130 research outputs found

### Practical implementation and error bounds of integer-type general algorithm for higher order differential equations

In our preceding paper, we have proposed an algorithm for obtaining
finite-norm solutions of higher-order linear ordinary differential equations of
the Fuchsian type [\sum_m p_m (x) (d/dx)^m] f(x) = 0 (where p_m is a polynomial
with rational-number-valued coefficients), by using only the four arithmetical
operations on integers, and we proved its validity. For any nonnegative integer
k, it is guaranteed mathematically that this method can produce all the
solutions satisfying \int |f(x)|^2 (x^2+1)^k dx < \infty, under some
conditions. We materialize this algorithm in practical procedures. An
interger-type quasi-orthogonalization used there can suppress the explosion of
calculations. Moreover, we give an upper limit of the errors. We also give some
results of numerical experiments and compare them with the corresponding exact
analytical solutions, which show that the proposed algorithm is successful in
yielding solutions with high accuracy (using only arithmetical operations on
integers).Comment: Comparison with existing method is adde

### General theory for integer-type algorithm for higher order differential equations

Based on functional analysis, we propose an algorithm for finite-norm
solutions of higher-order linear Fuchsian-type ordinary differential equations
(ODEs) P(x,d/dx)f(x)=0 with P(x,d/dx):=[\sum_m p_m (x) (d/dx)^m] by using only
the four arithmetical operations on integers. This algorithm is based on a
band-diagonal matrix representation of the differential operator P(x,d/dx),
though it is quite different from the usual Galerkin methods. This
representation is made for the respective CONSs of the input Hilbert space H
and the output Hilbert space H' of P(x,d/dx). This band-diagonal matrix enables
the construction of a recursive algorithm for solving the ODE. However, a
solution of the simultaneous linear equations represented by this matrix does
not necessarily correspond to the true solution of ODE. We show that when this
solution is an l^2 sequence, it corresponds to the true solution of ODE. We
invent a method based on an integer-type algorithm for extracting only l^2
components. Further, the concrete choice of Hilbert spaces H and H' is also
given for our algorithm when p_m is a polynomial or a rational function with
rational coefficients. We check how our algorithm works based on several
numerical demonstrations related to special functions, where the results show
that the accuracy of our method is extremely high.Comment: Errors concerning numbering of figures are fixe

### The absolutely continuous spectrum of one-dimensional Schr"odinger operators

This paper deals with general structural properties of one-dimensional
Schr"odinger operators with some absolutely continuous spectrum. The basic
result says that the omega limit points of the potential under the shift map
are reflectionless on the support of the absolutely continuous part of the
spectral measure. This implies an Oracle Theorem for such potentials and
Denisov-Rakhmanov type theorems.
In the discrete case, for Jacobi operators, these issues were discussed in my
recent paper [19]. The treatment of the continuous case in the present paper
depends on the same basic ideas.Comment: references added; a few very minor change

### The Erpenbeck high frequency instability theorem for ZND detonations

The rigorous study of spectral stability for strong detonations was begun by
J.J. Erpenbeck in [Er1]. Working with the Zeldovitch-von Neumann-D\"oring (ZND)
model, which assumes a finite reaction rate but ignores effects like viscosity
corresponding to second order derivatives, he used a normal mode analysis to
define a stability function V(\tau,\eps) whose zeros in $\Re \tau>0$
correspond to multidimensional perturbations of a steady detonation profile
that grow exponentially in time. Later in a remarkable paper [Er3] he provided
strong evidence, by a combination of formal and rigorous arguments, that for
certain classes of steady ZND profiles, unstable zeros of $V$ exist for
perturbations of sufficiently large transverse wavenumber \eps, even when the
von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in
the sense defined (nearly twenty years later) by Majda. In spite of a great
deal of later numerical work devoted to computing the zeros of V(\tau,\eps),
the paper \cite{Er3} remains the only work we know of that presents a detailed
and convincing theoretical argument for detecting them.
The analysis in [Er3] points the way toward, but does not constitute, a
mathematical proof that such unstable zeros exist. In this paper we identify
the mathematical issues left unresolved in [Er3] and provide proofs, together
with certain simplifications and extensions, of the main conclusions about
stability and instability of detonations contained in that paper.
The main mathematical problem, and our principal focus here, is to determine
the precise asymptotic behavior as \eps\to \infty of solutions to a linear
system of ODEs in $x$, depending on \eps and a complex frequency $\tau$ as
parameters, with turning points $x_*$ on the half-line $[0,\infty)$

### On the class SI of J-contractive functions intertwining solutions of linear differential equations

In the PhD thesis of the second author under the supervision of the third
author was defined the class SI of J-contractive functions, depending on a
parameter and arising as transfer functions of overdetermined conservative 2D
systems invariant in one direction. In this paper we extend and solve in the
class SI, a number of problems originally set for the class SC of functions
contractive in the open right-half plane, and unitary on the imaginary line
with respect to some preassigned signature matrix J. The problems we consider
include the Schur algorithm, the partial realization problem and the
Nevanlinna-Pick interpolation problem. The arguments rely on a correspondence
between elements in a given subclass of SI and elements in SC. Another
important tool in the arguments is a new result pertaining to the classical
tangential Schur algorithm.Comment: 46 page

### Orbital stability: analysis meets geometry

We present an introduction to the orbital stability of relative equilibria of
Hamiltonian dynamical systems on (finite and infinite dimensional) Banach
spaces. A convenient formulation of the theory of Hamiltonian dynamics with
symmetry and the corresponding momentum maps is proposed that allows us to
highlight the interplay between (symplectic) geometry and (functional) analysis
in the proofs of orbital stability of relative equilibria via the so-called
energy-momentum method. The theory is illustrated with examples from finite
dimensional systems, as well as from Hamiltonian PDE's, such as solitons,
standing and plane waves for the nonlinear Schr{\"o}dinger equation, for the
wave equation, and for the Manakov system

### Cantor and band spectra for periodic quantum graphs with magnetic fields

We provide an exhaustive spectral analysis of the two-dimensional periodic
square graph lattice with a magnetic field. We show that the spectrum consists
of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum
of a certain discrete operator under the discriminant (Lyapunov function) of a
suitable Kronig-Penney Hamiltonian. In particular, between any two Dirichlet
eigenvalues the spectrum is a Cantor set for an irrational flux, and is
absolutely continuous and has a band structure for a rational flux. The
Dirichlet eigenvalues can be isolated or embedded, subject to the choice of
parameters. Conditions for both possibilities are given. We show that
generically there are infinitely many gaps in the spectrum, and the
Bethe-Sommerfeld conjecture fails in this case.Comment: Misprints correcte

### Solving analytic differential equations in polynomial time over unbounded domains

In this paper we consider the computational complexity of solving initial-value problems de ned with analytic ordinary diferential
equations (ODEs) over unbounded domains of Rn and Cn, under the Computable Analysis setting. We show that the solution can be computed in polynomial time over its maximal interval of de nition, provided it satis es a very generous bound on its growth, and that the function admits an analytic extension to the complex plane

### A functional model, eigenvalues, and finite singular critical points for indefinite Sturm-Liouville operators

Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator
are studied under the assumption that the weight function has one turning
point. An abstract approach to the problem is given via a functional model for
indefinite Sturm-Liouville operators. Algebraic multiplicities of eigenvalues
are obtained. Also, operators with finite singular critical points are
considered.Comment: 38 pages, Proposition 2.2 and its proof corrected, Remarks 2.5, 3.4,
and 3.12 extended, details added in subsections 2.3 and 4.2, section 6
rearranged, typos corrected, references adde

### Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlev\'e Property

The integrability (solvability via an associated single-valued linear
problem) of a differential equation is closely related to the singularity
structure of its solutions. In particular, there is strong evidence that all
integrable equations have the Painlev\'e property, that is, all solutions are
single-valued around all movable singularities. In this expository article, we
review methods for analysing such singularity structure. In particular, we
describe well known techniques of nonlinear regular-singular-type analysis,
i.e. the Painlev\'e tests for ordinary and partial differential equations. Then
we discuss methods of obtaining sufficiency conditions for the Painlev\'e
property. Recently, extensions of \textit{irregular} singularity analysis to
nonlinear equations have been achieved. Also, new asymptotic limits of
differential equations preserving the Painlev\'e property have been found. We
discuss these also.Comment: 40 pages in LaTeX2e. To appear in the Proceedings of the CIMPA Summer
School on "Nonlinear Systems," Pondicherry, India, January 1996, (eds) B.
Grammaticos and K. Tamizhman

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