• 대한수학회지
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• 제41권2호
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• pp.319-344
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• 2004

In this paper, we define conditional integrals on abstract Wiener and Hilbert spaces and then obtain a formula for evaluating the integrals. We use this formula to establish the existence of conditional Feynman integrals for the classes $A^{q}$(B) and $A^{q}$(H) of functions on abstract Wiener and Hilbert spaces and then specialize this result to provide the fundamental solution to the Schrodinger equation with the forced harmonic oscillator.tor.

• Journal of applied mathematics & informatics
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• 제28권3_4호
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• pp.931-938
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• 2010

The disentangling map from the commutative algebra to the noncommutative algebra of operators is the essential operation of Feynman's operational calculus for noncommuting operators. Thus formulas which simplify this operation are meaningful to the subject. In a recent paper the procedure for "methods for iterative disentangling" has been established in the setting of Feynman's operational calculus for time independent operators $A_1$, $\cdots$, $A_n$ and associated probability measures${\mu}_1$, $\cdots$, ${\mu}_n$. The main purpose for this paper is to extend the procedure for methods for iterative disentangling to time dependent operators.

• 충청수학회지
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• 제8권1호
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• pp.119-130
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• 1995

I study the extension of the definition of the analytic operator-valued Feynman integral to potentials given by a class of signed measures described in terms of additive functionals associated with regular Dirichlet forms.

• 대한수학회지
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• 제28권1호
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• pp.99-125
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• 1991

• East Asian mathematical journal
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• 제14권2호
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• pp.237-247
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• 1998

In this paper, we prove a translation theorem for the analytic Feynman integral for functions in $\mathcal{F}_{A_1,A_2}$(B) and show how this integral can be modified so as to be translation invariant.

• 충청수학회지
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• 제9권1호
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• pp.129-136
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• 1996

We will investigate conditions of smooth measure ${\mu}$ for which analytic operator-valued Feynman integral associated with ${\mu}$ exist.

• 대한수학회보
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• 제33권3호
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• pp.465-475
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• 1996

Fix t > 0. Denote by $C^t$ the space of $R$-valued continuous functions x on [0,t]. Let $C_0^t$ be the Wiener space - $C_0^t = {x \in C^t : x(0) = 0}$ - equipped with Wiener measure m. Let F be a function from $C^t to C$.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제14권4호
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• pp.317-333
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• 2007

In 2002, the author and professor Ryu introduced the concept of analogue of Wiener measure. In this paper, we prove the existence theorem of Fourier-Feynman transform on analogue of Wiener measure in $L_2-norm$ sense.

• 대한수학회보
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• 제48권3호
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• pp.655-672
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• 2011

Let $C^r$[0,t] be the function space of the vector-valued continuous paths x : [0,t] ${\rightarrow}$ $R^r$ and define $X_t$ : $C^r$[0,t] ${\rightarrow}$ $R^{(n+1)r}$ and $Y_t$ : $C^r$[0,t] ${\rightarrow}$ $R^{nr}$ by $X_t(x)$ = (x($t_0$), x($t_1$), ..., x($t_{n-1}$), x($t_n$)) and $Y_t$(x) = (x($t_0$), x($t_1$), ..., x($t_{n-1}$)), respectively, where 0 = $t_0$ < $t_1$ < ... < $t_n$ = t. In the present paper, with the conditioning functions $X_t$ and $Y_t$, we introduce two simple formulas for the conditional expectations over $C^r$[0,t], an analogue of the r-dimensional Wiener space. We establish evaluation formulas for the analogues of the analytic Wiener and Feynman integrals for the function $G(x)=\exp{{\int}_0^t{\theta}(s,x(s))d{\eta}(s)}{\psi}(x(t))$, where ${\theta}(s,{\cdot})$ and are the Fourier-Stieltjes transforms of the complex Borel measures on ${\mathbb{R}}^r$. Using the simple formulas, we evaluate the analogues of the conditional analytic Wiener and Feynman integrals of the functional G.

• 대한화학회지
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• 제17권6호
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• pp.395-405
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• 1973

Integral Hellmann-Feynman formula를 사용하여 $KNiF_3$의 cubic crystal splitting 10Dq를 first principle로부터 계산하였다. Covalency parameter들과 필요한 적분치들은 Sugano와 Shulman의 계산치를 사용하였다. 계산치 7100$cm^{-1}$는 실험치 7250$cm^{-1}$와 대단히 잘 일치하였다. 고차섭동에너지 보정치는 10 Dq 자체와 같은 order of magnitude를 가지며 따라서 first principle로부터 10 Dq를 계산하는데 있어서 반드시 고려되어야 할 몫이라는 것을 발견하였다. 또한 point charge potential이 crystal field potential의 압도적인 부분을 차지하는 것을 발견하였다.