Following the classical definition of factorization of matrix-functions, we introduce a definition of factorization for functional operators with involutive rotation on the unit circle. Partial indices are defined and...Following the classical definition of factorization of matrix-functions, we introduce a definition of factorization for functional operators with involutive rotation on the unit circle. Partial indices are defined and their uniqueness is proven. In previous works, the main research method for the study scalar singular integral operators and Riemann boundary value problems with Carlemann shift were operator identities, which allowed to eliminate shift and to reduce scalar problems to matrix problems without shift. In this study, the operator identities were used for the opposite purpose: to transform operators of multiplication by matrix-functions into scalar operators with Carlemann linear-fractional shift.展开更多
The observables of continuous eigenvalues are de?ned in an in?nite-dimensional ket space. The complete set of such eigenstates demands a spectrum density factor, for example, for the photons in the free space and elec...The observables of continuous eigenvalues are de?ned in an in?nite-dimensional ket space. The complete set of such eigenstates demands a spectrum density factor, for example, for the photons in the free space and electrons in the vacuum. From the derivation of the Casimir force without an arti?cial regulator we determine the explicit expression of the spectrum density factor for the photon ?eld to be an exponential function. The undetermined constant in the function is ?xed by the experimental data for the Lamb shift. With that, we predict that there exists a correction to the Casimir force.展开更多
文摘Following the classical definition of factorization of matrix-functions, we introduce a definition of factorization for functional operators with involutive rotation on the unit circle. Partial indices are defined and their uniqueness is proven. In previous works, the main research method for the study scalar singular integral operators and Riemann boundary value problems with Carlemann shift were operator identities, which allowed to eliminate shift and to reduce scalar problems to matrix problems without shift. In this study, the operator identities were used for the opposite purpose: to transform operators of multiplication by matrix-functions into scalar operators with Carlemann linear-fractional shift.
基金Supported by the National Natural Science Foundation of China under Grant No.11774316the Natural Science Foundation of Zhejiang Province under Grant No.Z15A050001
文摘The observables of continuous eigenvalues are de?ned in an in?nite-dimensional ket space. The complete set of such eigenstates demands a spectrum density factor, for example, for the photons in the free space and electrons in the vacuum. From the derivation of the Casimir force without an arti?cial regulator we determine the explicit expression of the spectrum density factor for the photon ?eld to be an exponential function. The undetermined constant in the function is ?xed by the experimental data for the Lamb shift. With that, we predict that there exists a correction to the Casimir force.
