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.
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Kraußhar, R.S., Vieira, N. The Schrödinger equation on cylinders and the n-torus. J. Evol. Equ. 11, 215–237 (2011). https://doi.org/10.1007/s00028-010-0089-4
Mathematics Subject Classification (2000)
- Primary 30G35
- Secondary 35J10
- Schrödinger equation on manifolds
- Regularized parabolic-type Dirac operators
- Hypoelliptic equations
- Hodge decomposition