The Schrödinger equation on cylinders and the n-torus
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In this paper, we study the solutions to the Schrödinger equation on some conformally flat cylinders and on the n-torus. First, we apply an appropriate regularization procedure. Using the Clifford algebra calculus with an appropriate Witt basis, the solutions can be expressed as multiperiodic eigensolutions to the regularized parabolic-type Dirac operator. We study their fundamental properties, give representation formulas of all these solutions in terms of multiperiodic generalizations of the elliptic functions in the context of the regularized parabolic-type Dirac operator. Furthermore, we also develop some integral representation formulas. In particular, we set up a Green type integral formula for the solutions to the homogeneous regularized Schrödinger equation on cylinders and n-tori. Then, we treat the inhomogeneous Schrödinger equation with prescribed boundary conditions in Lipschitz domains on these manifolds. We present an L p -decomposition where one of the components is the kernel of the first-order differential operator that factorizes the cylindrical (resp. toroidal) Schrödinger operator. Finally, we study the behavior of our results in the limit case where the regularization parameter tends to zero.
Mathematics Subject Classification (2000)Primary 30G35 Secondary 35J10 35C15
KeywordsSchrödinger equation on manifolds Regularized parabolic-type Dirac operators Hypoelliptic equations Regularization Hodge decomposition
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- 1.Artino, R., Barros-Neto, J.: Hypoelliptic Boundary-Value Problems, Lectures Notes in Pure and Applied Mathematics 53, Marcel Dekker: New York; (1980).Google Scholar
- 5.Delanghe, R., Sommen, F., Souçek, V., Clifford algebras and spinor-valued functions, Kluwer Academic Publishers, 1992.Google Scholar
- 6.Dix, D.: D., Application of Clifford analysis to inverse scattering for the linear hierarchy in several space dimensions, In: Ryan, J. (ed.), pp. 260–282, CRC Press, Boca Raton, FL (1995).Google Scholar
- 7.Gürlebeck, K., Sprößig, W.: Quaternionic and Clifford calculus for physicists and engineers, John Wiley and Sons, (1997).Google Scholar
- 10.Kraußhar, R.S.: Generalized Analytic Automorphic Forms in Hypercomplex Spaces, Frontiers in Mathematics, Birkhäuser, Basel, (2004).Google Scholar
- 12.Shapiro, M., Kravchenko, V.V.: Integral Representation for spatial models of mathematical physics, Pitman research notes in mathematics series 351, Harlow, Longman, (1996).Google Scholar
- 13.Tao, T.: Nonlinear Dispersive Equations, Local and Global Analysis, CBMS Regional Conference Series in Mathematics 106, American Mathematical Society: Providence, RI, (2006).Google Scholar
- 14.Velo, V.: Mathematical Aspects of the nonlinear Schrödinger Equation, Proceedings of the Euroconference on nonlinear Klein-Gordon and Schrödinger systems: theory and applications, Vázquez, Luis et al.(ed.), pp. 39–67 World Scientific, Singapore (1996).Google Scholar
- 15.Vieira, N.: Theory of the parabolic Dirac operator and its applications to non-linear differential equations, PhD Thesis, Univerity of Aveiro, (2009).Google Scholar