Submanifold Differential Operators in D-Module Theory I: Schrödinger Operators

For this quarter of century, quantum differential operators in a lower dimensional submanifold embedded or immersed in real n-dimensional euclidean space E n have been studied as physical models, which are realized as restriction of the operators in E n to the submanifold. For this decade, the Dirac operators in the submanifold have been investigated, 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 sequel articles, we will give mathematical construction of the differential operators on a submanifold in E n in terms of D-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 E n. Recently it becomes recognized that the Dirac operators play important roles in geometry e.g., differential, algebraic, arithmetic geometry and so on. Pinkall gave an invited talk in the international congress of mathematicians in 1998 on the relation between immersed surfaces in three and/or four dimensional euclidean space E n , (n = 3, 4) and Dirac operators, which was worked with Pedit [PP]. They constructed quaternion differential geometry and reduced the Dirac operators, which exhibit the geometrical properties of the surface. The Dirac operators of E n (n = 3, 4) also had been discovered by Konopelchenko in studies on geometrical interpretation of soliton theory [KO1, KO2, KT] and by me in the framework of the quantum physics [M3, M4]. Further on case of E 3 , Friedrich obtained it by investigation of spin bundle [FR]. My Dirac operator is purely constructed in analytic category as we will show and is directly related to index theorems [M2, M3, TM]. Thus I believe that it is important to reformulate my works in the framework of pure mathematics and to translate them for mathematicians. In this and squeal article [II], I will mathematically formulate the canonical Schrödinger operator and Dirac operator on a submanifold in E n. Indeed, there have appeared similar studies [DES, FH, RB] only on the Schrödinger operator case but it does not look enough to overcome several obstacle between physics and mathematics. The submanifold quantum mechanics, which I called, is opened by Jensen and Koppe in 1971 [JK] and rediscovered by da Costa …