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REFERENCE LINKING PLATFORM OF KOREA S&T JOURNALS
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Journal of the Korean Mathematical Society
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Journal DOI :
The Korean Mathematical Society
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Volume & Issues
Volume 38, Issue 6 - Nov 2001
Volume 38, Issue 5 - Sep 2001
Volume 38, Issue 4 - Jul 2001
Volume 38, Issue 3 - May 2001
Volume 38, Issue 2 - Mar 2001
Volume 38, Issue 1 - Jan 2001
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FEYNMAN′S OPERATIONAL CALCULI FOR TIME DEPENDENT NONCOMMUTING OPERATORS
Brian Jefferies ;
Journal of the Korean Mathematical Society, volume 38, issue 2, 2001, Pages 193~226
We study Feynman's Operational Calculus for operator-valued functions of time and for measures which are not necessarily probability measures; we also permit the presence of certain unbounded operators. further, we relate the disentangling map defined within the solutions of evolution equations and, finally, remark on the application of stability results to the present paper.
FEYNMAN-KAC SEMIGROUPS, MARTINGALES AND WAVE OPERATORS
Van Casteren, Jan A. ;
Journal of the Korean Mathematical Society, volume 38, issue 2, 2001, Pages 227~274
In this paper we intended to discuss the following topics: (1) Notation, generalities, Markov processes. The close relationship between (generators of) Markov processes and the martingale problem is exhibited. A link between the Korovkin property and generators of Feller semigroups is established. (2) Feynman-Kac semigroups: 0-order regular perturbations, pinned Markov measures. A basic representation via distributions of Markov processes is depicted. (3) Dirichlet semigroups: 0-order singular perturbations, harmonic functions, multiplicative functionals. Here a representation theorem of solutions to the heat equation is depicted in terms of the distributions of the underlying Markov process and a suitable stopping time. (4) Sets of finite capacity, wave operators, and related results. In this section a number of results are presented concerning the completeness of scattering systems (and its spectral consequences). (5) Some (abstract) problems related to Neumann semigroups: 1st order perturbations. In this section some rather abstract problems are presented, which lie on the borderline between first order perturbations together with their boundary limits (Neumann type boundary conditions and) and reflected Markov processes.
WHITE NOISE APPROACH TO FEYNMAN INTEGRALS
Hida, Takeyuki ;
Journal of the Korean Mathematical Society, volume 38, issue 2, 2001, Pages 275~281
The trajectory of a classical dynamics is determined by the least action principle. As soon as we come to quantum dynamics, we have to consider all possible trajectories which are proposed to be a sum of the classical trajectory and Brownian fluctuation. Thus, the action involves the square of the derivative B(t) (white noise) of a Brownian motion B(t). The square is a typical example of a generalized white noise functional. The Feynman propagator should therefore be an average of a certain generalized white noise functional. This idea can be applied to a large class of dynamics with various kinds of Lagrangians.
THE HEISENBERG INEQUALITY ON ABSTRACT WIENER SPACES
Lee, Yuh-Jia ;
Journal of the Korean Mathematical Society, volume 38, issue 2, 2001, Pages 283~296
The Heisenberg inequality associated with the uncertainty principle is extended to an infinite dimensional abstract Wiener space (H, B) with an abstract Wiener measure p
) and T
L(B, H), it is shown that (※Equations, See Full-text), where F(sub)
is the Fourier-Wiener transform of
. The conditions when the equality holds also discussed.
COHERENT SATE REPRESENTATION AND UNITARITY CONDITION IN WHITE NOISE CALCULUS
Obata, Nobuaki ;
Journal of the Korean Mathematical Society, volume 38, issue 2, 2001, Pages 297~309
White noise distribution theory over the complex Gaussian space is established on the basis of the recently developed white noise operator theory. Unitarity condition for a white noise operator is discussed by means of the operator symbol and complex Gaussian integration. Concerning the overcompleteness of the exponential vectors, a coherent sate representation of a white noise function is uniquely specified from the diagonal coherent state representation of the associated multiplication operator.
CONSTRUCTION OF SOME PROCESSES ON THE WIENER SPACE ASSOCIATED TO SECOND ORDER OPERATORS
Cruzeiro, A.B. ;
Journal of the Korean Mathematical Society, volume 38, issue 2, 2001, Pages 311~319
We show that it is possible to associate diffusion processes to second order perturbations of the Ornstein-Uhlenbeck operator L on the Wiener space of the form L = L + 1/2∑L
where the ξ(sub)
are "tangent processes" (i.e., semimartingales with antisymmetric diffusion coefficients).
A FEYNMAN FUNCTIONAL FOR THE GLOBAL POSITIONING SYSTEM
Facio, Brian-De ;
Journal of the Korean Mathematical Society, volume 38, issue 2, 2001, Pages 321~336
A Feynman functional formulation is given for the Global Positioning System, GPS. Both the sequential and analytic Feynman functionals are presented for the classical, exterior, gravity problems which included rigid body rotations, special relativity and some general relativity corrections. A mathematically rigorous approach is introduced whose solutions exist, are unique and which depend continuously on the intial data. This formulation is convergent and has the finite approximation property. It is emphasized that all of the problems studied are classical (not quantum) evolution systems.
NORM CONVERGENCE OF THE LIE-TROTTER-KATO PRODUCT FORMULA AND IMAGINARY-TIME PATH INTEGRAL
Ichinose, Takashi ;
Journal of the Korean Mathematical Society, volume 38, issue 2, 2001, Pages 337~348
The unitary Lie-Trotter-Kato product formula gives in a simplest way a meaning to the Feynman path integral for the Schroding-er equation. In this note we want to survey some of recent results on the norm convergence of the selfadjoint Lie-Trotter Kato product formula for the Schrodinger operator -1/2Δ + V(x) and for the sum of two selfadjoint operators A and B. As one of the applications, a remark is mentioned about an approximation therewith to the fundamental solution for the imaginary-time Schrodinger equation.
INTEGRATION STRUCTURES FOR THE OPERATOR-VALUED FEYNMAN INTEGRAL
Jefferies, Brian ;
Journal of the Korean Mathematical Society, volume 38, issue 2, 2001, Pages 349~363
The analytic in mass operator-valued Feynman integral is related to integration with respect to unbounded set functions formed from the semigroup obtained by analytic continuation of the heat semigroup and the spectral measure of multiplication by characteristics functions.
PATH INTEGRALS ASSOCIATED WITH STRUM-LIOUVILLE OPERATORS
Thomas, Erik G.F. ;
Journal of the Korean Mathematical Society, volume 38, issue 2, 2001, Pages 365~383
We consider path integrals, more precisely projective systems of Fresnel distributions, associated with Sturm-Liouville boundary value problems.
FEYNMAN INTEGRALS, DIFFUSION PROCESSES AND QUANTUM SYMPLECTIC TWO-FORMS
Zambrini, Jean-Claude ;
Journal of the Korean Mathematical Society, volume 38, issue 2, 2001, Pages 385~408
This is an introduction to a stochastic version of E. Cartan′s symplectic mechanics. A class of time-symmetric("Bernstein") diffusion processes is used to deform stochastically the exterior derivative of the Poincare-Cartan one-form on the extended phase space. The resulting symplectic tow-form is shown to contain the (a.e.) dynamical laws of the diffusions. This can be regarded as a geometrization of Feynman′s path integral approach to quantum theory; when Planck′s constant reduce to zero, we recover Cartan′s mechanics. The underlying strategy is the one of "Euclidean Quantum Mechanics".
A FUBINI THEOREM FOR ANALYTIC FEYNMAN INTEGRALS WITH APPLICATIONS
Huffman, Timothy ; Skoug, David ; Storvick, David ;
Journal of the Korean Mathematical Society, volume 38, issue 2, 2001, Pages 409~420
In this paper we establish a Fubini theorem for various analytic Wiener and Feynman integrals. We then proceed to obtain several integration formulas as corollaries.
INTEGRATION FORMULAS INVOLVING FOURIER-FEYNMAN TRANSFORMS VIA A FUBINI THEOREM
Huffman, Timothy ; Skoug, David ; Storvick, David ;
Journal of the Korean Mathematical Society, volume 38, issue 2, 2001, Pages 421~435
In this paper we use a general Fubini theorem established in  to obtain several Feynman integration formulas involving analytic Fourier-Feynman transforms. Included in these formulas is a general Parseval's relation.
FUNCTIONAL INTEGRATION, KONTSEVICH INTEGRAL AND FORMAL INTEGRATION
Kauffman, Louis H. ;
Journal of the Korean Mathematical Society, volume 38, issue 2, 2001, Pages 437~468
This paper is an exposition of the relationship between Witten's functional integral and Vassiliev invariants.
STOCHASTIC MEHLER KERNELS VIA OSCILLATORY PATH INTEGRALS
Truman, Aubrey ; Zastawniak, Tomasz ;
Journal of the Korean Mathematical Society, volume 38, issue 2, 2001, Pages 469~483
The configuration space and phase space oscillatory path integrals are computed in the case of the stochastic Schrodinger equation for the harmonic oscillator with a stochastic term of the form (K
(sub)t)(x) o dW(sub)t, where K is either the position operator or the momentum operator, and W(sub)t is the Wiener process. In this way formulae are derived for the stochastic analogues of the Mehler kernel.
ANALYTIC FOURIER-FEYNMAN TRANSFORM AND FIRST VARIATION ON ABSTRACT WIENER SPACE
Chang, Kun-Soo ; Song, Teuk-Seob ; Yoo, Il ;
Journal of the Korean Mathematical Society, volume 38, issue 2, 2001, Pages 485~501
In this paper we express analytic Feynman integral of the first variation of a functional F in terms of analytic Feynman integral of the product F with a linear factor and obtain an integration by parts formula of the analytic Feynman integral of functionals on abstract Wiener space. We find the Fourier-Feynman transform for the product of functionals in the Fresnel class F(B) with n linear factors.