Zeitschrift f¨ur Analysis und ihre Anwendungen

Proposition 1. For any two particular solutions f0 and f1 of (2), where f0 is assumed to be nonvanishing, the function w = f0∂z(f −1 0 f1) is a solution of the equation wz = −∂zf0 f0 w (3) which is equivalent to the system ∂xu −∂yv = −∂xf0 f0 u + ∂yf0 f0 v, ∂yu + ∂xv = ∂yf0

Proposition 2. Let f0 be a nonvanishing particular solution of (2) and w be a solution of (3). Then the function f = Sw is a solution of (2).

Proposition 3. Let f be a solution of (2). Then SPf = f + Cf0, where C is an arbitrary real constant.

Proposition 4. [11] Let W be a solution of (9). Then the function w = i ˙W is a solution of (3). The (F, G)-integral is defined as follows [2]: Z Γ Wd(F,G)z = F(z1) Re Z Γ G∗Wdz + G(z1) Re Z Γ F ∗Wdz, where Γ is a rectifiable curve leading from z0 to z1, F ∗= − 2F

Proposition 5. Let w be a solution of (3). Then the function W(z) = − Z z z0 iwd(F,G)z = − 1 f0(z) Re Z z z0 if0wdz −if0(z) Re Z z z0 w f0 dz

Theorem 1. Let f0 be a nonvanishing particular solution of (2)and Φ = u+iv be an 1 f2 0 -analytic function. Then W = u f0 +ivf0 is a solution of (9), w = i · W = i(Wz + ∂zf0 f0 W) is a solution of (3) and f = Sw is a solution of (2). We have an inverse result also.

Theorem 2. Let f0 be a nonvanishing particular solution of (2) and f be another solution of (2). Then w = Pf is a solution of (3), the function W(z) = − R z z0 iwd(F,G)z is a solution of (9), and Φ = f0 Re W + i f0 Im W is a 1 f2 0 -analytic function. Thus due to Theorem 1 we are able to convert p-analytic functions into solutions of the Schr¨odinger equation and due to Theorem 2 we have an inverse result.