SCHRODINGER EQUATIONS WITH TIME-DEPENDENT STRONG MAGNETIC FIELDS

Time dependent d-dimensional Schrodinger equations i partial derivative(t)u = H(t)u, H(t) = -(partial derivative(x) - iA(t, x))(2) + V(t, x) are considered in the Hilbert space G = L-2(R-d) of square integrable functions. V (t, x) and A(t, x) are assumed to be almost critically singular with respect to the spatial variables x is an element of R-d both locally and at infinity for the operator H(t) to be essentially selfadjoint on C-0(infinity) (R-d). In particular, when the magnetic fields B(t, x) produced by A(t, x) are very strong at infinity, V (t, x) can explode to the negative infinity like -theta vertical bar B(t, x)vertical bar - C(vertical bar x vertical bar(2) + 1) for some theta < 1 and C > 0. It is shown that such equations uniquely generate unitary propagators in G under suitable conditions on the size and singularities of the time derivatives of the potentials (V) over dot(t, x) and (A) over dot(t, x).