Kato's Theorem Research Articles

The Well‐Posedness of Solutions for a Generalized Shallow Water Wave Equation

A nonlinear partial differential equation containing the famous Camassa‐Holm and Degasperis‐Procesi equations as special cases is investigated. The Kato theorem for abstract differential equations is applied to establish the local well‐posedness of solutions for the equation in the Sobolev space Hs(R) with s > 3/2. Although the H1‐norm of the solutions to the nonlinear model does not remain constant, the existence of its weak solutions in the lower‐order Sobolev space Hs with 1 ≤ s ≤ 3/2 is proved under the assumptions u0 ∈ Hs and .

SCHRÖDINGER EQUATIONS WITH TIME-DEPENDENT UNBOUNDED SINGULAR POTENTIALS

We consider time-dependent perturbations by unbounded potentials of Schrödinger operators with scalar and magnetic potentials which are almost critical for the selfadjointness. We show that the corresponding time-dependent Schrödinger equations generate a unique unitary propagator if perturbations of scalar and magnetic potentials are differentiable with respect to the time variable and they increase at the spatial infinity at most quadratically and at most linearly, respectively, where both have mild local singularities. We use time-dependent gauge transforms and apply Kato's abstract theorem on evolution equations.

The Local Strong and Weak Solutions for a Nonlinear Dissipative Camassa‐Holm Equation

Using the Kato theorem for abstract differential equations, the local well‐posedness of the solution for a nonlinear dissipative Camassa‐Holm equation is established in space C([0, T), Hs(R))∩C1([0, T), Hs−1(R)) with s > 3/2. In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space Hs(R) with 1 ≤ s ≤ 3/2 is developed.

Discrete approximation of quantum stochastic models

We develop a general technique for proving convergence of repeated quantum interactions to the solution of a quantum stochastic differential equation. The wide applicability of the method is illustrated in a variety of examples. Our main theorem, which is based on the Trotter–Kato theorem, is not restricted to a specific noise model and does not require boundedness of the limit coefficients.

Approximation and limit theorems for quantum stochastic models with unbounded coefficients

We prove a limit theorem for quantum stochastic differential equations with unbounded coefficients which extends the Trotter–Kato theorem for contraction semigroups. From this theorem, general results on the convergence of approximations and singular perturbations are obtained. The results are illustrated in several examples of physical interest.

Blow-up of solution of an initial boundary value problem for a generalized Camassa–Holm equation

In this Letter, we study the following initial boundary value problem for a generalized Camassa–Holm equation { u t − u x x t + 3 u u x − 2 u x u x x − u u x x x + k ( u − u x x ) x = 0 , t ⩾ 0 , x ∈ [ 0 , 1 ] , u ( 0 , t ) = u ( 1 , t ) = u x ( 0 , t ) = u x ( 1 , t ) = 0 , t ⩾ 0 , u ( 0 , x ) = u 0 ( x ) , x ∈ [ 0 , 1 ] , where k is a real constant. We establish local well-posedness of this closed-loop system by using Kato's theorem for abstract quasilinear evolution equation of hyperbolic type. Then, by using multiplier technique, we obtain a conservation law which enable us to present a blow-up result.

Kinetic energy and ground-state electron densities as fingerprints of wavefunction entanglement in two-electron spin-compensated atomic models

We take as a starting point the ground-state electron density in two-electron model atoms in which Coulomb confinement in the He atom is first replaced by harmonic restoring forces. Switching off electron–electron interactions, one readily constructs a third-order differential equation for the ground-state electron density, as in the recent work of March and Ludena (2004 Phys. Lett. A 330 16). We then switch on two different model interactions, first in the so-called Hookean atom going back to Kestner and Sinanoglu (1962 Phys. Rev. 128 2687), in which e2/r12 is retained as in He, and secondly in the Moshinsky (1968 Am. J. Phys. 36 52) atom in which Kr212/2 is switched on. Some analyticity properties of the low-order linear homogeneous differential equations which result are next studied. He-like atomic ions are then treated in the limit of large atomic number Z. In this latter case, one identifies both the electron–nuclear cusp, or equivalently Kato's theorem, and the corresponding electron–electron cusp in the ground-state spatial wavefunction Ψ(r1, r2). A final comment concerns quantum information and entanglement in relation to the recent work of Amovilli and March (2004 Phys. Rev. A 69 054302).

On the proof by reductio ad absurdum of the Hohenberg–Kohn theorem for ensembles of fractionally occupied states of Coulomb systems

AbstractIt is demonstrated that the original reductio ad absurdum proof of the generalization of the Hohenberg–Kohn theorem for ensembles of fractionally occupied states for isolated many‐electron Coulomb systems with Coulomb‐type external potentials by Gross and colleagues is self‐contradictory, since the to‐be‐refuted assumption (negation) regarding the ensemble one‐electron densities and the assumption regarding the external potentials are logically incompatible to each other due to the Kato electron‐nuclear cusp theorem. It is proved, however, that the Kato theorem itself provides a satisfactory proof of this theorem. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Abstract Cauchy Problems for Quasi‐Linear Evolution Equations in the Sense of Hadamard

This paper is devoted to the well‐posedness of abstract Cauchy problems for quasi‐linear evolution equations. The notion of Hadamard well‐posedness is considered, and a new type of stability condition is introduced from the viewpoint of the theory of finite difference approximations. The result obtained here generalizes not only some results on abstract Cauchy problems closely related with the theory of integrated semigroups or regularized semigroups but also the Kato theorem on quasi‐linear evolution equations. An application to some quasi‐linear partial differential equation of weakly hyperbolic type is also given. 2000 Mathematics Subject Classification 34G20, 47J25 (primary), 47D60, 47D62 (secondary).

Approximation of local convoluted semigroups

We extend the Trotter–Kato theorem on C0-semigroups to local convoluted semigroups on dual spaces and apply these results to the general Banach space setting. Compared to known results we obtain weaker convergence assumptions on the resolvent.

Cohomologie non ramifiée sur une courbe p -adique lisse (Unramified Cohomology on a Smooth p -Adic Curve)

Let k be a local field with residue characteristic p. Let n be a prime-to-p integer. Consider a smooth, irreducible k-curve X. In this article, we study the subgroup of H3(k(X), μn⊗2) consisting of classes which are unramified on X. It is known to be isomorphic to H0(XZar, R3π*μn⊗2), where π: Xet → XZar is the canonical map. A purely topological description of this group is given, using the Berkovich analytification Xan of X. More precisely, denote by Y a smooth and irreducible Berkovich-analytic k-curve, by Ytop the underlying topological space, by Yet the etale-analytic site and by πan the canonical map Yet → Ytop. Let Δ be the skeleton of Y (it is a closed subset defined by Berkovich which is locally a finite graph). Then we show (th. 42) that H0(Ytop, R3πan*μn⊗2) is naturally isomorphic (through sort of a pointwise evaluation of cohomology classes) to the group of harmonic cochains defined on Δ with values in ℤ/n. Now, if X is a smooth algebraick-curve the natural map H0(XZar, R3π*μn⊗2) → H0(Xantop, R3πan*μn⊗2) is shown to be an isomorphism (th. 5.2). It is a new formulation (for the case where X is proper) and a generalization (to open curves) of a result which was due to Kato. Moreover we use here some steps of Kato's proof, but not his whole result. So our method gives, for projective curves, a (partially) new proof of Kato's theorem.

Some aspects of the Trotter – Kato theorem

Abstract - The classical proof of the Trotter - Kato theorem also provides rate estimates for smooth initial data. We discuss the possibilities to reduce the so-called smoothness gap for special types of semigroups as, for instance, analytic semigroups. Furthermore, we show that the assumption of stability of the approximating semigroups can be relaxed if at the same time the consistency requirement is replaced by a stronger assumption. Finally, we give a proof for a ‘folk’ theorem for nonhomogeneous problems which repeatedly was proved under stronger assumptions than necessary.

Comment on calculations of excited states by the density functional theory

Valone and Capitani [Phys. Rev. A 23 (1981) 2127] proposed a method for calculating a single selected excited state by finding the ground state energy of the square of a shifted Hamiltonian. Here we prove that this approach can not enable one to calculate excited states on the basis of the Hohenberg and Kohn density functional theory. The justification for the use of the density functional theory for a single excited state on the basis of Kato's cusp theorem is briefly discussed.

Thomas-Fermi Theory of an Inhomogeneous Electron Liquid Generalized to Incorporate Density Gradients

Motivated by exact results for many closed shells in a bare Coulomb field, a generalization of the Thomas-Fermi statistical model is proposed. This generalization includes density gradients in the density-potential relation, and offers the possibility of avoiding the singularity (of the original method) in the density at an atomic nucleus and of embodying Kato's theorem.

Excited states in density functional theory

Density functional theory for a single excited state is presented using Kato's theorem and the concept of adiabatic connection. The degenerate case is also detailed. The optimized potential method is generalized. The generalized Krieger, Li, and Iafrate (KLI) approximation is derived. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 681–691, 1998

LpContraction Semigroups for Vector Valued Functions

We discuss semigroups acting on vector valued functions. We consider a comparison theorem between two semigroups: a semigroup acting on scalar valued functions and a semigroup acting on vector valued functions. The criterion is given by the abstract Kato theorem. By using this, we give a sufficient condition for the criterion in the setting of square field operator. We also consider the essential self-adjointness of a perturbed semigroup.

The One-Electron Energies of a Neutral Zinc Atom Evaluated Using Nonlinear Field Theoretical Methods: Comparison with Hartree–Fock Calculations

The Method of Coherent Structures (MCS) has been used, as an example of a sufficiently complex atomic system, to determine the one-electron eigenvalues of a neutral zinc atom. An ensemble procedure is used in MCS to properly account for spatial correlations even for a relatively small number of particles. The MCS is a semiclassical approach, but the essentially Fermionic character of the electrons emerges via physical boundary conditions on the choice of classical field. We interpret the classical field as a coherent state envelope which partly determines the effective potential in which quantum fluctuations take place. We find that three distinct régimes of behaviour of the classical field exist depending on the magnitude of the effective Coulomb repulsion, μ. When μ<μ0, a critical value, those classical fields which correspond to certain discrete values of μ are physically acceptable normalisable bound charge states. When μ<μ0 and does not correspond to the discrete values, classical solutions exhibit a damped oscillatory behaviour typical of charged "ring" waves. For μ>μ0, corresponding to an excess of electronic charge, only a small fraction of the charge, represented by a few oscillations about the nucleus, remains attracted to this centre while the remainder escapes from the nucleus. Parameters representing the repulsive energy strength and energy shifts agree well with trends in the Slater parameter F 0(2s,2p) as a function of atomic number Z and also Moseley's Law. Agreement is also obtained with the first-principles form of charge density at small distances from the nucleus as given by Kato's Theorem and also at large distances as outlined by March. The usefulness and validity of this approach has been shown by comparing such energies with those determined by a Hartree–Fock–Slater (HF) method which is well known to satisfactorily account for many specific details of the level structure of atoms. Good agreement with HF has been obtained for both shell positions and energy ordering of states within a shell.

The generalization and applications of Kato's perturbation theorem

In this paper we generalize Kato's perturbation theory to weakly compact perturbations. As applications of this theory, we discuss the distribution of the essential spectrum of linear transport operators.

Bound-state position and momentum densities and Slater sum for closed shells in a bare Coulomb field.

Some generalizations are effected here of the work of Heilmann and Lieb [Phys. Rev. A 52, 3628 (1995)], who summed the squares of all the normalized bound-state wave functions for the hydrogen atom. One of their main results is shown to be equivalent to a spatial generalization of Kato's theorem. Their asymptotic evaluation of the above sum for large r is used to obtain a property of the bound-state Slater sum in the high-temperature limit. The corresponding momentum space density is also briefly discussed. \textcopyright{} 1996 The American Physical Society.

A nonlinear field theoretical method of determining the one-electron energies of a neutral zinc atom

The method of coherent structures has been used to determine the one-electron energies of a neutral zinc atom. Spatial correlations have been incorporated with an ensemble technique and the fermionic statistics restored to this semiclassical approach via the choice of classical field. This latter we interpret as a coherent state envelope which determines the effective potential for the calculation of quantum energies. We have introduced a parameter, μ, which is a measure of the Coulomb repulsion. Only when μ < μ 0, a critical value, do we obtain physically acceptable normalisable bound charge states for certain discrete values of μ. We have calculated shell energies and energy orderings within each shell and obtained good agreement with those calculated using a standard Hartree-Fock method. The coefficients required to do this agree well with trends in the Slater parameter F 0 (2s, 2p) as a function of Z as well as Moseley's law. Agreement is also obtained with the first principles form of the charge density as given by Kato's theorem and at larger distances from the nucleus.