Nodal Sets of Schrödinger Eigenfunctions in Forbidden Regions

Abstract

Abstract. This note concerns the nodal sets of eigenfunctions of semiclassical Schrödinger operators acting on compact, smooth, Riemannian manifolds, with no boundary. We prove that if H is a separating hypersurface that lies inside the classically forbidden region, then H cannot persist as a component of the zero set of infinitely many eigenfunctions. In addition, on real analytic surfaces, we obtain sharp upper bounds for the number of intersections of the zero sets of the Schrödinger eigenfunctions with a fixed curve that lies inside the classically forbidden region.Let (M, g) be a smooth, compact, Riemannian manifold with no boundary. Write ∆ g for the Laplace operator, and given any smooth potential V ∈ C ∞ (M ; R), consider the Schrödinger operator acting on L 2 (M ) defined aswhere h ∈ (0, 1]. Let E ∈ R be a regular value for the total energy function p(x, ξ) = |ξ| 2 gx + V (x) defined on T * M , and write Ω E for the the classically forbidden region Ω E := {x ∈ M : V (x) > E}.In this paper we study the nodal sets of Schrödinger eigenfunctions (with energy close to E) inside the classically forbidden region, in the semiclassical limit h → 0 + . Consider L 2 -normalized Schrödinger eigenfunctions {φ h } withThere is a large literature devoted to the study of the zero sets of Laplace eigenfunctions, Z φ h = {x ∈ M : φ h (x) = 0}, on compact manifolds. We refer the reader to [Z2] for a detailed list of references. The Hausdorff measure of the zero sets, their distribution properties, the number of nodal domains and their inner radius, have been extensively studied (although many open problems remain, even for surfaces). More generally, it is natural to study the properties of zero sets of Schrödinger eigenfunctions inside the classically allowed region where V < E. Many of the known results in the homogeneous case where V = 0 extend to Schrödinger eigenfunctions in the allowable region (see [Jin]). In contrast, very little is known about the zero sets of Schrödinger eigenfunctions inside the classically forbidden region where V > E. In dimension one, it is known that the eigenfunctions of the Harmonic Oscillator have no zeros in the forbidden region and in recent work, Hanin-Zelditch-Zhou [HZZ] have proved that in any higher dimension Y

Reverse Agmon Estimates in Forbidden Regions

“…Moreover, the Schwartz kernel Q(t, s, h) extends holomorphically in the outgoing t variable to Q C (t, s) with (t, s) ∈ H C ρ * × γ where ρ * > 0 is a suitable tube radius independent of h (see Remark 6 (ii) and (iii) below). It then follows from a potential layer formula [CT16] (14), that the eigenfunction restriction u h • q(t) also holomorphically continues to (u h • q) C (t) in the strip H C ρ * . Setting Q(h, ρ * ) := max (r(s),q C (t))∈γ×H C ρ * (|Q C (t, s)|, |∂ ν(s) Q C (t, s))|) and using the formula in (4.4) combined with some potential layer analysis, in [CT16] (16), it is proved that…”

“…In [CT16] Theorem 4, the authors show that by choosing ρ * > 0 sufficiently small (see Remark 6 (iii)) one can show that Q(h, ρ * ) = O(1).…”

“…We then give applications to lower bounds for L p -restrictions of eigenfunctions to hypersurfaces in the forbidden region (so-called goodness estimates in the terminology of Toth and Zelditch [TZ09]). Finally, we apply these rather explicit bounds to improve on the nodal intersection bounds of Canzani and Toth [CT16] for a large class of hypersurfaces in forbidden regions. We now describe our results in more detail.…”

“…Finally, in section 4, we give an application of (1.8) to nodal intersection bounds in forbidden regions. In [CT16], Canzani and Toth prove that for any C ω separating hypersurface H in the forbidden region, with Z u h = {x ∈ M; u h (x) = 0}, #{Z u h ∩ H} ≤ C ′ H h −1 . While this rate in h is easily seen to be sharp in general, the constant C ′ H > 0 is not explicitly controlled in [CT16].…”

“…In [CT16], Canzani and Toth prove that for any C ω separating hypersurface H in the forbidden region, with Z u h = {x ∈ M; u h (x) = 0}, #{Z u h ∩ H} ≤ C ′ H h −1 . While this rate in h is easily seen to be sharp in general, the constant C ′ H > 0 is not explicitly controlled in [CT16]. The lower bound in (1.8) allows us to give a more concrete upper bound for C ′ H in the cases where H is a separating hypersurface that is smoothly isotopic to a level set of the defining function y n in the forbidden region.…”