Quaternionic factorization of the Schrödinger operator and its applications to some first-order systems of mathematical physics

Kravchenko, Viktor G.; Kravchenko, Vladislav V.

doi:10.1088/0305-4470/36/44/008

Cited by 22 publications

(32 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let a and b be defined by (6) and h be a scalar continuously differentiable function in˜ . For an arbitrary scalar function ∈ C [2,1] (˜ ) the equality holds…”

Section: Corollarymentioning

confidence: 99%

See 2 more Smart Citations

On a factorization of the Schrödinger and Klein–Gordon operators

Cerejeiras

Kähler

Kravchenko

2008

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…Let a and b be defined by (6) and h be a scalar continuously differentiable function in˜ . For an arbitrary scalar function ∈ C [2,1] (˜ ) the equality holds…”

Section: Corollarymentioning

confidence: 99%

“…For a = 0, = 0, = 0 Theorem 1 was proved in [5] and was intensively used in [6,7,12,18,[23][24][25] and others.…”

Section: Remarkmentioning

confidence: 99%

See 1 more Smart Citation

On a factorization of the Schrödinger and Klein–Gordon operators

Cerejeiras

Kähler

Kravchenko

2008

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…These include for instance the stationary Lamé-, Navier-Stokes-, Maxwell-and Schrödinger equation and many others. See also [4,6,7,11] or [12]. Indeed, this kind of L 2 -space decomposition (when applicable) represents one of the most central aspects of complex and hypercomplex analysis.…”

Section: Introductionmentioning

confidence: 99%

The Schrödinger equation on cylinders and the n-torus

Kraußhar

Vieira

2010

J. Evol. Equ.

View full text Add to dashboard Cite

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.

show abstract

“…However, since it is a theory for elliptic partial differential equations it works for the stationary case only. In this paper we use two extra basis elements (which forms a Witt basis) in order to construct a parabolic Dirac operator which factorizes the Schroedinger operator and which contains only partial derivatives (thus, avoiding the use of fractional derivatives) (for the stationary case, see also [6]). …”

Section: Introductionmentioning

confidence: 99%