Averaged Pointwise Bounds for Deformations of Schrodinger Eigenfunctions

作者：Eswarathasan Suresh^*; Toth John A

来源：Annales Henri Poincare, 2013, 14(3): 611-637.

DOI：10.1007/s00023-012-0198-4

Let (M, g) be an n-dimensional, compact Riemannian manifold and P-0((h) over bar)= -(h) over bar (2)Delta(g) + V(x) be a semiclassical Schrodinger operator with (h) over bar is an element of (0, (h) over bar (0)]. Let E((h) over bar) is an element of [E - o(1), E + o(1)] and (phi((h) over bar))((h) over bar is an element of(0, (h) over bar0)] be a family of L-2 -normalized eigenfunctions of P-0((h) over bar) with P-0((h) over bar)phi((h) over bar) = E((h) over bar)phi((h) over bar). We consider magnetic deformations of P-0((h) over bar) of the form P-u((h) over bar) = -Delta(omega u) ((h) over bar) + V (x), where Delta(omega u) ((h) over bar) = ((h) over bard + i omega(u)(x))*((h) over bard + i omega(u)(x)). Here, u is a k-dimensional parameter running over B-k(epsilon) (the ball of radius epsilon), and the family of the magnetic potentials (omega(u))(u is an element of B)(k)((epsilon)) satisfies the admissibility condition given in Definition 1.1. This condition implies that k >= n and is generic under this assumption. Consider the corresponding family of deformations of (phi((h) over bar))((h) over bar is an element of(0,(h) over bar0)], given by (phi(u)((h) over bar))(h)is an element of(0,(h) over bar0], where
phi((u))(h) := e(-it0Pu ((h) over bar/(h) over bar) phi((h) over bar)
for vertical bar t(0)vertical bar is an element of (0, epsilon); the latter functions are themselves eigenfunctions of the h-elliptic operators Q(u)((h) over bar) := e(-it0Pu((h) over bar)/(h) over bar) P-0((h) over bar )e(it0Pu((h) over bar)/(h) over bar) with eigenvalue E((h) over bar) and Q(0)((h) over bar) = P-0((h) over bar). Our main result, Theorem 1.2, states that for epsilon > 0 small, there are constants C-j = C-j(M, V, omega , epsilon) > 0 with j = 1, 2 such that
C-1 <= integral(Bk(epsilon)) vertical bar phi((u))((h) over bar) (x)vertical bar(2) du <= C-2,
uniformly for x is an element of M and (h) over barh is an element of (0, h(0)]. We also give an application to eigenfunction restriction bounds in Theorem 1.3.

• 出版日期2013-4