• 대한수학회지
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• 제38권2호
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• pp.337-348
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• 2001

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.

• 대한수학회논문집
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• 제20권3호
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• pp.521-529
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• 2005

We prove that the dimension of the space of positive (bounded, respectively) solutions for the Schrodinger operator whose potential q is nonnegative on a graph with q-regular ends is equal to the number of ends (q-nonparabolic ends, respectively).

• East Asian mathematical journal
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• 제35권1호
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• pp.1-7
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• 2019

We consider the initial value problem of the Schrodinger equation for an interesting Cauchy-Euler type operator ${\mathfrak{R}}$ on ${\mathbb{C}}^n$ that is an analogue of the harmonic oscillator in ${\mathbb{R}}^n$. We get an appropriate $L^1-L^{\infty}$ dispersive estimate for the solution of the initial value problem.

• 대한수학회보
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• 제24권1호
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• pp.23-26
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• 1987

The aim of this note is to study some properties of Schrodinger operators, the magnetic case, $H_{0}$ (a)=1/2(-i.del.-a)$^{2}$; H(a)= $H_{0}$ (a)+V, where a=( $a_{1}$,.., $a_{n}$ ).mem. $L^{2}$$_{loc}$ and V is a potential energy. Also, we are interested in solutions, .psi., of H(a).psi.=E.psi. in the sense that (.psi., $e^{-tH}$(a).PSI.)= $e^{-tE}$(.psi.,.PSI.) for all .PSI..mem. $C_{0}$ $^{\infty}$( $R^{n}$ ) (see B. Simon [1]). In section 2, under some conditions, we find that a semibounded quadratic form of H9a) exists and that the Schrodinger operator H(a) with Re V.geq.0 is accretive on a form domain Q( $H_{0}$ (a)). But, it is well-known that the Schrodinger operator H=1/2.DELTA.+V with Re V.geq.0 is accretive on $C_{0}$ $^{\infty}$( $R^{n}$ ) in N Okazawa [4]. In section 3, we want to discuss $L^{p}$ estimates of Schrodinger semigroups.ups.

• 대한수학회지
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• 제38권2호
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• pp.469-483
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• 2001

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$\psi$(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.

• 대한수학회보
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• 제34권1호
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• pp.103-114
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• 1997

The asymptotic Dirichlet problem for harmonic functions on a noncompact complete Riemannian manifold has a long history. It is to find the harmonic function satisfying the given Dirichlet boundary condition at infinity. By now, it is well understood [A, AS, Ch, S], when M is a Cartan-Hadamard manifold with sectional curvature $-b^2 \leq K_M \leq -a^2 < 0$. (By a Cartan-Hadamard manifold, we mean a complete simply connected manifold of non-positive sectional curvature.)

• East Asian mathematical journal
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• 제40권5호
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• pp.515-526
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• 2024

We consider the initial value problem of the Schrödinger equation for an Euler operator 𝓡 on ℂⁿ that is an analogue of the harmonic oscillator in ℝⁿ. We get some regularity results of the Schrödinger equation on Fock spaces.

• 대한수학회보
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• 제37권2호
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• pp.353-360
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• 2000

The fundamental gap between the lowest two Dirich-let eigenvalues for a Schr dinger operator HR={{{{ { { d}^{2 } } over { { dx}^{2 } } }}}}+V(x) on L({{{{ LEFT | -R,R RIGHT | }}}}) is compared with the gap for a same operator Hs with a different domain {{{{ LEFT [ -S,S RIGHT ] }}}} and the difference is exponentially small when the potential has a large barrier.

• 전자공학회논문지A
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• 제33A권7호
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• pp.223-229
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• 1996

a simpel coordinate transformaton method is suggested that converts Schr$\"{o}$dinger's equation involving a position-dependent effective mass in a heterostructure to an equation involving a positon-independent effective mass. This method enables the conceptual study of the effect of the positon-dependent effective mass inserted between the divergence operator and the gradient operator in Schr$\"{o}$dinger's equation. It is also shown that the characteristics such as a transmission coefficient in various heterostructures involving a position-dependent effective masses can be obtained iwth ease by the suggested method.

• 대한수학회논문집
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• 제25권4호
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• pp.609-614
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• 2010

We prove that if graphs of bounded degree are roughly isometric to each other, then the spaces of bounded energy finite solutions of the Schr$\ddot{o}$dinger operator on the graphs are isomorphic to each other. This is a direct generalization of the results of Soardi [5] and of Lee [3].