INVERSE SPECTRAL ANALYSIS WITH PARTIAL INFORMATION ON THE POTENTIAL, I. THE CASE OF AN A.C. COMPONENT IN THE SPECTRUM
INVERSE SPECTRAL ANALYSIS WITH PARTIAL INFORMATION ON THE POTENTIAL, I. THE CASE OF AN A.C. COMPONENT IN THE SPECTRUM
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发表时间：
1997
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通讯作者：
F. Gesztesy;B. Simon
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作者：
F. Gesztesy;B. Simon
我们考虑L(R)中的算子−D2dx2+V，唯一的假设是V是±∞处的极限点，且L2中的−D2dx2+V((0，∞))的谱中有一些绝对连续的成分S+.证明了Von(−∞，0)完全由已知的Von(0，∞)和右入射散射的反射系数R+(λ)和能量λ∈S决定，其中S⊆S+有正的勒贝格测度。众所周知，知道正能量下的反射系数并不能足够快地确定薛定谔算符−d dx2+V(V(X)→0)的势V，但这也需要束缚态能量和相关的归一化常数。这在单孤子势中表现得最为明显，其中R+(λ)≡0，λ≥0)，即使存在由孤子中心和宽度参数化为参数的这种势族。最近有一连串的论文[2，3，4，6，12，18，19]表明，如果V是已知的A.E.并且至少在R上存在第一个矩的意义上，当|x|→∞足够快地消失时，则不需要正规常数，甚至不需要束缚态能量(其中一些文献仅限于假设V在右半线上消失的情况)。我们在这里的目的是要注意，这是一个非常普遍和非常基本的现象的特例：不要求V像|x|→∞一样具有简单的渐近性。相反，唯一有意义的是，V是已知的A.E.关于(0，∞)1991年数学学科分类。主34A55、34B20、34L25；次34L40。
We consider operators− d2 dx2 +V in L(R) with the sole hypothesis that V is limit point at ±∞ and that − d2 dx2 + V in L2((0,∞)) has some absolutely continuous component S+ in its spectrum. We prove that V on (−∞, 0) is completely determined by knowledge of V on (0,∞) and by the reflection coefficient R+(λ) for scattering from right incidence and energies λ ∈ S, where S ⊆ S+ has positive Lebesgue measure. It is well known [15] that knowledge of the reflection coefficient at positive energies does not determine the potential V of a Schrödinger operator − d dx2 + V (V (x) → 0 sufficiently rapidly as |x| → ∞), but that one also needs bound state energies and associated norming constants. This is most dramatically seen in one-soliton potentials where R+(λ) ≡ 0, λ ≥ 0, even though there is a two-parameter family of such potentials parametrized by the center and width of the soliton. There has been a recent rash of papers [2, 3, 4, 6, 12, 18, 19] showing that if V is known a.e. on a half-line and vanishes sufficiently fast as |x| → ∞ in the sense that at least its first moment on R exists, then the norming constants and even the bound state energies are not needed (some of these papers are limited to the case where V is assumed to vanish on the right half-line). Our goal here is to note that this is a special case of a very general and very elementary phenomenon: It is not required that V has simple asymptotics as |x| → ∞. Rather, all that is significant is that V be known a.e. on (0,∞) 1991 Mathematics Subject Classification. Primary 34A55, 34B20, 34L25; Secondary 34L40.