Self-adjoint Integral Operator Research Articles

Functional difference equations and eigenfunctions of a Schrödinger operator with δ' -interaction on a circular conical surface.

Eigenfunctions and their asymptotic behaviour at large distances for the Laplace operator with singular potential, the support of which is on a circular conical surface in three-dimensional space, are studied. Within the framework of incomplete separation of variables an integral representation of the Kontorovich-Lebedev (KL) type for the eigenfunctions is obtained in terms of solution of an auxiliary functional difference equation with a meromorphic potential. Solutions of the functional difference equation are studied by reducing it to an integral equation with a bounded self-adjoint integral operator. To calculate the leading term of the asymptotics of eigenfunctions, the KL integral representation is transformed to a Sommerfeld-type integral which is well adapted to application of the saddle point technique. Outside a small angular vicinity of the so-called singular directions the asymptotic expression takes on an elementary form of exponent decreasing in distance. However, in an asymptotically small neighbourhood of singular directions, the leading term of the asymptotics also depends on a special function closely related to the function of parabolic cylinder (Weber function).

Quantizing time: Interacting clocks and systems

This article generalizes the conditional probability interpretation of time in which time evolution is realized through entanglement between a clock and a system of interest. This formalism is based upon conditioning a solution to the Wheeler-DeWitt equation on a subsystem of the Universe, serving as a clock, being in a state corresponding to a time t. Doing so assigns a conditional state to the rest of the Universe |ψS(t)⟩, referred to as the system. We demonstrate that when the total Hamiltonian appearing in the Wheeler-DeWitt equation contains an interaction term coupling the clock and system, the conditional state |ψS(t)⟩ satisfies a time-nonlocal Schrödinger equation in which the system Hamiltonian is replaced with a self-adjoint integral operator. This time-nonlocal Schrödinger equation is solved perturbatively and three examples of clock-system interactions are examined. One example considered supposes that the clock and system interact via Newtonian gravity, which leads to the system's Hamiltonian developing corrections on the order of G/c4 and inversely proportional to the distance between the clock and system.

Self-adjoint integral operator for bounded nonlocal transport.

An integral operator is developed to describe nonlocal transport in a one-dimensional system bounded on both ends by material walls. The "jump" distributions associated with nonlocal transport are taken to be Lévy α-stable distributions, which become naturally truncated by the bounding walls. The truncation process results in the operator containing a self-consistent, convective inward transport term (pinch). The properties of the integral operator as functions of the Lévy distribution parameter set [α,γ] and the wall conductivity are presented. The integral operator continuously recovers the features of local transport when α=2. The self-adjoint formulation allows for an accurate description of spatial variation in the Lévy parameters in the nonlocal system. Spatial variation in the Lévy parameters is shown to result in internally generated flows. Examples of cold-pulse propagation in nonlocal systems illustrate the capabilities of the methodology.

Mercer’s Theorem on General Domains: On the Interaction between Measures, Kernels, and RKHSs

Given a compact metric space X and a strictly positive Borel measure ν on X, Mercer’s classical theorem states that the spectral decomposition of a positive self-adjoint integral operator T k :L 2(ν)→L 2(ν) of a continuous k yields a series representation of k in terms of the eigenvalues and -functions of T k . An immediate consequence of this representation is that k is a (reproducing) kernel and that its reproducing kernel Hilbert space can also be described by these eigenvalues and -functions. It is well known that Mercer’s theorem has found important applications in various branches of mathematics, including probability theory and statistics. In particular, for some applications in the latter areas, however, it would be highly convenient to have a form of Mercer’s theorem for more general spaces X and kernels k. Unfortunately, all extensions of Mercer’s theorem in this direction either stick too closely to the original topological structure of X and k, or replace the absolute and uniform convergence by weaker notions of convergence that are not strong enough for many statistical applications. In this work, we fill this gap by establishing several Mercer type series representations for k that, on the one hand, make only very mild assumptions on X and k, and, on the other hand, provide convergence results that are strong enough for interesting applications in, e.g., statistical learning theory. To illustrate the latter, we first use these series representations to describe ranges of fractional powers of T k in terms of interpolation spaces and investigate under which conditions these interpolation spaces are contained in L ∞(ν). For these two results, we then discuss applications related to the analysis of so-called least squares support vector machines, which are a state-of-the-art learning algorithm. Besides these results, we further use the obtained Mercer representations to show that every self-adjoint nuclear operator L 2(ν)→L 2(ν) is an integral operator whose representing function k is the difference of two (reproducing) kernels.

On the unique solvability of a problem on water waves above submerged bodies

Time-harmonic motion of an ideal unbounded fluid in the presence of rigid bodies located under fluid’s free surface is considered. New criteria for the unique solvability of the corresponding linear boundary-value problem are suggested. These criteria are based on the introduction of two compact self-adjoint integral operators and on the investigation of their eigenvalues and eigenfunctions. For the two-dimensional problem, an algorithm is developed for finding solutions to the homogeneous problem (the so-called trapped modes). Examples of numerical computations illustrating the theoretical results are given. Bibliography: 18 titles.

ON THE NORM OF A CERTAIN SELF-ADJOINT INTEGRAL OPERATOR AND APPLICATIONS TO BILINEAR INTEGRAL INEQUALITIES

In this paper, the norm of a bounded self-adjoint integral operator $T:$ $% L^{2}(0,\infty )\rightarrow L^{2}(0,\infty )$ is obtained. As applications, a new bilinear integral inequality with a best constant factor and some particular cases such as Hilbert- type inequalities are considered.

ON THE NORM OF A CERTAIN SELF-ADJOINT INTEGRAL OPERATOR AND APPLICATIONS TO BILINEAR INTEGRAL INEQUALITIES

In this paper, the norm of a bounded self-adjoint integral operator $T:$ $% L^{2}(0,\infty )\rightarrow L^{2}(0,\infty )$ is obtained. As applications, a new bilinear integral inequality with a best constant factor and some particular cases such as Hilbert- type inequalities are considered.

On equiconvergence of spectral expansions of self-adjoint integral operators

The paper gives simple sufficient conditions on the kernel of the self-adjoint integral operator considered ensuring the equiconvergence of expansions in eigenfunctions and associated functions with the ordinary trigonometric Fourier series expansions. The presented arguments correct inaccuracies in [1].

On the Norm of a Self-Adjoint Operator and a New Bilinear Integral Inequality

In this paper, the expression of the norm of a self-adjoint integral operator T : L 2(0,∞) → L 2(0,∞) is obtained. As applications, a new bilinear integral inequality with a best constant factor is established and some particular cases are considered.

Quadratic functionals and small ball probabilities for the $m$-fold integrated Brownian motion

Let the Gaussian process Xm(t) be the m-fold integrated Brownian motion for positive integer m. The Laplace transform of the quadratic functional of Xm(t) is found by using an appropriate self-adjoint integral operator. The result is then used to show the power of a general connection between small ball probabilities for the Gaussian process. The connection is discovered by introducing an independent random shift. The interplay between our results and the principal eigenvalues for nonuniform elliptic generators on an unbounded domain is also discussed.

The quasi-diffusive approximation in transport theory: Local solutions

The one velocity, plane geometry integral neutron transport equation is transformed into a system of two equations, one of them being the equation of continuity and the other a generalized Fick`s law, in which the usual diffusion coefficient is replaced by a self-adjoint integral operator. As the kernel of this operator is very close to the Green function of a diffusion equation, an approximate inversion by means of a second order differential operator allows to transform these equations into a purely differential system which is shown to be equivalent, in the simplest case, to a diffusion-like equation. The method, the principles of which have been exposed in a previous paper, is here extended and applied to a variety of problems. If the inversion is properly performed, the quasi-diffusive solutions turn out to be quite accurate, even in the vicinity of the interface between different material regions, where elementary diffusion theory usually fails. 16 refs., 3 tabs.

Space-variant models of visual acuity using self-adjoint integral operators.

The purpose of this paper is to outline a more general approach to visual acuity experiments than the classical methods borrowed from the optical sciences. A theory based on integral operators with symmetric kernels replaces the standard use of filters constructed from windowed sines and cosines. This more general approach allows greater latitude in the range of phenomena to be modelled. It also permits all the standard techniques used in Fourier expansion of a stimulus and response to be generalized to a space-variant system such as the human visual field.

Space-variant models of visual acuity using self-adjoint integral operators

Space-variant models of visual acuity using self-adjoint integral operators

The Cauchy and the Szegö kernels on multiply connected regions

LetD be a bounded plane region whose boundary ∂D is Dini-smooth. An operatorB:L 2(∂D)→L 2(∂D) that has been considered by Kerzman and Stein is investigated. This operator is a compact self-adjoint integral operator whose kernel β (z, ζ) is bounded on ∂Dx∂D and has a geometric description involving chords inD. With the aid ofB the Szego kernel can be expressed in terms of the Cauchy kernel. Here, the operatorB is recovered from the classical theory of kernel functions resulting in extension of the work of Kerzman and Stein. In particular, the eigenvalue problem associated with the operatorB is studied and its complete equivalence with a very classical problem due to Bergman, Schiffer and Singh is established.

Variational structure of the Vlasov equation

The variational structure of the Vlasov–Maxwell integral equations is derived for a plasma equilibrium having two ignorable coordinates. It is shown that the kernel of the Maxwell equations is a self-adjoint integral operator. This operator may also be represented as a differential equation of arbitrary order. This representation is useful when the differential operator is truncated to finite order, yielding a system of intrinsically self-adjoint differential equations.

Heat Transfer in Composite Solids with Heat Generation

Steady-state heat transfer problems have been considered in a composite solid comprising two materials, one, a slab, which forms the bulk of the interior and the other, a plate, which forms a thin layer around the boundary. Through the use of appropriate Green’s functions, it is shown that the boundary value problem can be converted into a Fredholm integral equation of the second kind. The integral operator in the integral equation is shown to be self-adjoint under an appropriate inner product. Solutions have been obtained for the integral equation by expansion in terms of eigenfunctions of the self-adjoint integral operator, from which the solution to the boundary value problem is constructed. Two problems have been considered, for the first of which the eigenvalues and eigenvectors of the self-adjoint operator were analytically obtained; for the second, the spectral decomposition was obtained numerically by expansion in a convenient basis set. Detailed numerical computations have been made for the second problem using various types of heat source functions. The calculations are relatively easy and inexpensive for the examples considered. These examples, we believe, are sufficiently diverse to constitute a rather stringent test of the numerical merits of the eigenvalue technique used.

熱中性子化の特性スペクトル

The behavior of the neutron energy spectrum in a bare homogeneous moderator after the introduction of a burst of neutrons has been studied theoretically. The problem is reduced to an eigenvalue problem for a self-adjoint integral operator governing the neutron thermalization.In an infinite system, it has been shown that the Maxwellian component of the spectrum does not couple with higher modes which are orthogonal to the former and contribute nothing to the actual neutron density when integrated. The asymptotic neutron spectrum and the diffusion cooling are formulated most conveniently in the present scheme.The formalism is a general one and may be applied to any scattering model with due labor of computations. Numerical calculations has been done for heavy gaseous moderators.