Singular Potentials Research Articles

From spectral cluster to uniform resolvent estimates on compact manifolds

It is well known that uniform resolvent estimates imply spectral cluster estimates. We show that the converse is also true in some cases. In particular, Sogge's universal spectral cluster estimates for the Laplace–Beltrami operator on closed Riemannian manifolds directly imply uniform resolvent estimates outside a parabolic region, without any reference to parametrices. The method is purely functional analytic and takes full advantage of the known spectral cluster bounds. This yields new resolvent estimates for manifolds with boundary or with low-regularity metrics, among other examples. Moreover, we show that the resolvent estimates are stable under perturbations and use this to establish uniform Sobolev and spectral cluster inequalities for Schrödinger operators with singular potentials.

On the absence of global weak solutions for a nonlinear time-fractional Schrödinger equation

In this paper, an initial value problem for a nonlinear time-fractional Schrödinger equation with a singular logarithmic potential term is investigated. The considered problem involves the left/forward Hadamard-Caputo fractional derivative with respect to the time variable. Using the test function method with a judicious choice of the test function, we obtain sufficient criteria for the absence of global weak solutions.

Semilinear elliptic Schrödinger equations involving singular potentials and source terms

Let Ω⊂RN (N>2) be a C2 bounded domain and Σ⊂Ω be a compact, C2 submanifold without boundary, of dimension k with 0≤k<N−2. Put Lμ=Δ+μdΣ−2 in Ω∖Σ, where dΣ(x)=dist(x,Σ) and μ is a parameter. We study the boundary value problem (P) −Lμu=g(u)+τ in Ω∖Σ with condition u=ν on ∂Ω∪Σ, where g:R→R is a nondecreasing, continuous function and τ and ν are positive measures. The interplay between the inverse-square potential dΣ−2, the nature of the source term g(u) and the measure data τ,ν yields substantial difficulties in the research of the problem. We perform a deep analysis based on delicate estimate on the Green kernel and Martin kernel and fine topologies induced by appropriate capacities to establish various necessary and sufficient conditions for the existence of a solution in different cases.

Wave Equation for Sturm–Liouville Operator with Singular Intermediate Coefficient and Potential

Wave Equation for Sturm–Liouville Operator with Singular Intermediate Coefficient and Potential

Multiple Solutions to Cylindrically Symmetric Curl-Curl Problems and Related Schrödinger Equations with Singular Potentials

Multiple Solutions to Cylindrically Symmetric Curl-Curl Problems and Related Schrödinger Equations with Singular Potentials

Wave equation for Sturm-Liouville operator with singular potentials

The paper is denoted to the initial-boundary value problem for the wave equation with the Sturm-Liouville operator with irregular (distributive) potentials. To obtain a solution to the equation, the separation method and asymptotics of the eigenvalues and eigenfunctions of the Sturm-Liouville operator are used. Homogeneous and inhomogeneous cases of the equation are considered. Next, existence, uniqueness, and consistency theorems for a very weak solution of the wave equation with singular coefficients are proved.

Solving the inverse Sturm–Liouville problem with singular potential and with polynomials in the boundary conditions

Solving the inverse Sturm–Liouville problem with singular potential and with polynomials in the boundary conditions

Global minimizers of a large class of anisotropic attractive‐repulsive interaction energies in 2D

AbstractWe study a large family of Riesz‐type singular interaction potentials with anisotropy in two dimensions. Their associated global energy minimizers are given by explicit formulas whose supports are determined by ellipses under certain assumptions. More precisely, by parameterizing the strength of the anisotropic part we characterize the sharp range in which these explicit ellipse‐supported configurations are the global minimizers based on linear convexity arguments. Moreover, for certain anisotropic parts, we prove that for large values of the parameter the global minimizer is only given by vertically concentrated measures corresponding to one dimensional minimizers. We also show that these ellipse‐supported configurations generically do not collapse to a vertically concentrated measure at the critical value for convexity, leading to an interesting gap of the parameters in between. In this intermediate range, we conclude by infinitesimal concavity that any superlevel set of any local minimizer in a suitable sense does not have interior points. Furthermore, for certain anisotropic parts, their support cannot contain any vertical segment for a restricted range of parameters, and moreover the global minimizers are expected to exhibit a zigzag behavior. All these results hold for the limiting case of the logarithmic repulsive potential, extending and generalizing previous results in the literature. Various examples of anisotropic parts leading to even more complex behavior are numerically explored.

Global solvability for nonlinear wave equations with singular potential

In this work we consider the so-called Strauss conjecture for 3D nonlinear wave equation with singular potential and provide a positive answer concerning the global existence part. The key point is to exploit the additional decay of the solution away from the light cone. For this purpose we derive appropriate conformal type energy estimates under repulsively condition on the potential. However, it is not obvious whether functions in a domain of the Friedrichs extension of the minus Laplacian with singular potential have H2-regularity or not. To clarify this issue, we make use of the essential self-adjointness of the Schrödinger operator to obtain the desired property.

Exact solutions to fall of particle to singular potential: classical versus quantum cases

Exact solutions describing a fall of a particle to the centre of a non-regularized singular potential in classical and quantum cases are obtained and compared. We inspect the quantum problem with the help of the conventional Schrödinger’s equation. During the fall, the wave function spatial localization area contracts into a single zero-dimensional point. For the fall-admitting potentials, the Hamiltonian is non-Hermitian. Because of that, the wave function norm occurs time-dependent. It demands an extension to this case of the continuity equation and rules for mean value calculations. Surprisingly, the quantum and classical solutions exhibit striking similarities. In particular, both are self-similar as the particle energy equals zero. The characteristic spatial scales of the quantum and classical self-similar solutions obey the same temporal dependence. We present arguments indicating that these self-similar solutions are attractors to a broader class of solutions, describing the fall at finite energy of the particle.

Multi–component Cahn–Hilliard Systems with Singular Potentials: Theoretical Results

We consider a system of nonlinear diffusion equations modelling (isothermal) phase segregation of an ideal mixture of Nge 2 components occupying a bounded region Omega subset mathbb {R}^{d},dle 3. Our system is subject to a constant mobility matrix of coefficients, a free energy functional given in terms of singular entropy generated potentials and localized capillarity effects. We prove well-posedness and regularity results which generalize the ones obtained by Elliott and Luckhaus (IMA Preprint Ser 887, 1991). In particular, if dle 2, we derive the uniform strict separation of solutions from the singular points of the (entropy) nonlinearity. Then, even if d=3, we prove the existence of a global (regular) attractor as well as we establish the convergence of solutions to single equilibria. If d=3, this convergence requires the validity of the asymptotic strict separation property. This work constitutes the first part of an extended three-part study involving the phase behavior of multi-component systems, with a second part addressing the presence of nonlocal capillarity effects, and a final part concerning the numerical study of such systems along with some relevant application.

Global well-posedness and convergence to equilibrium for the Abels-Garcke-Grün model with nonlocal free energy

We investigate the nonlocal version of the Abels-Garcke-Grün (AGG) system, which describes the motion of a mixture of two viscous incompressible fluids. This consists of the incompressible Navier-Stokes-Cahn-Hilliard system characterized by concentration-dependent density and viscosity, and an additional flux term due to interface diffusion. In particular, the Cahn-Hilliard dynamics of the concentration (phase-field) is governed by the aggregation/diffusion competition of the nonlocal Helmholtz free energy with singular (logarithmic) potential and constant mobility. We first prove the existence of global strong solutions in general two-dimensional bounded domains and their uniqueness when the initial datum is strictly separated from the pure phases. The key points are a novel well-posedness result of strong solutions to the nonlocal convective Cahn-Hilliard equation with singular potential and constant mobility under minimal integral assumption on the incompressible velocity field, and a new two-dimensional interpolation estimate for the L4(Ω) control of the pressure in the stationary Stokes problem. Secondly, we show that any weak solution, whose existence was already known, is globally defined, enjoys the propagation of regularity and converges towards an equilibrium (i.e., a stationary solution) as t→∞. Furthermore, we demonstrate the uniqueness of strong solutions and their continuous dependence with respect to general (not necessarily separated) initial data in the case of matched densities and unmatched viscosities (i.e., the nonlocal model H with variable viscosity, singular potential and constant mobility). Finally, we provide a stability estimate between the strong solutions to the nonlocal AGG model and the nonlocal Model H in terms of the difference of densities.

Reconstruction of the Schrödinger Operator with a Singular Potential on Half-Line by Its Prescribed Essential Spectrum

Reconstruction of the Schrödinger Operator with a Singular Potential on Half-Line by Its Prescribed Essential Spectrum

Bound States and Heat Kernels for Fractional-Type Schrödinger Operators with Singular Potentials

We consider non-local Schrödinger operators H=-L-V in L^2(mathbb {R}^d), d geqslant 1, where the kinetic terms L are pseudo-differential operators which are perturbations of the fractional Laplacian by bounded non-local operators and V is the fractional Hardy potential. We prove pointwise estimates of eigenfunctions corresponding to negative eigenvalues and upper finite-time horizon estimates for heat kernels. We also analyze the relation between the matching lower estimates of the heat kernel and the ground state near the origin. Our results cover the relativistic Schrödinger operator with Coulomb potential.

Best constants in bipolar Lp Hardy-type inequalities

In this work we prove sharp Lp versions of the multipolar Hardy inequalities in [8,10], in the case of a bipolar potential and p≥2. Our results are sharp and minimizers do exist in the energy space. New features appear when p>2 compared to the linear case p=2 at the level of criticality of the p-Laplacian −Δp perturbed by a singular Hardy bipolar potential.

Cahn–Hilliard–Brinkman model for tumor growth with possibly singular potentials

We analyze a phase field model for tumor growth consisting of a Cahn–Hilliard–Brinkman system, ruling the evolution of the tumor mass, coupled with an advection-reaction-diffusion equation for a chemical species acting as a nutrient. The main novelty of the paper concerns the discussion of the existence of weak solutions to the system covering all the meaningful cases for the nonlinear potentials; in particular, the typical choices given by the regular, the logarithmic, and the double obstacle potentials are admitted in our treatise. Compared to previous results related to similar models, we suggest, instead of the classical no-flux condition, a Dirichlet boundary condition for the chemical potential appearing in the Cahn–Hilliard-type equation. Besides, abstract growth conditions for the source terms that may depend on the solution variables are postulated.

Quantitative unique continuation for spectral subspaces of Schrödinger operators with singular potentials

Recent (scale-free) quantitative unique continuation estimates for spectral subspaces of Schrödinger operators are extended to allow singular potentials such as certain Lp-functions. The proof is based on accordingly adapted Carleman estimates. Applications include Wegner and initial length scale estimates for random Schrödinger operators and control theory for the controlled heat equation with singular heat generation term.

Semilinear Schrödinger equations with a critical scale of the singular electromagnetic field

We solve semilinear Schrödinger equations with a singular magnetic potential of the Coulomb scale |x|−1 and a singular electric potential of the scale |x|−2.

Conditionally exactly solvable Dirac potential, including x 1/3 pseudoscalar interaction

We study the solution of the 1D Dirac equation for a pseudoscalar potential, which includes an interaction term proportional to and a term with a fixed strength. This is a conditionally exactly solvable singular potential, symmetric with respect to the origin. A feature of the potential is that the effective potential for the Schrödinger equation, to which the Dirac equation can be reduced, splits into two known potentials: the first Stillinger potential and the second Exton potential for the positive and negative semiaxes, respectively. We present the solution and analyze the energy spectrum and wave functions of the bound states. Our results reveal notable differences between the Schrödinger and Dirac behaviors.

Critical Kirchhoff-type equation with singular potential

In this paper, we deal with the following Kirchhoff-type equation: \begin{equation*} -\bigg(1 +\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx\bigg) \Delta u +\frac{A}{|x|^{\alpha}}u =f(u),\quad x\in\mathbb{R}^{3}, \end{equation*} where $A&gt; 0$ is a real parameter and $\alpha\in(0,1)\cup ({4}/{3},2)$. Remark that $f(u)=|u|^{2_{\alpha}^{*}-2}u +\lambda|u|^{q-2}u +|u|^{4}u$, where $\lambda&gt; 0$, $q\in(2_{\alpha}^{*},6)$, $2_{\alpha}^{*}=2+{4\alpha}/({4-\alpha})$ is the embedding bottom index, and $6$ is the embedding top index and Sobolev critical exponent. We point out that the nonlinearity $f$ is the almost ``optimal'' choice. First, for $\alpha\in({4}/{3},2)$, applying the generalized version of Lions-type theorem and the Nehari manifold, we show the existence of nonnegative Nehari-type ground sate solution for above equation. Second, for $\alpha\in(0,1)$, using the generalized version of Lions-type theorem and the Poho\v{z}aev manifold, we establish the existence of nonnegative Poho\v{z}aev-type ground state solution for above equation. Based on our new generalized version of Lions-type theorem, our works extend the results in Li-Su [Z. Angew. Math. Phys. {\bf 66} (2015)].