On the basis property of eigenfunctions of the Frankl problem with a nonlocal parity condition of the second kind

In the present paper, we write out the eigenvalues and the corresponding eigenfunctions of the modified Frankl problem with a nonlocal parity condition of the second kind. We analyze the completeness, the basis property, and the minimality of the eigenfunctions depending on the parameter of the problem.     

Basis property of eigenfunctions of the Frankl problem with a nonlocal evenness condition and with a discontinuity in the solution gradient

In the present paper, we write out the eigenfunctions of the Frankl problem with a nonlocal evenness condition and with a discontinuity of the normal derivative of the solution on the line of change of type of the equation. We show that these eigenfunctions form a Riesz basis in the elliptic part of the domain. In addition, we prove the Riesz basis property on [0, π/2] of the system of cosines occurring in the expressions for the eigenfunctions. Earlier, the Riesz basis property was proved for the eigenfunctions of the Frankl problem with a nonlocal evenness condition and with continuous solution gradient.     

On the completeness of eigenfunctions of the Frankl problem with the oddness condition

Modified Frankl problems with the oddness condition are considered for the Lavrent’ev-Bitsadze equation. The eigenvalues and eigenfunctions of these problems are found. The completeness of these eigenfunctions in the elliptic part of the domain is proved.     

Basis properties of a fourth order differential operator with spectral parameter in the boundary condition

We consider a fourth order eigenvalue problem containing a spectral parameter both in the equation and in the boundary condition. The oscillation properties of eigenfunctions are studied and asymptotic formulae for eigenvalues and eigenfunctions are deduced. The basis properties in L _p (0; l); p ∈ (1;∞); of the system of eigenfunctions are investigated.     

Problem with lacking Tricomi condition on a characteristic and an analog of the Frankl condition on a segment of the degeneration line for a class of mixed type equations

For the Gellerstedt equation with a singular coefficient, we consider a boundary value problem that differs from the Tricomi problem in that the boundary characteristic AC is arbitrarily divided into two parts AC ₀ and C ₀ C and the Tricomi condition is posed on the first of them, while the second part C ₀ C is free of boundary conditions. The lacking Tricomi condition is equivalently replaced by an analog of the Frankl condition on a segment of the degeneration line. The well-posedness of this problem is proved.     

Basis property of the eigenfunctions of a generalized gasdynamic Frankl problem with a nonlocal evenness condition and with a discontinuity in the solution gradient

We find the eigenfunctions of a generalized Frankl problem with the use of Bessel functions. We prove that these eigenfunctions form a Riesz basis in the space L ₂(D ₊), where D ₊ is the elliptic part of the domain. In addition, we prove the Riesz basis property of a trigonometric function system and the completeness of this system in the space L ₂(0, π/2).     

On the basis property of root functions of a periodic problem with an integral perturbation of the boundary condition

We consider the spectral problem for the Schrödinger operator with an integral perturbation in the periodic boundary conditions. The unperturbed problem is assumed to have multiple eigenvalues and a system of eigenfunctions forming a Riesz basis in L ₂(0, 1). We show that the basis property of systems of root functions of the problem can change under arbitrarily small changes in the kernel of the integral perturbation.     

A criterion for the existence of decaying solutions in the problem on a resonator with a cylindrical waveguide

For the Helmholtz equation Δu + k ² u = 0 in a domain Ω with a cylindrical outlet Q ₊ = ω × ?₊ to infinity, we construct a fictitious scattering operator $\mathfrak{S}$ that is unitary in L ₂(ω) and establish a bijection between the lineal of decaying solutions of the Dirichlet problem in Ω and the subspace of eigenfunctions of $\mathfrak{S}$ corresponding to the eigenvalue 1 and orthogonal to the eigenfunctions with eigenvalues λ_n ≤ k ² of the Dirichlet problem for the Laplace operator on the cross-section ω.     

Relations among eigenvalues of Sturm-Liouville problems with different types of leading coefficient functions

For any Sturm-Liouville problem with a separable boundary condition and whose leading coefficient function changes sign (exactly once), we first give a geometric characterization of its eigenvalues λn using the eigenvalues of some corresponding problems with a definite leading coefficient function. Consequences of this characterization include simple proofs of the existence of the λn's, their Prüfer angle characterization, and a way for determining their indices from the zeros of their eigenfunctions. Then, interlacing relations among the λn's and the eigenvalues of the corresponding problems are obtained. Using these relations, a simple proof of asymptotic formulas for the λn's is given.     

Spectrum of a non-self-adjoint operator associated with the periodic heat equation

We study the spectrum of the linear operator L=−θ_∂−?θ_∂(sinθθ_∂) subject to the periodic boundary conditions on θ∈[−π,π]. We prove that the operator is closed in with the domain in for |?|<2, its spectrum consists of an infinite sequence of isolated eigenvalues and the set of corresponding eigenfunctions is complete. By using numerical approximations of eigenvalues and eigenfunctions, we show that all eigenvalues are simple, located on the imaginary axis and the angle between two subsequent eigenfunctions tends to zero for larger eigenvalues. As a result, the complete set of linearly independent eigenfunctions does not form a basis in .     

Mathematical aspects of ‘t Hooft’s eigenvalue problem in two-dimensional quantum chromodynamics

We consider the eigenvalue problem of t′ Hooft for the meson spectrum in 2-dimensional QCD. Various alternative formulations are discussed, and their equivalence is proved. Then, a variational characterization of the eigenfunctions and the eigenvalues is derived yielding that the spectrum is discrete and consists of denumerably many positive eigenvalues tending to infinity. The corresponding eigenfunctions are real analytic, and form a complete system in L₂ Finally, the number of nodes of each eigenfunctions is estimated.     

The Cauchy process and the Steklov problem

Let X_t be a Cauchy process in . We investigate some of the fine spectral theoretic properties of the semigroup of this process killed upon leaving a domain D. We establish a connection between the semigroup of this process and a mixed boundary value problem for the Laplacian in one dimension higher, known as the “Mixed Steklov Problem.” Using this we derive a variational characterization for the eigenvalues of the Cauchy process in D. This characterization leads to many detailed properties of the eigenvalues and eigenfunctions for the Cauchy process inspired by those for Brownian motion. Our results are new even in the simplest geometric setting of the interval (−1,1) where we obtain more precise information on the size of the second and third eigenvalues and on the geometry of their corresponding eigenfunctions. Such results, although trivial for the Laplacian, take considerable work to prove for the Cauchy processes and remain open for general symmetric α-stable processes. Along the way we present other general properties of the eigenfunctions, such as real analyticity, which even though well known in the case of the Laplacian, are not available for more general symmetric α-stable processes.     

Upper spectral bounds and a posteriori error analysis of several mixed finite element approximations for the Stokes eigenvalue problem

This paper discusses conforming mixed finite element approximations for the Stokes eigenvalue problem. Firstly, several mixed finite element identities are proved. Based on these identities, the following new results are given: (1) It is proved that the numerical eigenvalues obtained by mini-element, P1-P1 element and Q1-Q1 element approximate the exact eigenvalues from above. (2) As for the P1-P1 , Q1-Q1 and Q1-P0 element eigenvalues, the asymptotically exact a posteriori error indicators are presented. (3) The reliable and efficient a posteriori error estimator proposed by Verfürth is applied to mini-element eigenfunctions. Finally, numerical experiments are carried out to verify the theoretical analysis.     

On the ratio of different norms of caloric functions and related eigenfunctions of an integral operator

We investigate the ratio of L 1 and L 2 norms of the Cauchy problem solutions of heat equations with compact support initial data.The related asymptotic behavior of the eigenvalues and eigenfunctions of certain integral operators is obtained.     

On non-self-adjoint Sturm-Liouville operators in the space of vector functions

In this article we obtain asymptotic formulas for the eigenvalues and eigenfunctions of the non-self-adjoint operator generated in L ₂ ^m [0, 1] by the Sturm-Liouville equation with m × m matrix potential and the boundary conditions which, in the scalar case (m = 1), are strongly regular. Using these asymptotic formulas, we find a condition on the potential for which the root functions of this operator form a Riesz basis.     

On the asymptotics of eigenfunctions of Sturm-Liouville operators with distribution potentials

We consider a Sturm-Liouville operator in the space L ₂[0, π] and derive asymptotic formulas for the eigenvalues and eigenfunctions of this operator for the case of Dirichlet-Neumann boundary conditions. The leading and second terms of the asymptotics are found in closed form.     

Numerical analysis of the spectrum of the Orr-Sommerfeld problem

A high-accuracy method for computing the eigenvalues λ _n and the eigenfunctions of the Orr-Sommerfeld operator is developed. The solution is represented as a combination of power series expansions, and the latter are then matched. The convergence rate of the expansions is analyzed by applying the theory of recurrence equations. For the Couette and Poiseuille flows in a channel, the behavior of the spectrum as the Reynolds number R increases is studied in detail. For the Couette flow, it is shown that the eigenvalues λ _n regarded as functions of R have a countable set of branch points R _k > 0 at which the eigenvalues have a multiplicity of 2. The first ten of these points are presented within ten decimals.     

<Emphasis Type="Italic">C</Emphasis><Superscript>0</Superscript>IPG adaptive algorithms for the biharmonic eigenvalue problem

This paper focuses on C ⁰IPG adaptive algorithms for the biharmonic eigenvalue problem with the clamped boundary condition. We prove the reliability and efficiency of the a posteriori error indicator of the approximating eigenfunctions and analyze the reliability of the a posteriori error indicator of the approximating eigenvalues. We present two adaptive algorithms, and numerical experiments indicate that both algorithms are efficient.     

The localization for eigenfunctions of the dirichlet problem in thin polyhedra near the vertices

Under some geometric assumptions, we show that eigenfunctions of the Dirichlet problem for the Laplace operator in an n-dimensional thin polyhedron localize near one of its vertices. We construct and justify asymptotics for the eigenvalues and eigenfunctions. For waveguides, which are thin layers between periodic polyhedral surfaces, we establish the presence of gaps and find asymptotics for their geometric characteristics.     

One‐dimensional approximations of the eigenvalue problem of curved rods

In this work we analyse the asymptotic behaviour of eigenvalues and eigenfunctions of the linearized elasticity eigenvalue problem of curved rod‐like bodies with respect to the small thickness ? of the rod. We show that the eigenfunctions and scaled eigenvalues converge, as ? tends to zero, toward eigenpairs of the eigenvalue problem associated to the one‐dimensional curved rod model which is posed on the middle curve of the rod. Because of the auxiliary function appearing in the model, describing the rotation angle of the cross‐sections, the limit eigenvalue problem is non‐classical. This problem is transformed into a classical eigenvalue problem with eigenfunctions being inextensible displacements, but the corresponding linear operator is not a differential operator. Copyright © 2001 John Wiley & Sons, Ltd.