On an estimate of solutions of a linear homogeneous Volterra integro-differential equation of the first order in the critical case on the half-line

We obtain sufficient conditions for a certain estimate to hold on the half-line for all solutions of a linear homogeneous Volterra integro-differential equation of the first order in the critical case. We present some corollaries concerning the absolute integrability of powers of the solutions on the half-line and the convergence of solutions to zero (including the exponential and power-law convergence) as the independent variable tends to infinity. Illustrative examples are given.     

On bounded solutions of a nonlinear Schrödinger equation on the half-line

We consider the problem on nonzero solutions of the Schrödinger equation on the half-line with potential that implicitly depends on the wave function via a nonlinear ordinary differential equation of the second order under zero boundary conditions for the wave function and the condition that the potential is zero at the beginning of the interval and its derivative is zero at infinity. The problem is reduced to the analysis and investigation of solutions of the Cauchy problem for a system of two nonlinear second-order ordinary differential equations with initial conditions depending on two parameters. We show that if the solution of the Cauchy problem for some parameter values can be extended to the entire half-line, then there exists a nonzero solution of the original problem with finitely many zeros.     

On the Skorokhod mapping for equations with reflection and possible jump-like exit from a boundary

For a solution of a reflection problem on a half-line similar to the Skorokhod reflection problem but with possible jump-like exit from zero, we obtain an explicit formula and study its properties. We also construct a Wiener process on a half-line with Wentzell boundary condition as a strong solution of a certain stochastic differential equation.     

On the improperness sets of families of linear differential systems

We consider families of linear differential systems continuously depending on a real parameter with continuous (or piecewise continuous) coefficients on the half-line. The improperness set of such a family is defined as the set of all parameter values for which the corresponding systems in the family are Lyapunov improper. We show that a subset of the real axis is the improperness set of some family if and only if it is a G _δσ -set. The result remains valid for families in which the matrices of the systems are bounded on the half-line. Almost the same result holds for families in which the parameter occurs only as a factor multiplying the system matrix: their improperness sets are the G _δσ -sets not containing zero. For families of the last kind with bounded coefficient matrix, we show that their improperness set is an arbitrary open subset of the real line.     

Remark on the theory of Sergeev frequencies of zeros,signs, and roots for solutions of linear differential equations: II

The theorem that claims that the spectra (ranges) of upper and lower Sergeev frequencies of zeros, signs, and roots of a linear differential equation of order > 2 with continuous coefficients belong to the class of Suslin sets on the nonnegative half-line of the extended numerical line is inverted for the spectra of upper frequencies of third-order equations under the assumption that the spectra contain zero. In addition, we construct examples of third-order equations with continuous coefficients whose Lebesgue sets of the upper Sergeev frequency of signs belong to the exact first Borel class, and the Lebesgue sets of upper Sergeev frequencies of zeros and roots belong to the exact second Borel class.     

The Initial-Boundary Value Problem for the Korteweg–de Vries Equation

We prove local well-posedness of the initial-boundary value problem for the Korteweg–de Vries equation on right half-line, left half-line, and line segment, in the low regularity setting. This is accomplished by introducing an analytic family of boundary forcing operators.     

Estimates for the wandering rate of solutions of a linear differential equation via its coefficients

For nontrivial solutions of a linear nonautonomous differential equation with integrally small coefficients, we improve earlier-known upper bounds for the wandering rate. In particular, our estimates imply that the upper bound of the range of the wandering rate for equations of arbitrary order tends to zero as all of their coefficients uniformly (on the time half-line) tend to zero at infinity.     

(Modified) Fredholm Determinants for Operators with Matrix-Valued Semi-Separable Integral Kernels Revisited

We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Greens functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) Fredholm determinants in terms of solutions of closely associated Volterra integral equations, and as a result offers a natural way to compute such determinants.We illustrate our approach by identifying classical objects such as the Jost function for half-line Schrödinger operators and the inverse transmission coe.cient for Schrödinger operators on the real line as Fredholm determinants, and rederiving the well-known expressions for them in due course. We also apply our formalism to Floquet theory of Schrödinger operators, and upon identifying the connection between the Floquet discriminant and underlying Fredholm determinants, we derive new representations of the Floquet discriminant.Finally, we rederive the explicit formula for the 2-modified Fredholm determinant corresponding to a convolution integral operator, whose kernel is associated with a symbol given by a rational function, in a straghtforward manner. This determinant formula represents a Wiener-Hopf analog of Days formula for the determinant associated with finite Toeplitz matrices generated by the Laurent expansion of a rational function.     

Theta line bundles and the determinant of the Hodge bundle

We give an expression of the determinant of the push forward of a symmetric line bundle on a complex abelian fibration, in terms of the pull back of the relative dualizing sheaf via the zero section.

     

Resonances for Dirac operators on the half-line

We consider the 1D Dirac operator on the half-line with compactly supported potentials. We study resonances as the poles of scattering matrix or equivalently as the zeros of modified Fredholm determinant. We obtain the following properties of the resonances: (1) asymptotics of counting function, (2) estimates on the resonances and the forbidden domain.     

Sparse potentials with fractional Hausdorff dimension

We construct non-random bounded discrete half-line Schrödinger operators which have purely singular continuous spectral measures with fractional Hausdorff dimension (in some interval of energies). To do this we use suitable sparse potentials. Our results also apply to whole line operators, as well as to certain random operators. In the latter case we prove and compute an exact dimension of the spectral measures.     

Degenerate boundary conditions for the diffusion operator

We describe all degenerate two-point boundary conditions possible in a homogeneous spectral problem for the diffusion operator. We show that the case in which the characteristic determinant is identically zero is impossible for the nonsymmetric diffusion operator and that the only possible degenerate boundary conditions are the Cauchy conditions. For the symmetric diffusion operator, the characteristic determinant is zero if and only if the boundary conditions are falsely periodic boundary conditions; the characteristic determinant is identically a nonzero constant if and only if the boundary conditions are generalized Cauchy conditions.     

Optimal deterministic algorithms for some variants of Online Quota Traveling Salesman Problem

This paper is concerned with the Online Quota Traveling Salesman Problem. Depending on the symmetry of the metric and the requirement for the salesman to return to the origin, four variants are analyzed. We present optimal deterministic algorithms for each variant defined on a general space, a real line, or a half-line. As a byproduct, an improved lower bound for a variant of Online TSP on a half-line is also obtained.     

Generalized logarithmic exponent of linear differential systems in the topology of uniform convergence

We show that the generalized logarithmic exponent of an n-dimensional homogeneous linear differential system with continuous bounded coefficients on the half-line treated as a function of its right-hand side on the set of all such systems equipped with the topology of uniform convergence belongs exactly to the second Baire class if n ≥ 2 and to the zero Baire class if n = 1.     

Estimates of 1D resonances in terms of potentials

We discuss resonances for Schrödinger operators with compactly supported potentials on the line and the half-line. We estimate the sum of the negative power of all resonances and eigenvalues in terms of the norm of the potential and the diameter of its support. The proof is based on harmonic analysis and Carleson measures arguments.     

Scaling dynamics for solvable coagulation equations with dust and gel

We study limiting behavior of rescaled size distributions that evolve by Smoluchowski's rate equations for coagulation, with rate kernel K=2, x+y or xy. We find that the dynamics naturally extend to probability distributions on the half-line with zero and infinity appended, representing populations of clusters of zero and infinite size. The “scaling attractor” (set of subsequential limits) is compact and has a Levy-Khintchine-type representation that linearizes the dynamics and allows one to establish several signatures of chaos. In particular, for any given solution trajectory, there is a dense family of initial distributions (with the same initial tail) that yield scaling trajectories that shadow the given one for all time. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Special functions as subordinated semigroups on the real line

We give examples of convolution semigroups on the positive half-line and on the real line. Such semigroups are expressed in terms of special functions which arise in classical differential equations.     

Exponential dichotomy,integral separation and diagonalizability of linear systems of ordinary differential equations

This paper is concerned with linear systems of ordinary differential equations. A criterion for integral separation in terms of exponential dichotomy is given. As corollaries we obtain the roughness theorem for integral separation and the new result that an upper triangular system on a half-line is integrally separated if and only if the system corresponding to its diagonal is. We then show that a diagonal system on a half-line is integrally separated if and only if a certain perturbed system is diagonalizable. Using this result we are able to deduce that the interior of the set of diagonalizable systems on a half-line is the set of systems with integral separation and that the interior for the whole line is the set of systems which are integrally separated on both half-lines.     

Continual version of the Perron effect of change of values of the characteristic exponents

We prove the existence of a perturbed two-dimensional system of ordinary differential equations such that its linear approximation has arbitrarily prescribed negative characteristic exponents, the perturbation is of arbitrarily prescribed higher order of smallness in a neighborhood of the origin, all of its nontrivial solutions are infinitely extendible to the right, and the whole set of their Lyapunov exponents is contained in the positive half-line, is bounded, and has positive Lebesgue measure. In the general case, we also obtain explicit representations of the exponents of these solutions via their initial values.     

Existence of solutions of a boundary value problem on the half-line to second order nonlinear delay differential equations

By using the Schauder fixed point theorem, we establish a result for the existence of solutions of a boundary value problem on the half-line to second order nonlinear delay differential equations. We also present the application of our result to the special case of second order nonlinear ordinary differential equations as well as to a specific class of second order nonlinear delay differential equations. Moreover, we give a general example which demonstrates the applicability of our result. Received: 10 May 2004