Singular perturbations in the interaction representation

A general method for treating highly singular perturbations V of self-adjoint operators H in Hilbert space is applied to the case of perturbations of $\text{(?}\text{id}\text{dx}\text{)}{}^{\mathrm{n}}$ in L₂ ($\text{R}$¹ by (multiplications by) distributions. A self-adjoint operator H_V that agrees with H + V in the usual sense when V is sufficiently regular, and is moreover a continuous function of V, within the class of distributions under consideration, in the strong operator topology for unbounded self-adjoint operators, is shown to exist. This operator H_V need not be semi-bounded, or determined by a sesquilinear form associated with H + V. The method proceeds by construction of the corresponding unitary propagator in the interaction representation, essentially e^?itHVe^itH, which is shown to be expressible as a uniformly convergent perturbative series for small times.     

A parameter-uniform numerical method for a singularly perturbed two parameter elliptic problem

In this paper, a class of singularly perturbed elliptic partial differential equations posed on a rectangular domain is studied. The differential equation contains two singular perturbation parameters. The solutions of these singularly perturbed problems are decomposed into a sum of regular, boundary layer and corner layer components. Parameter-explicit bounds on the derivatives of each of these components are derived. A numerical algorithm based on an upwind finite difference operator and a tensor product of piecewise-uniform Shishkin meshes is analysed. Parameter-uniform asymptotic error bounds for the numerical approximations are established.     

A second order scheme for a time-dependent,singularly perturbed convection–diffusion equation

We consider a numerical scheme for a one-dimensional, time-dependent, singularly perturbed convection–diffusion problem. The problem is discretized in space by a standard finite element method on a Bakhvalov–Shishkin type mesh. The space error is measured in an L² norm. For the time integration, the implicit midpoint rule is used. The fully discrete scheme is shown to be convergent of order 2 in space and time, uniformly in the singular perturbation parameter.     

Singularly perturbed problems in partial differential equations: a survey

This survey paper contains a surprisingly large amount of material on singularly perturbed partial differential equations and indeed can serve as an introduction to some of the ideas and methods of the singular perturbation theory. Starting from Prandtl's work a large amount of work has been done in the area of singular perturbations. This paper limits its coverage to some standard singular perturbation models considered by various workers and the methods developed by numerous researchers after 1980–2000. In this review we have covered singularly perturbed partial differential equations. About ODEs the survey has already been done by us [see M.K. Kadalbajoo, K.C. Patidar, Appl. Math. Comput. 130 (2002) 457–510].     

On the theory of semigroups of operators on locally convex spaces

The behavior of strongly continuous one-parameter semigroups of operators on locally convex spaces is considered. The emphasis is placed on semigroups that grow too rapidly to be treated by classical Laplace transform methods.A space

of continuous E-valued functions is defined for a locally convex space E, and the generalized resolvent $\text{R}$ of an operator A on E is defined as an operator on

. It is noted that $\text{R}$ may exist when the classical resolvent (λ ? A)^?1 fails to exist. Conditions on $\text{R}$ are given that are necessary and sufficient to guarantee that A is the generator of a semigroup T(t). The action of $\text{R}$ is characterized by convolution against the semigroup, and the semigroup is computed as the limit of $\text{R}$ acting on an approximate identity.Conditions on an operator B are introduced that are sufficient to guarantee that A + B is the generator of a semigroup whenever A is. A formula is given for the perturbed semigroup.Two characterizations of semigroups that can be extended holomorphically into some sector of the complex plane are given. One is in terms of the growth of the derivative $\text{(}\text{d}\text{dt}\text{) T(t)}$ as t approaches 0, the other is in terms of the behavior of $\text{R}$ⁿ, the powers of the generalized resolvent.Throughout, the generalized resolvent plays a role analogous to the role of the classical resolvent in the work of Hille, Phillips, Yosida, Miyadera, and others.     

Singularly perturbed self-adjoint operators in scales of Hilbert spaces

Finite-rank perturbations of a semibounded self-adjoint operator A are studied in a scale of Hilbert spaces associated with A. The notion of quasispace of boundary values is used to describe self-adjoint operator realizations of regular and singular perturbations of the operator A by the same formula. As an application, the one-dimensional Schrödinger operator with generalized zero-range potential is studied in the Sobolev space W ₂ ^p (?), p ∈ ?.     

Commutative properties of singularly perturbed operators

Suppose the self-adjoint operatorA in the Hilbert spaceH commutes with the bounded operatorS. Suppose another self-adjoint operatorā is singularly perturbed with respect toA, i.e., it is identical toA on a certain dense set inH. We study the following question: Under what conditions doesā also commute withS? In addition, we consider the case whenS is unbounded and also the case whenS is replaced by a singularly perturbed operator S. As application, we consider the Laplacian inL ₂(R ^q ) that is singularly perturbed by a set of δ functions and commutes with the symmetrization operator inR ^q ,q=2, 3, or with regular representations of arbitrary isometric transformations inR ^q ,q≤3.     

Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation

For a singularly perturbed parabolic convection-diffusion equation, the conditioning and stability of finite difference schemes on uniform meshes are analyzed. It is shown that a convergent standard monotone finite difference scheme on a uniform mesh is not ?-uniformly well conditioned or ?-uniformly stable to perturbations of the data of the grid problem (here, ? is a perturbation parameter, ? ∈ (0, 1]). An alternative finite difference scheme is proposed, namely, a scheme in which the discrete solution is decomposed into regular and singular components that solve grid subproblems considered on uniform meshes. It is shown that this solution decomposition scheme converges ?-uniformly in the maximum norm at an O(N ^?1lnN + N ₀ ^?1 ) rate, where N + 1 and N ₀ + 1 are the numbers of grid nodes in x and t, respectively. This scheme is ?-uniformly well conditioned and ?-uniformly stable to perturbations of the data of the grid problem. The condition number of the solution decomposition scheme is of order O(δ^?2lnδ^?1 + δ ₀ ^?1 ); i.e., up to a logarithmic factor, it is the same as that of a classical scheme on uniform meshes in the case of a regular problem. Here, δ = N ^?1lnN and δ₀ = N ₀ ^?1 are the accuracies of the discrete solution in x and t, respectively.     

Analytic perturbation of generalized inverses

We investigate the analytic perturbation of generalized inverses. Firstly we analyze the analytic perturbation of the Drazin generalized inverse (also known as reduced resolvent in operator theory). Our approach is based on spectral theory of linear operators as well as on a new notion of group reduced resolvent. It allows us to treat regular and singular perturbations in a unified framework. We provide an algorithm for computing the coefficients of the Laurent series of the perturbed Drazin generalized inverse. In particular, the regular part coefficients can be efficiently calculated by recursive formulae. Finally we apply the obtained results to the perturbation analysis of the Moore–Penrose generalized inverse in the real domain.     

On the C(R) stability of uncertain singularly perturbed systems

In this paper, a simple criterion for the C(R) stability of uncertain singularly perturbed systems is proposed. Such a criterion can be easily checked by some algebraic inequality. The upper bound of the singular perturbation parameter ε is also given by estimating the unique positive zero of specific function. Finally, a numerical example is provided to illustrate the main result.     

On finite rank perturbations of definitizable operators

It was shown by P. Jonas and H. Langer that a selfadjoint definitizable operator A in a Krein space remains definitizable after a finite rank perturbation in resolvent sense if the perturbed operator B is selfadjoint and the resolvent set ρ(B) is nonempty. It is the aim of this note to prove a more general variant of this perturbation result where the assumption on ρ(B) is dropped. As an application a class of singular ordinary differential operators with indefinite weight functions is studied.     

Local analysis near a folded saddle-node singularity

Folded saddle-nodes occur generically in one parameter families of singularly perturbed systems with two slow variables. We show that these folded singularities are the organizing centers for two main delay phenomena in singular perturbation problems: canards and delayed Hopf bifurcations. We combine techniques from geometric singular perturbation theory—the blow-up technique—and from delayed Hopf bifurcation theory—complex time path analysis—to analyze the flow near such folded saddle-nodes. In particular, we show the existence of canards as intersections of stable and unstable slow manifolds. To derive these canard results, we extend the singularly perturbed vector field into the complex domain and study it along elliptic paths. This enables us to extend the invariant slow manifolds beyond points where normal hyperbolicity is lost. Furthermore, we define a way-in/way-out function describing the maximal delay expected for generic solutions passing through a folded saddle-node singularity. Branch points associated with the change from a complex to a real eigenvalue structure in the variational equation along the critical (slow) manifold make our analysis significantly different from the classical delayed Hopf bifurcation analysis where these eigenvalues are complex only.     

Some apparent connection between Baxter and averaging operators

A formula for the resolvent R(λ, T) of a Baxter operator T, on a complex Banach algebra $\text{A}$ with identity e, is obtained. With the parameter θ ≠ 0 and e, but under some restriction, this formula is analogous to that for the resolvent of an averaging operator. A counterexample is given, which shows that such a Baxter operator is not averaging in general. When θ is regular in $\text{A}$, a simple representation of T in terms of summation and averaging operators is obtained.     

Low energy scattering for nonlinear Klein-Gordon equations

The scattering operator which belongs to a pair of PDEs consisting of the Klein-Gordon equation and a perturbation of it by a power-like nonlinearity z.hfl;(u) is studied. It is shown that this operator can be defined on a whole neighbourhood of the origin in energy space if $\text{z.hfl;(u) = ±¦u¦}{}^{\mathrm{p\; ?\; 1}}\text{u}\text{or}\text{±¦u¦}{}^{\mathrm{p}}$, where $\text{1+4}\text{(n ? 1)}\text{<p <}\text{1 + 4}\text{(n ? 2)}$ and the space dimension n ? 2 is arbitrary.     

Zur Stabilität diskreter Teilspektren singulärer Differentialoperatoren zweiter Ordnung gegenüber Störungen der Koeffizienten und der Definitionsgebiete

The stability ofL ²-eigenvalues and associated eigenspaces of singular second order differential operators of Schrödinger-type is shown for asymptotic perturbations of the coefficients and the domain of definition. The perturbations involved are more general than those studied in [3] and [5], because we do not postulate the convergence of the coefficients “from above” or of the domains “from inside” or “from outside”. Moreover, the domain of definition is allowed to be perturbed in its interior. The underlying abstract perturbation theory was established in a previous paper [9].     

A parameter-uniform numerical method for time-dependent singularly perturbed differential-difference equations

A numerical study is made for solving a class of time-dependent singularly perturbed convection–diffusion problems with retarded terms which often arise in computational neuroscience. To approximate the retarded terms, a Taylor’s series expansion has been used and the resulting time-dependent singularly perturbed differential equation is approximated using parameter-uniform numerical methods comprised of a standard implicit finite difference scheme to discretize in the temporal direction on a uniform mesh by means of Rothe’s method and a B-spline collocation method in the spatial direction on a piecewise-uniform mesh of Shishkin type. The method is shown to be accurate of order O(M⁻¹ + N⁻² ln³N), where M and N are the number of mesh points used in the temporal direction and in the spatial direction respectively. An extensive amount of analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter. Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations. Comparisons of the numerical solutions are performed with an upwind and midpoint upwind finite difference scheme on a piecewise-uniform mesh to demonstrate the efficiency of the method.     

Modal controllability and weighted shifts

In this paper is characterized the situation wherein the properties of modal controllability are valid for the abstract linear autonomous control system $\text{x}\text{?}\text{= Ax + bu}$, where A is a weighted shift operator and b is a vector in separable Banach space. When A is a basis operator on l^p, it is shown that the control system is modal controllable. It is proved that the same result is valid for a unicellular weighted shift if the system is strictly cyclic or exactly controllable.     

A finite element method with a large mesh-width for a stiff two-point boundary value problem

We describe a finite element method for computation of numerical approximations of the solution of the second order singularly perturbed two-point boundary value problem on [?1, 1]

$\text{? u″ + pu′ = f, u(?1) = u(1) = 0, 0 < ? ∠ 1, (′ =}\text{d}\text{dx}\text{)}$

On a quasi-uniform mesh we construct exponentially fitted trial spaces which consist of piece-wise polynomials and of exponentials which fit locally to the singular solution of the equation or its adjoint. We discretise the Galerkin form for the boundary problem using such exponentially fitted trial spaces. We derive rigorous bounds for the error of discretisation with respect to the energy norm and we obtain superconvergence at the mesh-points, the error depending on ?, the mesh-width and the degree of the piece-wise polynomials.     

High order parameter-robust numerical method for singularly perturbed reaction-diffusion problems

We present a high order parameter-robust finite difference method for singularly perturbed reaction-diffusion problems. The problem is discretized using a suitable combination of fourth order compact difference scheme and central difference scheme on generalized Shishkin mesh. The convergence analysis is given and the method is proved to be almost fourth order uniformly convergent in maximum norm with respect to singular perturbation parameter ε. Numerical experiments are conducted to demonstrate the theoretical results.     

Eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations

udy the perturbation theory of structured matrices under structured rank one perturbations, and then focus on several classes of complex matrices. Generic Jordan structures of perturbed matrices are identified. It is shown that the perturbation behavior of the Jordan structures in the case of singular J-Hamiltonian matrices is substantially different from the corresponding theory for unstructured generic rank one perturbation as it has been studied in [18, 28, 30, 31]. Thus a generic structured perturbation would not be generic if considered as an unstructured perturbation. In other settings of structured matrices, the generic perturbation behavior of the Jordan structures, within the confines imposed by the structure, follows the pattern of that of unstructured perturbations.