Let X be a spin manifold and Y a riemannian manifold. Let ψ(x,y) be a spinor over X depending on y∈Y. I define the Dirac-Schrödinger equation:
Dxψ=Δyψ+V(ψ)
where Dx is the Dirac operator over X, Δy is the Laplacian operator over Y, and V is a potential.
Can we find solutions of the Dirac-Schrödinger equation?