Absence of positive eigenvalues for Schrödinger operators with rough potentials
speaker: David Jerison (Massachusetts Institute of Technology)abstract: In these lectures, David Jerison discussed joint work with A. D. Ionescu appearing in GAFA 2003. There the authors prove that certain Schrödinger operators with short-range, but rough potentials do not have embedded eigenvalues. For example, consider the eigenvalue equation $$ \Delta u + Vu = -\lambda u $$ in Euclidean space. If $\lambda >0$, $V$ belongs to $L{n2}(Rn)$, $u$ belongs to $L2(Rn)$ (the key decay at infinity that gets the uniqueness process started), and $\nabla u$ belongs to $L2$ on compact subsets, then $u$ is identically zero. In other words, there are no embedded eigenvalues $\lambda$ in interior of the continuous spectrum $[0,\infty)$ of the operator $\Delta + V$.
The first step is to show that this theorem follows from a Carleman inequality. Let $1p+ 1p'=1$ and $1p - 1p'= 2n$. The Carleman inequality says for all $\phi \in C\infty0(Rn)$, $$ \
Wm \phi \
{L{p'}(Rn)} \le C \
Wm(\Delta + 1)\phi \
{Lp(Rn)} $$ for a family of weight functions $Wm$ consisting of truncated powers of $
x
$, namely, $Wm(x) = \min (
x
m, Rm)$ for some sequence $Rm \to \infty$ as $m\to \infty$. The key feature of Carleman inequalities is that the constant $C$ is independent of $m$. The Carleman inequality is true provided $Rm$ tends to infinity sufficiently fast. The proof of the Carleman inequality is long and involves an explicit construction of an inverse for the operator $\Delta + 1$ on the range of $(\Delta +1)(C0\infty)$ using separation of variables, Bessel functions and spherical harmonics. An important tool is symbol-type estimates for the Bessel and Legendre functions across the entire range of values of the order and argument. This eventually reduces matters to more typical oscillatory integral bounds. Another tool developed here is optimal operator bounds from $Lp(S{n-1})$ to $L{p'}(S{n-1})$ of sums $$ \sum{k=N}{2N} ak Hk $$ where $Hk$ is the projection onto spherical harmonics of degree $k$. These bounds take advantage not only of the size of the coefficients $ak$, but also their cancellation properties.
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