Related papers: On Green's Functions and Positive, Self-Adjoint, E…
The main objective of the work is to provide sharp two-sided estimates of $\lambda$-Green function of hyperbolic Brownian motion of a half-space. We strongly rely on recent results obtained by K. Bogus and J. Malecki [3], regarding precise…
Using three different notions of generalized principal eigenvalue of linear second order elliptic operators in unbounded domains, we derive necessary and sufficient conditions for the validity of the maximum principle, as well as for the…
The purpose of this article is to establish upper and lower estimates for the integral kernel of the semigroup exp(-tP) associated to a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The…
We construct Green's function for second order elliptic operators of the form $Lu=-\nabla \cdot (\mathbf{A} \nabla u + \boldsymbol{b} u)+ \boldsymbol c \cdot \nabla u+ du$ in a domain and obtain pointwise bounds, as well as Lorentz space…
We study the resolvent kernel (Green's function) of magnetic Dirac operators on a half-plane with boundary conditions interpolating between infinite mass and zigzag cases, excluding the latter. We show that these kernels have all the…
We establish an eigenfunctional theorem for positive operators, evocative of the Krein--Rutman theorem. A more general version gives a joint eigenfunctional for commuting operators.
The Hamiltonian $H={1\over2} p^2+{1\over2}m^2x^2+gx^2(ix)^\delta$ with $\delta,g\geq0$ is non-Hermitian, but the energy levels are real and positive as a consequence of ${\cal PT}$ symmetry. The quantum mechanical theory described by $H$ is…
We obtain pointwise lower bounds for heat kernels of higher order differential operators with Dirichlet boundary conditions on bounded domains in $\R^N$. The bounds exhibit explicitly the nature of the spatial decay of the heat kernel close…
This paper presents the basic ideas and properties of elliptic functions and elliptic integrals as an expository essay. It explores some of their numerous consequences and includes applications to some problems such as the simple pendulum,…
The aim of this paper is to show certain properties of the Green's functions related to the Hill's equation coupled with different two point boundary value conditions. We will obtain the expression of the Green's function of Neumann,…
We propose a new data-driven approach for learning the fundamental solutions (Green's functions) of various linear partial differential equations (PDEs) given sample pairs of input-output functions. Building off the theory of functional…
The generating function method is applied to the trace of the heat kernel and the one-loop effective action derived from the covariant perturbation theory. The basis of curvature invariants of second order for the heat kernel (Green…
We explicitly construct a heat kernel as a Neumann series for certain function spaces, such as $L^{1}$, $L^{2}$, and Hilbert spaces, associated to a locally compact Hausdorff space $\mathfrak{X}$ with Borel $\sigma$-algebra $\mathcal{B}$,…
We obtain two-sided estimates for the heat kernel (or the fundamental function) associated with the following fractional Schr\"odinger operator with negative Hardy potential $$\Delta^{\alpha/2} -\lambda |x|^{-\alpha}$$ on $\RR^d$, where…
In previous works, we used a so-called deformation formula in order to study, in particular, the Borel summability of the heat kernel of some operators. A goal of this paper is to collect miscellaneous remarks related to these works. Here…
In general, for higher order elliptic equations and boundary value problems like the biharmonic equation and the linear clamped plate boundary value problem neither a maximum principle nor a comparison principle or -- equivalently -- a…
Stieltjes boundary problems generalize the customary class of well-posed two-point boundary value problems in three independent directions, regarding the specification of the boundary conditions: (1) They allow more than two evaluation…
We consider a class of homogeneous partial differential operators on a finite-dimensional vector space and study their associated heat kernels. The heat kernels for this general class of operators are seen to arise naturally as the limiting…
It is shown that positivity in $(0,1)\times (0,1)$ of Green function of positively defined fourth-order ordinary differential operator (with separated boundary conditions) is a criterium of sign-regularity of this operator.
Motivated by recent interest on Kirchhoff-type equations, in this short note we utilize a classical, yet very powerful, tool of nonlinear functional analysis in order to investigate the existence of positive eigenvalues of systems of…