Perfectly Matched Layer Technique For Quantum Scattering Problem

Consider the Schrodinger equation with electromagnetic potentials inR²where i = y(-1)^1/2,A(x) = (A₁(x), A₂(x)) is the magnetic potential, V(x) is the electric potential, k∈R¹ is the wave number, |u(x)|² is the probability density of finding the particle at the position x.We assume that A₁(x),A₂(x) and V(x) are complex valued and satisfyandwhereα> 3/2, C_Aj,j = 1,2, C_b are all positive constants.We denote that uⁱ is the incident field, u^? is the scattered field, u = uⁱ + u^? is total field which satisfies Schrodinger equation (1), andω∈S¹ is the direction of propagation. We assume that the incident field is given by the time-harmonic probability waveand that u^? satisfies the radiation conditionWe outline that the direct scattering problem is to determine u from the knowledge of uⁱ,(A(x), V(x)), Schrodinger equation (1) and radiation condition (4).We prove the following main theorem:Theorem .1. Assume that A(x) = (A₁(x),A₂(x)) and V(x) have compact support in R². Then there exists a unique solution to (1),(4).Let . Assume that A(x),V(x) and u(x) are dependent on r and independent onθas x∈R²\K[0, R], R>0; then we deduce thatwhere(2) and (3) have to be replaced byandwhereα> 3/2, C_aj,j = 1,2, C_b are all positive constants. Plus, we deduce thatDenote u_s(r) and u_H(r) the solutions of (5) and (8) separatively. We prove the following main theorem:Theorem .2. Let (6) and (7) be satisfied. Thenfor almost every real k, where C is a constant and independent of r. DenoteΩ_ab = K(0, b)\K[0, a], where a < b, andΓ_a = (?)K(0, a). We assume that suppA_j(x),suppV(x) (?) K[Q,R],j = 1,2, consideringwe introduce the PML equationwhere Finally, we prove the following theorems:Theorem .3. Berenger problem (9) has a unique solution u^B in H¹ (K(0, b)) for almost every k∈R¹.Theorem .4. PML Technique of Schrodinger equation with electromagnetic potentials with compact support is convergent.