentials. Our result covers the Coulomb potential, and in this case we obtain the analogue of the Gell-Mann–Brueckner formula c_{1}rho log left( rho right) +c_{2}rho in the high density limit. We do this by refining the analysis of our bosonization method to deal with singular potentials, and to capture the exchange contribution which is absent in the purely bosonic picture.