Submanifold Differential Operators in $\Cal D$-Module Theory I : Schrödinger Operators

Shigeki Matsutani

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Submanifold Differential Operators in $\Cal D$-Module Theory I : Schrödinger Operators

Shigeki Matsutani

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Abstract

For this quarter of century, differential operators in a lower dimensional submanifold embedded or immersed in real $n$-dimensional euclidean space $\EE^n$ have been studied as quantum mechanical models, which are realized as restriction of the operators in $\EE^n$ to the submanifold. For this decade, the Dirac operators in the submanifold have been investigated in such a scheme , which are identified with operators of the Frenet-Serret relation for a space curve case and of the generalized Weierstrass relation for a conformal surface case. These Dirac operators are concerned well in the differential geometry, since they completely represent the submanifolds. In this and a future series of articles, we will give mathematical construction of the differential operators on a submanifold in $\EE^n$ in terms of $\DMod$-module theory and rewrite recent results of the Dirac operators mathematically. In this article, we will formulate Schrödinger operators in a low-dimensional submanifold in $\EE^n$.

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BibTeX RIS

Shigeki Matsutani. 2000-01-09. Submanifold Differential Operators in $\Cal D$-Module Theory I : Schrödinger Operators. https://arxiv.org/abs/math/9910051

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