Allen Devinatz

Local smoothing and convergence properties of Schrödinger type equations

Abstract The initial value problem for the Schrodinger type equation ∂u ∂t = iP(D)u , u (0, x ) = u 0 ( x ), x ∈ R n , is considered. Here P ( ξ ) is a principal type polynomial of order m ( i.e ., ¦▽P(ξ)¦ ⩾ C(1 + ¦ξ¦) m − 1 , ¦ξ¦ ⩾ R) , or a certain type of other real symbol such as P(ξ) = ¦ξ¦ m , m > 1 . The following results are proved: (a) If u 0 ∈ L 2 ( R n ), then for any fixed R > 0 write u 0 = u 1 + u 2 , where u 1 and u 2 are square integrable and the Fourier transform of u 1 , vanishes for ¦ξ¦ ⩾ R and that of u 2 vanishes for ¦ξ¦ ⩽ R . If u 1 ( t , x ) = e itP ( D ) u 1 ( x ), u 2 ( t , x ) = e itP ( D ) u 2 ( x ), then u 1 is real analytic in R × R n and satisfies for every multi-index α, sup t, x ¦D α u 1 (t, x)¦ ⩽ C α ∥u 0 ∥ L 2 , while (1+|x| 2 ) − 1 2 u 2 (t,x)∈L 2 (R t ;H (m−1) 2 (R x n )) . (b) If u 0 ∈ L 2, S ( R n ), S >0 then for every t ≠0, u ( t ,·)∈ H loc ( m −1) S ( R n ). (c) if u 0 ∈H S (R n ), S> 1 2 then for a.e. x ∈ R n , u ( t , x )→ u 0 ( x ) as t →0.

Spectral and scattering theory for the adiabatic oscillator and related potentials

We consider the Schrodinger operator H=−Δ+V (r) on Rn, where V (r) =a sin(brα)/rβ+VS(r), VS(r) being a short range potential and α≳0, β≳0. Under suitable restrictions on α, β, but always including α=β=1, we show that the absolutely continuous spectrum of H is the essential spectrum of H, which is [0,∞), and the absolutely continuous part of H is unitarily equivalent to −Δ. We use these results to show the existence and completeness of the Mo/ller wave operators. Our results are obtained by establishing the asymptotic behavior of solutions of the equation Hu=zu for complex values of z.

A limiting absorption principle for Schrödinger operators with Von Neumann-Wigner type potentials

In this paper we prove the main step in establishing a limiting absorption principle for von Neumann-Wigner type Schrödinger Hamiltonians of the form −Δ+csinb|x|/|x|+V(x), whereV(x) is a short range potential. The first fundamental step is to obtain a limiting absorption principal for the “free” operator −Δ+csinb.|x|/|x|. The free operator is unitarily equivalent to a direct sum of ordinary differential operators. We obtain uniform estimates for the resolvents of these ordinary differential operators. by obtaining uniform estimates for the Weyl-Green kernels of these resolvents. In turn, these latter estimates require uniform estimates on the Wronskians of certain generalized eigen-solutions of these differential operators.

The existence of wave operators for oscillating potentials

The existence of the Mo/ller wave operators is proved for Hamiltonians of the form H=−Δ+a sinbrα/rβ+V(x), where V is a short range potential, generally noncentral, and α and β take on suitable values including the case α=1, β≳ 1/2 .

On an inequality of Tosio Kato for degenerate-elliptic operators

Let Ω be a domain in Rn and T = ∑j,k = 1n(∂j − ibj(x)) ajk(x)(∂k − ibk(x)), where the ajk and the bj are real valued functions in C1(Ω), and the matrix (ajk(x)) is symmetric and positive definite for every x ϵ Ω. If T0 is the same as T but with bj = 0, j = 1,…, n, and if u and Tu are in Lloc1(Ω), then T. Kato has established the distributional inequality T0 ¦ u ¦ ⩾ Re[(sign ū) Tu]. He then used this result to obtain selfadjointness results for perturbed operators of the form T − q on Rn. In this paper we shall obtain Katos inequality for degenerate-elliptic operators with real coefficients. We then use this to get selfadjointness results for second order degenerate-elliptic operators on Rn.

On the Uniqueness ofL2-solutions in half-space of certain differential equations

Square integrable solutions to the equation{−∂2/∂y2 + P(Dx)+b(y)−λ}u(x, y) = f(x, y) are considered in the half-spacey>0, x ∈ℝn, whereP(Dx) is a constant coefficient operator. Under suitable conditions on limy→0u(x, y), b(y), f(x, y) and λ, it is shown that suppu = suppf. This generalizes a result due to Walter Littman.

Schrödinger Operators with Oscillating Potentials

Dae to the pioneering contributions of Kato-Kuroda [10], [11] and to the more recent works of Agmon [l] and Enss [6], a spectral and scattering theory for Schrodinger operators with short range potentials is now well established. An interesting example of a potential which does not belong to this class is the Wigner-von- Neumann [17] potential. This potential is the sum of a short range potential and of an oscillating one which is of the form,

A Schrödinger Operator with an Oscillating Potential

{P_o}\left( x \right) = c\frac{{\sin b{{\left| x \right|}^\alpha }}}{{{{\left| x \right|}^\beta }}},\alpha ,\beta > 0