Differential operator related to the generalized superradiance integral equation

In this paper we consider a class of problems which are generalized versions of the three-dimensional superradiance integral equation. A commuting differential operator will be found for this generalized problem. For the three-dimensional superradiance problem an alternative set of complete eigenfunctions will also be provided. The kernel for the superradiance problem when restricted to one-dimension is the same as appeared in the works of Slepian, Landau and Pollak (cf. Slepian and Pollak (1961) [1], Landau and Pollak (1961, 1962) [2] and [3], Slepian (1964, 1978) [4] and [5]). The uniqueness of the differential operator commuting with that kernel is indicated.     

一类奇型Sturm-Liouville算子的逆问题

研究了奇型Sturm-Liouville算子的逆问题.对于固定的n∈N,证明了Sturm-Liouville问题(1.3)-(1.5)的第n个特征值λ_n(q,H)关于H是严格单调增加的,及一组不同边界条件下的第n个特征值的谱集合{λ_n(q,H_k)}_(k=1)~(+∞)能够唯一确定(0,πr)上的势函数q(x).     

Spectral analysis for the Gauss problem on continued fractions

We present a new derivation of the formula appearing in Babenko (1978) and Mayer and Roepstorff (1987) that gives the probability distribution of τ⁻ⁿ${\tau }^{-n}$ in terms of the eigenvalues of a symmetric operator. Here τ$\tau$ is the well-known Gauss-map.     

Numerical solution of the inverse spectral problem for Bessel operators

We consider some inverse spectral problems associated with the singular Sturm-Liouville equation

     

Approximate method for eigensensitivity analysis of a defective matrix

Based on the exact modal expansion method, an arbitrary high-order approximate method is developed for calculating the second-order eigenvalue derivatives and the first-order eigenvector derivatives of a defective matrix. The numerical example shows the validity of the method. If the different eigenvalues μ(1),…,μ(q) of the matrix are arranged so that |μ(1)|≤?≤|μ(q)| and satisfy the condition that |μ(q₁)|<|μ(q₁+1)| for some q₁<q, and if the approximate method only uses the left and right principal eigenvectors associated with μ(1),…,μ(q₁), then associated with μ(h)(h≤q₁) the errors of the eigenvalue and eigenvector derivatives by the pth-order approximate method are nearly proportional to |μ(h)/μ(q₁+1)|^p+1.     

Spectral estimates of least energy solutions to the Brezis-Nirenberg problem with a variable coefficient

Let uε be a least energy solution to the Brezis-Nirenberg problem:

     

A fourth-order boundary value problem for a Sturm-Liouville type equation

An existence result of multiple solutions for a fourth-order Sturm-Liouville boundary value problem with variable parameters is established. As a consequence, three solutions for a boundary value problem with a fourth-order equation in a complete form are obtained. Our approach is based on variational methods.     

Analytic transient solutions of a cylindrical heat equation with a heat source

The method of separation of variables is applied in order to investigate the analytical solutions of a certain two-dimensional cylindrical heat equation. In the analysis presented here, the partial differential equation is directly transformed into ordinary differential equations. The closed-form transient temperature distributions and heat transfer rates are generalized for a linear combination of the products of Fourier-Bessel series of the exponential type. Relevant connections with some other closely-related recent works are also indicated.     

Eigenvalue Gap Theorems for a Class of Nonsymmetric Elliptic Operators on Convex Domains

Adapting the method of Andrews-Clutterbuck we prove an eigenvalue gap theorem for a class of non symmetric second order linear elliptic operators on a convex domain in euclidean space. The class of operators includes the Bakry-Emery laplacian with potential and any operator with second order term the laplacian whose first order terms have coefficients with compact support in the open domain. The eigenvalue gap is bounded below by the gap of an associated Sturm-Liouville problem on a closed interval.     

Spectral analysis of a two body problem with zero-range perturbation

We consider a class of singular, zero-range perturbations of the Hamiltonian of a quantum system composed by a test particle and a harmonic oscillator in dimension one, two and three and we study its spectrum. In fact we give a detailed characterization of point spectrum and its asymptotic behavior with respect to the parameters entering the Hamiltonian. We also partially describe the positive spectrum and scattering properties of the Hamiltonian.     

Asymptotic behavior for the principal eigenvalue of a reinforcement problem

In this paper, we consider the asymptotic behavior for the principal eigenvalue of an elliptic operator with piecewise constant coefficients. This problem was first studied by Friedman in 1980. We show how the geometric shape of the interface affects the asymptotic behavior for the principal eigenvalue. This is a refinement of the result by Friedman.     

Spectral analysis of the finite Hankel transform and circular prolate spheroidal wave functions

In this paper, we develop two practical methods for the computation of the eigenvalues as well as the eigenfunctions of the finite Hankel transform operator. These different eigenfunctions are called circular prolate spheroidal wave functions (CPSWFs). This work is motivated by the potential applications of the CPSWFs as well as the development of practical methods for computing their values. Also, in this work, we should prove that the CPSWFs form an orthonormal basis of the space of Hankel band-limited functions, an orthogonal basis of L²([0,1]) and an orthonormal system of L²([0,+∞[). Our computation of the CPSWFs and their associated eigenvalues is done by the use of two different methods. The first method is based on a suitable matrix representation of the finite Hankel transform operator. The second method is based on the use of an efficient quadrature method based on a special family of orthogonal polynomials. Also, we give two Maple programs that implement the previous two methods. Finally, we present some numerical results that illustrate the results of this work.     

Inequalities for eigenvalues of a clamped plate problem

Let be a connected bounded domain in an -dimensional Euclidean space . Assume that

are eigenvalues of a clamped plate problem or an eigenvalue problem for the Dirichlet biharmonic operator:

Then, we give an upper bound of the -th eigenvalue in terms of the first eigenvalues, which is independent of the domain , that is, we prove the following:

Further, a more explicit inequality of eigenvalues is also obtained.

     

On a singular boundary value problem arising in the theory of shallow membrane caps

We investigate the following singular boundary value problem which originates from the theory of shallow membrane caps,

     

Singular mixed boundary value problem

We study singular boundary value problems with mixed boundary conditions of the form

where , , f is a nonnegative function and satisfies the Carathéodory conditions on . Here, f can have a time singularity at t=0 and/or t=T and a space singularity at x=0 and/or y=0. We present conditions for the existence of solutions positive on [0,T) and having continuous first derivatives on [0,T].     

Almost-periodic solutions for a second order abstract Cauchy problem

Summary Let M be a number field of degree m with ring of integers \bZ_M. Let F\in\bZ_M[X,Y] be a form of degree n such that F(X,1) has distinct roots. Let\break G\in\bZ[X,Y] be an arbitrary polynomial of degree k. Assuming that k\le n-2m\pl 1 if all roots of F^{(i)}(X,1) (1\le i\le n) are complex and k\le n-4m\pl 1 otherwise, we provide an efficient algorithm for finding all solutions X,Y\in\bZ_M, \max\b(\overline{|X|},\overline{|Y|}\,\b)\ki C of the inequality \overline{\b|F(X,Y)\b|\!}\,\le c \cdot \overline{\b|G(X,Y)\b|\!}\,. We provide numerical examples with m=3 and C=10^{100}.