关键词: 修正随机梯度SymbolQC@, 修正点态随机梯度SymbolQC@_s, 修正点态随机梯度的共轭SymbolQC@^*_s, Skorohod积分δ, 计数算子N

Abstract: The paper considers the representation of the number operator N in continuous-time Guichardet-Fock space L2(Γ;η). Firstly, the gradient-Skorohod integral representation of N is given by using modified stochastic gradient SymbolQC@ and non-adaptive Skorohod integral δ:N=δSymbolQC@. Secondly, the representation of Bochner integral is given: N=∫_R+SymbolQC@^*sSymbolQC@sds in the sense of the inner product, by means of the family of isometric operator {SymbolQC@^*sSymbolQC@_s; s∈R₊}. Meanwhile, the spectrum of N is just the nonnegative integral N, and for any n≥0, the closed subspace L2(Γ⁽ⁿ⁾;η) of Guichardet-Fock space L2(Γ;η) is just the eigenspace corresponding to the eigenvalue n, and N has the spectrum representation: N=∑^∞_n=1nJ_n, where J_n:L2(Γ;η)→L2(Γ⁽ⁿ⁾;η), is the orthogonal projection.

Key words: modified stochastic gradient SymbolQC@, point-state modified stochastic gradient SymbolQC@s, adjoint of the point state modified stochastic gradient SymbolQC@^*s, Skorohod integral δ, number operator N