Related papers: On the Orr-Sommerfeld Equation with Linear Profile
The interface problem for the linear Schr\"odinger equation in one-dimensional piecewise homogeneous domains is examined by providing an explicit solution in each domain. The location of the interfaces is known and the continuity of the…
On a bounded smooth domain we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to the boundary of the domain. We derive global a priori bounds of the…
We develop a theory of sesquilinear forms over finite fields, investigating their representations via polynomials and coefficient matrices, along with classification results for these forms. Through their connection to quadratic forms, we…
We prove a limiting absorption principle for a generalized Helmholtz equation on an exterior domain with Dirichlet boundary conditions \begin{equation*} (L+\lambda)v=f, \qquad \lambda\in \mathbb{R} \end{equation*} under a Sommerfeld…
We derive explicit inequalities for sums of eigenvalues of one-dimensional Schr\"{o}dinger operators on the whole line. In the case of the perturbed harmonic oscillator, these bounds converge to the corresponding trace formula in the limit…
Let $q(x,t)$ satisfy the Dirichlet initial-boundary value problem for the nonlinear Schr\"odinger equation on the finite interval, $0 < x < L$, with $q_{0}(x) = q(x,0)$, $g_{0}(t) = q(0,t)$, $f_{0}(t) = q(L,t)$. Let $g_{1}(t)$ and…
In this work a Sturm-Liouville type problem with retarded argument which contains spectral parameter in the boundary conditions and with transmission conditions at the point of discontinuity are investigated. We obtained asymptotic formulas…
In recent developments, a general approach for solving Riemann--Hilbert problems numerically has been developed. We review this numerical framework, and apply it to the calculation of orthogonal polynomials on the real line. Combining this…
We analyze Schr\"odinger operators whose potential is given by a singular interaction supported on a sub-manifold of the ambient space. Under the assumption that the operator has at least two eigenvalues below its essential spectrum we…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
We study asymptotic behaviors of solutions to the Loewner-Nirenberg problem in finite cones and establish optimal asymptotic expansions in terms of the corresponding solutions in infinite cones. The spherical domains over which cones are…
For a natural extension of the circular unitary ensemble of order n, we study as n tends to infinity, the asymptotic behavior of the sequence of orthogonal polynomials with respect to the spectral measure. The last term of this sequence is…
The aim of this note is to study the spectrum of a linearized Liouville-type problem, characterizing the case in which the first eigenvalue is zero. Interestingly enough, we obtain also point-wise information on the associated first…
We derive the loop equation for the 1-matrix model with generic difference-type measure for eigenvalues and develop a recursive algebraic framework for solving it to an arbitrary order in the coupling constant in and beyond the planar…
Homogenization of a spectral problem in a bounded domain with a high contrast in both stiffness and density is considered. For a special critical scaling, two-scale asymptotic expansions for eigenvalues and eigenfunctions are constructed.…
A key quantity that occurs in the error analysis of several numerical methods for eigenvalue problems is the distance between the eigenvalue of interest and the next nearest eigenvalue. When we are interested in the smallest or fundamental…
In this paper we study the maximum principle, the existence of eigenvalue and the existence of solution for the Dirichlet problem for operators which are fully-nonlinear, elliptic but presenting some singularity or degeneracy which are…
The A-polynomial of a manifold whose boundary consists of a single torus is generalised to an eigenvalue variety of a manifold whose boundary consists of a finite number of tori, and the set of strongly detected boundary curves is…
In this paper, we establish the linear profile decomposition for the one dimensional fourth order Schr\"odinger equation $$ iu_t-\mu\Delta u+\Delta^2u=0, t\in\mathbb{R}, x\in\mathbb{R}, u(0,x)=f(x)\in L^2, $$ where $\mu\ge 0$. As an…
The primitive equations in a 3D infinite layer domain are considered with linearly growing initial data in the horizontal direction, which illustrates the global atmospheric rotating or straining flows. On the boundaries, Dirichlet, Neumann…