Hajime Ishihara

For a system of noninteracting Frenkel excitons in a one-dimensional lattice of size N with periodic boundary conditions, the third-order optical susceptibility ${\mathrm{\ensuremath{\chi}}}^{(3)}$ has been calculated rigorously in a nonlocal form with arbitrary dependence on external-field frequencies. Among the various terms in ${\mathrm{\ensuremath{\chi}}}^{(3)}$ (per unit volume), those explicitly proportional to N in the long-wavelength approximation have been shown to cancel out completely

We will overview the results in an informal approach to constructive reverse mathematics, that is reverse mathematics in Bishop’s constructive mathematics, especially focusing on compactness properties and continuous properties.

We study the size dependence of the nonlinear response of weakly confined excitons for the size region beyond the long wavelength approximation regime. The observed degenerate-four-wave mixing signal of GaAs thin layers exhibits an anomalous size dependence, where the signal is resonantly enhanced at a particular thickness region. The theoretical analysis elucidates that this enhancement is due to the size-resonant enhancement of the internal field with a spatial structure relevant to the nondip

Abstract This chapter proposes a base formal system for constructive reverse mathematics to classify various theorems in intuitionistic, constructive recursive, and classical mathematics by logical principles, function existence axioms, and their combinations. The system is weak enough to allow for a comparison of the results obtained in it with those obtained within classical reverse mathematics as well as to prove theorems in Bishop's constructive mathematics. The chapter also formalizes resul

Abstract The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We deal with principles which are equivalent to the statements “every mapping is sequentially nondiscontinuous”, “every sequentially nondiscontinuous mapping is sequentially continuous”, and “every sequentially continuous mapping is continuous”. As corollaries, we show that every mapping of a complete separable space is continuous in constructive recursive mathematics (the Kreise

Peculiar properties of nonlinear response, due to a resonant enhancement of internal field are predicted for the mesoscopic systems by means of a nonlocal theory of nonlinear response. In this theory, self-consistent motions of the internal field and the induced polarization, which are related nonlocally with each other, are determined by solving the equation system of Schr\"odinger and Maxwell's equations. A study with a model of ultrathin films consisting of one-dimensional Frenkel excitons ha

Abstract The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We show that every mapping is sequentially continuous if and only if it is sequentially nondiscontinuous and strongly extensional, and that “every mapping is strongly extensional”, “every sequentially nondiscontinuous mapping is sequentially continuous”, and a weak version of Markov's principle are equivalent. Also, assuming a consequence of Church's thesis, we prove a version of

A nonlocal formalism of the nonlinear optical-response field has been developed, in which the additional-boundary-condition theory for linear response has been extended. In this theory, Maxwell's equations in terms of site-represented susceptibility up to the third order are solved. Calculations using this theory have been performed for a one-dimensional Frenkel exciton model with hard-wall boundary conditions. As a result, it has been made clear that the nonlocal effect appears in the spectra e

Classically, weak Knig's lemma and Brouwer's fan theorem for detachable bars are equivalent. We give a direct constructive proof that the former implies the latter.

In this paper, we prove that, if we confine ourselves to some special classes of normed linear spaces, then the extension of linear functionals, with exact preservation of norm, is constructively provable.

The electric field associated with an exciton polariton in a slab of spatially dispersive medium has been calculated ``from first principles'' according to the general framework by Cho in which the argument of additional boundary condition (ABC) is completely avoided. As the exciton wave function, we have considered, in addition to the bulk component, the distortion of the type exp(-PZ) near the surfaces according to the analytic model of D'Andrea and Del Sole, and the slab thickness d has been

For the discussion of the size dependence of the third-order nonlinear susceptibility χ (3) for confined electronic systems, the result of numerical study is given for an exactly soluble model of noninteracting Frenkel excitons in a periodic chain. The case of pump-probe type nonlinear effect is explicitly treated in the long-wavelength approximation, and Im [χ (3) ] is evaluated as a function of chain size N, transfer energy b, and damping constants. For a given value of b(N), there is an enhan