Negative Laplacian Research Articles

Minimizers for the de Gennes–Cahn–Hilliard energy under strong anchoring conditions

AbstractIn this article, we use the Nehari manifold and the eigenvalue problem for the negative Laplacian with Dirichlet boundary condition to analytically study the minimizers for the de Gennes–Cahn–Hilliard energy with quartic double‐well potential and Dirichlet boundary condition on the bounded domain. Our analysis reveals a bifurcation phenomenon determined by the boundary value and a bifurcation parameter that describes the thickness of the transition layer that segregates the binary mixture's two phases. Specifically, when the boundary value aligns precisely with the average of the pure phases, and the bifurcation parameter surpasses or equals a critical threshold, the minimizer assumes a unique form, representing the homogeneous state. Conversely, when the bifurcation parameter falls below this critical value, two symmetric minimizers emerge. Should the boundary value be larger or smaller from the average of the pure phases, symmetry breaks, resulting in a unique minimizer. Furthermore, we derive bounds of these minimizers, incorporating boundary conditions and features of the de Gennes–Cahn–Hilliard energy.

QTAIM analysis for Metal - Metal and Metal- Non-metal Bonds in Tri-Osmium cluster

Electron density of the interaction of the compound's chemical bonding has been computed using topological analysis. The findings suggest that Os-Os, Os-N, and Os-H bonds have experienced bond critical points (BCP) with corresponding bonds paths (BP). The presence of N-C bridging atoms is significantly impacted by electron density of Os2-Os3 bond distribution therefore no critical point found and bond path. So, due to their positive electron density (b), negative laplacian 2(b), and positive the total energy density H(b) values, the Os-NC, Os-H, and Os-CO bonds all have transit closed-shell topological properties.

Stability of a Szegő-type asymptotics

We consider a multi-dimensional continuum Schrödinger operator H, which is given by a perturbation of the negative Laplacian by a compactly supported bounded potential. We show that for a fairly large class of test functions, the second-order Szegő-type asymptotics for the spatially truncated Fermi projection of H is independent of the potential and, thus, identical to the known asymptotics of the Laplacian.

New kind of electride sandwich complexes based on the cyclooctatetraene ligand M12(η8-C8H8)2M22 (M1 = Na, K and M2 = Ca, Mg): a theoretical study.

In the present study, the electronic structures of a series of binuclear sandwich complexes based on the cyclooctatetraene ligand M12(η8-C8H8)2M22 (M1 = Na, K and M2 = Ca, Mg) are studied theoretically. Each cyclooctatetraene ligand binds with the metal in the η8 binding mode. The M2-M2 bond length agrees well with the reported bimetallic covalent Ca2 and Mg2 bond lengths. The Wiberg bond index (WBI) also indicates the presence of covalent M2-M2 bonds, which gives additional stability to the complex. A non-nuclear attractor (NNA) is found in-between the M2-M2 bond and the negative Laplacian of the electron density is found at the NNA. Noncovalent interaction (NCI) plot shows that electron density is localized at the M2-M2 bond. Based on the performed analysis, we have concluded that the designed sandwich complexes are electrides. We herein report, for the first time, the electride sandwich complexes of the cyclooctatetraene ligand. Due to the presence of a diffuse electron system, the electride complexes exhibit higher values of the static second hyperpolarizability within the range of 2.6 × 105 to 1.4 × 106 a.u. Among the studied complexes, M12(η8-C8H8)2Ca2 exhibit a higher value of static second hyperpolarizability.

Heat flow on 1-forms under lower Ricci bounds. Functional inequalities, spectral theory, and heat kernel

We study the canonical heat flow (Ht)t≥0 on the cotangent module L2(T⁎M) over an RCD(K,∞) space (M,d,m), K∈R. We show Hess–Schrader–Uhlenbrock's inequality and, if (M,d,m) is also an RCD⁎(K,N) space, N∈(1,∞), Bakry–Ledoux's inequality for (Ht)t≥0 w.r.t. the heat flow (Pt)t≥0 on L2(M). Variable versions of these estimates are discussed as well. In conjunction with a study of logarithmic Sobolev inequalities for 1-forms, the previous inequalities yield various Lp-properties of (Ht)t≥0, p∈[1,∞].Then we establish explicit inclusions between the spectrum of its generator, the Hodge Laplacian Δ→, of the negative functional Laplacian −Δ, and of the Schrödinger operator −Δ+K. In the RCD⁎(K,N) case, we prove compactness of Δ→−1 if M is compact, and the independence of the Lp-spectrum of Δ→ on p∈[1,∞] under a volume growth condition.We terminate by giving an appropriate interpretation of a heat kernel for (Ht)t≥0. We show its existence in full generality without any local compactness or doubling, and derive fundamental estimates and properties of it.

Accumulation time of diffusion in a two-dimensional singularly perturbed domain

A general problem of current interest is the analysis of diffusion problems in singularly perturbed domains, within which small subdomains are removed from the domain interior and boundary conditions imposed on the resulting holes. One major application is to intracellular diffusion, where the holes could represent organelles or biochemical substrates. In this paper, we use a combination of matched asymptotic analysis and Green’s function methods to calculate the so-called accumulation time for relaxation to steady state. The standard measure of the relaxation rate is in terms of the principal non-zero eigenvalue of the negative Laplacian. However, this global measure does not account for possible differences in the relaxation rate at different spatial locations, is independent of the initial conditions, and relies on the assumption that the eigenvalues have sufficiently large spectral gaps. As previously established for diffusion-based morphogen gradient formation, the accumulation time provides a better measure of the relaxation process.

Prescribing Morse scalar curvatures: Critical points at infinity

Abstract The problem of prescribing conformally the scalar curvature of a closed Riemannian manifold as a given Morse function reduces to solving an elliptic partial differential equation with critical Sobolev exponent. Two ways of attacking this problem consist in subcritical approximations or negative pseudogradient flows. We show under a mild nondegeneracy assumption the equivalence of both approaches with respect to zero weak limits, in particular a one-to-one correspondence of zero weak limit finite energy subcritical blow-up solutions, zero weak limit critical points at infinity of negative type and sets of critical points with negative Laplacian of the function to be prescribed.

Linear and nonlinear non-Fredholm operators and their applications

<abstract><p>In this survey we discuss the recent results on the existence in the sense of sequences of solutions for certain elliptic problems containing the non-Fredholm operators. First of all, we deal with the solvability in the sense of sequences for some fourth order non-Fredholm operators, such that the methods of the spectral and scattering theory for Schrödinger type operators are used for the analysis. Moreover, we present the easily verifiable necessary condition of the preservation of the nonnegativity of the solutions of a system of parabolic equations in the case of the anomalous diffusion with the negative Laplacian in a fractional power in one dimension, which imposes the necessary form of such system of equations that must be studied mathematically. This class of systems of PDEs has a wide range of applications. We conclude the survey with several new results nowhere published concerning the solvability in the sense of sequences for the generalized Poisson type equation with a scalar potential.</p></abstract>

Solitary waves and excited states for Boson stars

We study the nonlinear, nonlocal, time-dependent partial differential equation [Formula: see text] which is known to describe the dynamics of quasi-relativistic boson stars in the mean-field limit. For positive mass parameter [Formula: see text] we establish existence of infinitely many (corresponding to distinct energies [Formula: see text]) traveling solitary waves, [Formula: see text], with speed [Formula: see text], where [Formula: see text] corresponds to the speed of light in our choice of units. These traveling solitary waves cannot be obtained by applying a Lorentz boost to a solitary wave at rest (with [Formula: see text]) because Lorentz covariance fails. Instead, we study a suitable variational problem for which the functions [Formula: see text] arise as solutions (called boosted excited states) to a Choquard-type equation in [Formula: see text], where the negative Laplacian is replaced by the pseudo-differential operator [Formula: see text] and an additional term [Formula: see text] enters. Moreover, we give a new proof for existence of boosted ground states. The results are based on perturbation methods in critical point theory.

The Schrödinger particle on the half-line with an attractive $$\delta $$-interaction: bound states and resonances

In this paper we provide a detailed description of the eigenvalue $ E_{D}(x_0)\leq 0$ (respectively $ E_{N}(x_0)\leq 0$) of the self-adjoint Hamiltonian operator representing the negative Laplacian on the positive half-line with a Dirichlet (resp. Neuman) boundary condition at the origin perturbed by an attractive Dirac distribution $-\lambda \delta(x-x_0)$ for any fixed value of the magnitude of the coupling constant. We also investigate the $\lambda$-dependence of both eigenvalues for any fixed value of $x_0$. Furthermore, we show that both systems exhibit resonances as poles of the analytic continuation of the resolvent. These results will be connected with the study of the ground state energy of two remarkable three-dimensional self-adjoint operators, studied in depth in Albeverio's monograph, perturbed by an attractive $\delta$-distribution supported on the spherical shell of radius $r_0$.

Halogen bonds and other noncovalent interactions in the crystal structures of trans-1,2-diiodo alkenes: an ab initio and QTAIM study.

A series of interatomic interactions interpretable as halogen bonds involving I…I, I…O, and I…C(π), as well as the noncovalent interactions I…H and O…O, were observed in the crystal structures of trans-1,2-diiodoolefins dimers according to ab initio calculations and the quantum theory of "atoms in molecules" (QTAIM) method. The interplay between each type of halogen bond and other noncovalent interactions was studied systematically in terms of bond length, electrostatic potential, and interaction energy, which are calculated via ab initio methods at the B3LYP-D3/6-311++G(d,p) and B3LYP-D3/def2-TZVP levels of theory. Characteristics and nature of the halogen bonds and other noncovalent interactions, including the topological properties of the electron density, the charge transfer, and their strengthening or weakening, were analyzed by means of both QTAIM and "natural bond order" (NBO). These computational methods provide additional insight into observed intermolecular interactions and are utilized to explain the differences seen in the crystal structures. Graphical abstract The contour map presents the regions of electronic concentration and depletion along each bond in one dimer. The blue points denote the BCPs. The blue lines denote positive Laplacian of electron density, which indicate the ionic interactions, van der Waals or intermolecular interactions, and the red lines denote negative Laplacian of electron density which indicate the covalent bonds.

Influence sandwich π–π stacking interactions in Benzaldoxime adsorption on the fullerene

Physisorption of Benzaldoxime derivatives on the Fullerene has been assayed at the M06/6-311++G (d, p) level of theory. The Anti-isomers are more stable than Syn-geometries in the range of –1.8 to –5.6 kcal mol–1. Global reactivity descriptors showed that donor and acceptor functional groups increase and decrease reactivity of the BO compound as electrophile, specially –NH2 and –NO2 groups. Changing of electron density in the molecular electrostatic potential maps and HOMO Anti -BOx––LUMOFullerene gaps values (0.009–0.540 eV) showing influence of these groups on the π–π stacking interactions. These non-covalent interactions stable formed donor complexes than acceptor complexes about –12.1 to –16.9 kJ mol–1. On the other hands, high density (ρ) and low negative Laplacian (∇2 ρ(r)) indexes in the formed BCPs imply strong sandwich stacking, especially –NH2 and –NO2 groups.

Stability of the Enhanced Area Law of the Entanglement Entropy

We consider a multi-dimensional continuum Schrödinger operator which is given by a perturbation of the negative Laplacian by a compactly supported potential. We establish both an upper bound and a lower bound on the bipartite entanglement entropy of the ground state of the corresponding quasi-free Fermi gas. The bounds prove that the scaling behaviour of the entanglement entropy remains a logarithmically enhanced area law as in the unperturbed case of the free Fermi gas. The central idea for the upper bound is to use a limiting absorption principle for such kinds of Schrödinger operators.

On Solvability in the Sense of Sequences for some Non-Fredholm Operators with Drift and Anomalous Diffusion

We study the solvability of certain linear nonhomogeneous elliptic equations and establish that, under some technical assumptions, the L2-convergence of the right-hand sides yields the existence and convergence of solutions in an appropriate Sobolev space. The problems involve differential operators with or without Fredholm property, in particular, the one-dimensional negative Laplacian in a fractional power, on the whole real line or on a finite interval with periodic boundary conditions. We prove that the presence of the transport term in these equations provides regularization of the solutions.

Investigation of bond topological and electrostatic properties of plumbagin molecule: An experimental and theoretical charge density study

Plumbagin is a naturally occurring derivative with several medicinal properties including antioxidants, antifungal, antimalarial, leprosy and antitumor properties; their structural and electrostatic properties are yet to be determined. The crystal structure of plumbagin has been solved; it shows that the compound crystallizes in P21/c space group with two molecules in the asymmetric unit. The electron density distribution of both molecules have determined from multipole model refinement. Among all the C–O bonds of both molecules (I & II), the OH group connected C–O bond and the methyl group connected C–C bond exhibits less electron density and the negative Laplacian of electron density as well. Further, the electrostatic potential (ESP) surface of two plumbagin molecules show some difference in their electronegative regions. The carbonyl O-atoms exhibit high electronegative ESP regions which are the key reactive locations of plumbagin when bind with the active site of target protein and in the ESP map, an aromatic π-cloud also observed in one of the molecule. The electron density distribution of O–H⋅⋅⋅O intermolecular interactions between the molecules I and II, reveals that these interactions are found very stronger than the other interactions in the crystal.

Propagation Property and Application to Inverse Scattering for Fractional Powers of Negative Laplacian

Propagation Property and Application to Inverse Scattering for Fractional Powers of Negative Laplacian

Lifshitz Tails for the Fractional Anderson Model

We consider the $d$-dimensional fractional Anderson model $(-\Delta)^\alpha+ V_\omega$ on $\ell^2(\mathbb Z^d)$ where $0<\alpha\leq 1$. Here $-\Delta$ is the negative discrete Laplacian and $V_\omega$ is the random Anderson potential consisting of iid random variables. We prove that the model exhibits Lifshitz tails at the lower edge of the spectrum with exponent $ d/ (2\alpha)$. To do so, we show among other things that the non-diagonal matrix elements of the negative discrete fractional Laplacian are negative and satisfy the two-sided bound $$ \frac{c_{\alpha,d}}{|n-m|^{d+2\alpha}} \leq -(-\Delta)^\alpha(n,m)\leq \frac{C_{\alpha,d}}{|n-m|^{d+2\alpha}} $$ for positive constants $c_{\alpha,d}$, $C_{\alpha,d}$ and all $n\neq m\in\mathbb Z^d$.

Experimental and theoretical charge density, intermolecular interactions and electrostatic properties of metronidazole.

Metronidazole is a radiosensitizer; it crystallizes in the monoclinic system with space group P21/c. The crystal structure of metronidazole has been determined from high-resolution X-ray diffraction measurements at 90 K with a resolution of (sin θ/λ)max = 1.12 Å-1. To understand the charge-density distribution and the electrostatic properties of metronidazole, a multipole model refinement was carried out using the Hansen-Coppens multipole formalism. The topological analysis of the electron density of metronidazole was performed using Bader's quantum theory of atoms in molecules to determine the electron density and the Laplacian of the electron density at the bond critical point of the molecule. The experimental results have been compared with the corresponding periodic theoretical calculation performed at the B3LYP/6-31G** level using CRYSTAL09. The topological analysis reveals that the N-O and C-NO2 exhibit less electron density as well as negative Laplacian of electron density. The molecular packing of crystal is stabilized by weak and strong inter- and intramolecular hydrogen bonding and H...H interactions. The topological analysis of O-H...N, C-H...O and H...H intra- and intermolecular interactions was also carried out. The electrostatic potential of metronidazole, calculated from the experiment, predicts the possible electrophilic and nucleophilic sites of the molecule; notably, the hydroxyl and the nitro groups exhibit large electronegative regions. The results have been compared with the corresponding theoretical results.

Prescribing Morse scalar curvatures: Subcritical blowing-up solutions

Prescribing conformally the scalar curvature of a Riemannian manifold as a given function consists in solving an elliptic PDE involving the critical Sobolev exponent. One way of attacking this problem consist in using subcritical approximations for the equation, gaining compactness properties. Together with the results in [30], we completely describe the blow-up phenomenon in case of uniformly bounded energy and zero weak limit in positive Yamabe class. In particular, for dimension greater or equal to five, Morse functions and with non-zero Laplacian at each critical point, we show that subsets of critical points with negative Laplacian are in one-to-one correspondence with such subcritical blowing-up solutions.

On the regularity and stability of the dual-phase-lag equation

In this paper we consider the following linear partial differential equation which is usually seen as an approximation to the dual-phase-lag heat equation proposed by Tzou. Ṫ+τqT̈+τq22T⃛=κ△T+κτT△Ṫ+κτT22△T̈on a bounded domain Ω in Rn with smooth boundary. We obtain analyticity for the associated C0−semigroup. Moreover, we also obtain exponential stability of the solutions by spectrum analysis and Hurwitz criterion under one of the following conditions:(i). τTτq>2−3; (ii). 2−3≥τTτq>(1+κτTλ1)2+(κτTλ1)2+(κτTλ1)3−(1+κτTλ1)κτTλ1(1+κτTλ1), where λ1 is the smallest eigenvalue of the negative Laplacian on Ω with Dirichlet boundary condition.