Positive Eigenfunction Research Articles

Approximation of positive solutions of symmetric eigenvalue problems with nonlinear dependence on the spectral parameter

A symmetric partial differential eigenvalue problem with nonlinear dependence on the spectral parameter arising in plasma physics is studied. We propose and justify new conditions for the existence of a positive eigenvalue and the corresponding positive eigenfunction. A finite element approximation of the problem preserving the property of positivity of solutions is constructed. The existence and convergence of approximate solutions are established.

Quantum backflow for a free-particle hermite wavepacket

Abstract Quantum backflow is the unexpected effect that wavepackets consisting of only positive momentum components can apparently move in the negative direction. This is usually described in terms of the backflow constant, which is a dimensionless quantity describing least upper bound on the amount of probability that can flow backwards during a given time interval. Backflow is usually calculated for wavepackets that can be written as a sum of positive momentum plane waves. Here we present a calculation of the backflow constant using the localised free particle hermite wavefunctions where equal weights of positive and negative momentum eigenfunctions occur. The resulting backflow constant is substantially smaller than the accepted value. The reasons for this are discussed and finally we draw conclusions about the calculation of backflow more generally.

Fine patterns for a nonlocal periodic-parabolic equation with spatial degeneracy

In this paper, we investigate several properties of the positive solution to the nonlocal periodic-parabolic problem{ut=d[∫ΩJ(x−y)u(y,t)dy−u(x,t)]+λu−b(x)q(t)up in Ω×[0,T],u(x,0)=u(x,T) in Ω, where Ω is a bounded smooth domain in RN (N≥1), λ>0 is a parameter, d>0 and p>1 are constants, q(t) is a positive and T-periodic continuous function, b(x) is a nonnegative continuous function, and is strictly positive only outside a domain Ω¯0⊂Ω. So the equation has a degeneracy over Ω0. In such a case, it is known that there is a unique positive solution uλ if and only if λ is between the principle eigenvalues λp(Ω) and λp(Ω0) of the associated nonlocal diffusion operator (over Ω and Ω0, respectively). We show that uλ is stable under perturbations of λ and b(x) in the equation, and when λ increases to λp(Ω0), uλ converges to ∞ uniformly over Ω¯×[0,T] but a clear pattern is exhibited by uλ/‖uλ‖∞:uλ(x,t)‖uλ‖∞→{0 for (x,t)∈(Ω¯∖Ω¯0)×[0,T],ϕ(x) for (x,t)∈Ω¯0×[0,T], where ϕ(x) is the normalised positive eigenfunction associated to λp(Ω0). The proof relies on a general regularity result for the solutions of these types of nonlocal equations, which is of independent interest.

On positive eigenfunctions of one nonlocal boundary problem

In this paper we present separate cases of Sturm–Liouville problem with nonlocal two-point boundary condition. The existence of positive eigenvalues and eigenfunctions is analysed. Obtained results we compare with results that are described in paper [1] and denote that in some cases they don’t coincide.

Fatou Theorem and Its Converse for Positive Eigenfunctions of the Laplace–Beltrami Operator on Harmonic NA Groups

Fatou Theorem and Its Converse for Positive Eigenfunctions of the Laplace–Beltrami Operator on Harmonic NA Groups

Asymptotic analysis of SIR epidemic model with nonlocal diffusion and generalized nonlinear incidence functional

We aim in this research to determine the global stability of equilibrium states for an SIR epidemic model with nonlocal diffusion and nonlinear incidence function in a heterogeneous environment. For achieving this result, we consider that the model parameters are Lipschitz continuous functions. At the infection free‐equilibrium, the eigenvalue problem has a principal simple eigenvalue corresponding to strictly positive eigenfunction which is expressed as . By a Lyapunov function approach, we show the global stability of drug‐free equilibrium for . For , and using persistence theory for dynamical systems, we show that the epidemic will persist and the unique endemic equilibrium state is globally stable by constructing a proper Lyapunov function.

A Ruelle–Perron–Frobenius theorem for expanding circle maps with an indifferent fixed point

We establish an original result for the thermodynamic formalism in the context of expanding circle transformations with an indifferent fixed point. For an observable whose modulus of continuity is linked to the dynamics near such a fixed point, by identifying an appropriate linear space to evaluate the action of the transfer operator, we show that there is a strictly positive eigenfunction associated with the maximal eigenvalue given as the exponential of the topological pressure. Taking into account also the corresponding eigenmeasure, the invariant probability thus obtained is proved to be the unique Gibbs-equilibrium state of the system.

Generalized multifractality at the spin quantum Hall transition: Percolation mapping and pure-scaling observables

This work extends the analysis of the generalized multifractality of critical eigenstates at the spin quantum Hall transition in two-dimensional disordered superconductors [J. F. Karcher et al, Annals of Physics, 435, 168584 (2021)]. A mapping to classical percolation is developed for a certain set of generalized-multifractality observables. In this way, exact analytical results for the corresponding exponents are obtained. Furthermore, a general construction of positive pure-scaling eigenfunction observables is presented, which permits a very efficient numerical determination of scaling exponents. In particular, all exponents corresponding to polynomial pure-scaling observables up to the order $q=5$ are found numerically. For the observables for which the percolation mapping is derived, analytical and numerical results are in perfect agreement with each other. The analytical and numerical results unambiguously demonstrate that the generalized parabolicity (i.e., proportionality to eigenvalues of the quadratic Casimir operator) does not hold for the spectrum of generalized-multifractality exponents. This excludes Wess-Zumino-Novikov-Witten models, and, more generally, any theories with local conformal invariance, as candidates for the fixed-point theory of the spin quantum Hall transition. The observable construction developed in this work paves a way to investigation of generalized multifractality at Anderson-localization critical points of various symmetry classes.

Nonlinear Eigenvalue Problems for the Dirichlet (p,2)-Laplacian

We consider a nonlinear eigenvalue problem driven by the Dirichlet (p,2)-Laplacian. The parametric reaction is a Carathéodory function which exhibits (p−1)-sublinear growth as x→+∞ and as x→0+. Using variational tools and truncation and comparison techniques, we prove a bifurcation-type theorem describing the “spectrum” as λ>0 varies. We also prove the existence of a smallest positive eigenfunction for every eigenvalue. Finally, we indicate how the result can be extended to (p,q)-equations (q≠2).

Positive eigenfunctions of a class of fractional Schrödinger operator with a potential well

In this paper, we are concerned with the following eigenvalue problem \begin{equation*} (-\Delta)^su+\lambda g(x)u=\alpha u,\ \ u\in H^s(\mathbb{R}^N),\ N\geq3, \end{equation*} where $s\in(0,1),\,\alpha,\lambda\in\mathbb{R}$ and \begin{equation*} g(x)\equiv0\ \text{on}\ \bar{\Omega},\ \ g(x)\in(0,1]\ \text{on}\ \mathbb{R}^N\backslash\bar{\Omega} \ \text{and}\ \lim_{|x|\rightarrow\infty}g(x)=1 \end{equation*} for some bounded open set $\Omega\subset\mathbb{R}^N$. We discuss the existence and some properties of the first two eigenvalues for this problem, which extend some classical results for semilinear Schrödinger equations to the nonlocal fractional setting.

Hidden positivity and a new approach to numerical computation of Hausdorff dimension: higher order methods

In 2018, the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In this paper, we extend this approach to incorporate high order approximation methods. We again rely on the fact that we can associate to the IFS a parametrized family of positive, linear, Perron–Frobenius operators L_s , an idea known in varying degrees of generality for many years. Although L_s is not compact in the setting we consider, it possesses a strictly positive C^m eigenfunction v_s with eigenvalue R(L_s) for arbitrary m and all other points z in the spectrum of L_s satisfy |z| \le b for some constant b < R(L_s) . Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value s=s_* for which R(L_s) =1 . This eigenvalue problem is then approximated by a collocation method at the extended Chebyshev points of each subinterval using continuous piecewise polynomials of arbitrary degree r . Using an extension of the Perron theory of positive matrices to matrices that map a cone K to its interior and explicit a priori bounds on the derivatives of the strictly positive eigenfunction v_s , we give rigorous upper and lower bounds for the Hausdorff dimension s_* , and these bounds converge rapidly to s_* as the mesh size decreases and/or the polynomial degree increases.

Shrinkage Priors on Complex-Valued Circular- Symmetric Autoregressive Processes

We investigate shrinkage priors on power spectral densities for complex-valued circular-symmetric autoregressive processes. We construct shrinkage predictive power spectral densities, which asymptotically dominate (i) the Bayesian predictive power spectral density based on the Jeffreys prior and (ii) the estimative power spectral density with the maximum likelihood estimator, where the Kullback-Leibler divergence from the true power spectral density to a predictive power spectral density is adopted as a risk. Furthermore, we propose general constructions of objective priors for Kähler parameter spaces by utilizing a positive continuous eigenfunction of the Laplace-Beltrami operator with a negative eigenvalue. We present numerical experiments on a complex-valued stationary autoregressive model of order 1.

A maximum principle for a fourth-order Dirichlet problem on smooth domains

Our main result is that for any bounded smooth domain Ω⊂ℝn there exists a positive-weight function w and an interval I such that for λ∈I and Δ2u=λwu+f in Ω with u=∂∂νu=0 on ∂Ω the following holds: if f is positive, then u is positive. The proofs are based on the construction of an appropriate weight function w with a corresponding strongly positive eigenfunction and on a converse of the Krein–Rutman theorem. For the Dirichlet bilaplace problem above with λ=0 the Boggio–Hadamard conjecture from around 1908 claimed that positivity is preserved on convex 2-dimensional domains and was disproved by counterexamples from Duffin and Garabedian some 40 years later. With w=1 not even the first eigenfunction is in general positive. So by adding a certain weight function our result shows a striking difference: not only is a corresponding eigenfunction positive but also a fourth-order “maximum principle” holds for some range of λ.

Existence of positive eigenfunctions to an anisotropic elliptic operator via the sub-supersolution method

Using the sub-supersolution method, we study the existence of positive solutions for the anisotropic problem 0.1 $$\begin{aligned} -\sum _{i=1}^N\frac{\partial }{\partial x_i}\left( \left| \frac{\partial u}{\partial x_i}\right| ^{p_i-2}\frac{\partial u}{\partial x_i}\right) =\lambda u^{q-1} \end{aligned}$$ where $$\Omega $$ is a bounded and regular domain of $${\mathbb {R}}^N$$ , $$q>1$$ , and $$\lambda >0$$ .

A weighted eigenvalue problem of the degenerate operator associated with infinity Laplacian

In this paper, we study a weighted eigenvalue problem of the degenerate operator associated with infinity Laplacian Δ∞hu+λa(x)|u|h−1u=0,inΩ,u=0,on∂Ω,where h>1,Δ∞hu=|Du|h−3Δ∞u is the h-homogeneous infinity Laplacian and a(x) is a positive continuous bounded function in Ω. We prove the existence of the principal eigenvalue and a corresponding positive eigenfunction. The approach to the weighted eigenvalue problem is based on the maximum principle and when a parameter is less than the principal eigenvalue, some existence and uniqueness results related to this problem are established.

Examining the Feasibility of the Sturm–Liouville Theory for Ross Recovery

Recent studies have suggested that it is feasible to recover a physical measure from a risk-neutral measure. Given a market state variable modeled as a Markov process, the key concept is to extract a unique positive eigenfunction of the generator of the Markov process. In this work, the feasibility of this recovery theory is examined. We prove that, under a restrictive integrability condition, recovery is feasible if and only if both endpoints of the state variable are limit-point. Several examples with explicit positive eigenfunctions are considered. However, in general, a physical measure cannot be recovered from a risk-neutral measure. We provide a financial and mathematical rationale for such recovery failure.

On the solvability of an indefinite nonlinear Kirchhoff equation via associated eigenvalue problems

We study the non-existence, existence and multiplicity of positive solutions to the following nonlinear Kirchhoff equation:{−M(∫RN|∇u|2dx)Δu+μV(x)u=Q(x)|u|p−2u+λf(x)u in RN,u∈H1(RN), where N≥3,2<p<2⁎:=2NN−2, M(t)=at+b(a,b>0), the potential V is a nonnegative function in RN and the weight function Q∈L∞(RN) with changes sign in Ω‾:={V=0}. We mainly prove the existence of at least two positive solutions in the cases that (i) 2<p<min⁡{4,2⁎} and 0<λ<[1−2[(4−p)/4]2/p]λ1(fΩ); (ii)p≥4,λ≥λ1(fΩ) and near λ1(fΩ) for μ>0 sufficiently large, where λ1(fΩ) is the first eigenvalue of −Δ in H01(Ω) with weight function fΩ:=f|Ω‾, whose corresponding positive principal eigenfunction is denoted by ϕ1. Furthermore, we also investigated the non-existence and existence of positive solutions if a,λ belongs to different intervals.

Global dynamics of a spatial heterogeneous viral infection model with intracellular delay and nonlocal diffusion

In this paper, we propose a spatial heterogeneous viral infection model, where heterogeneous parameters, the intracellular delay and nonlocal diffusion of free virions are considered. The global well-posedness, compactness and asymptotic smoothness of the semiflow generated by the system are established. It is shown that the principal eigenvalue problem of a perturbation of the nonlocal diffusion operator has a principal eigenvalue associated with a positive eigenfunction. The principal eigenvalue plays the same role as the basic reproduction number being defined as the spectral radius of the next generation operator. The existence of the unique chronic-infection steady state is established by the super-sub solution method. Furthermore, the uniform persistence of the model is investigated by using the persistence theory of infinite dimensional dynamical systems. By setting the eigenfunction as the integral kernel of Lyapunov functionals, the global threshold dynamics of the system is established. More precisely, the infection-free steady state is globally asymptotically stable if the basic reproduction number is less than one; while the chronic-infection steady state is globally asymptotically stable if the basic reproduction number is larger than one. Numerical simulations are carried out to illustrate the effects of intracellular delay and diffusion rate on the final concentrations of infected cells and free virions, respectively.

Ruelle operator for continuous potentials and DLR-Gibbs measures

In this work we study the Ruelle Operator associated to a continuous potential defined on a countable product of a compact metric space. We prove a generalization of Bowen's criterion for the uniqueness of the eigenmeasures and that one-sided one-dimensional DLR-Gibbs measures associated to a continuous translation invariant specifications are eigenmeasures of the transpose of the Ruelle operator. From the last claim one gets that for a continuous potential the concept of eigenprobability for the transpose of the Ruelle operator is equivalent to the concept of DLR probability. Bounded extensions of the Ruelle operator to the Lebesgue space of integrable functions, with respect to the eigenmeasures, are studied and the problem of existence of maximal positive eigenfunctions for them is considered. One of our main results in this direction is the existence of such positive eigenfunctions for Bowen's potential in the setting of a compact and metric alphabet. We also present a version of Dobrushin's Theorem in the setting of Thermodynamic Formalism.

On the first eigenvalue of the normalized 𝑝-Laplacian

We prove that if Ω \Omega is an open bounded domain with smooth and connected boundary, for every p ∈ ( 1 , + ∞ ) p \in (1, + \infty ) the first Dirichlet eigenvalue of the normalized p p -Laplacian is simple in the sense that two positive eigenfunctions are necessarily multiple of each other. We also give a (nonoptimal) lower bound for the eigenvalue in terms of the measure of Ω \Omega , and we address the open problem of proving a Faber–Krahn-type inequality with balls as optimal domains.