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1.
Square integrable solutions to the equation{–
2/y2 + P(Dx)+b(y)–}u(x, y) = f(x, y) are considered in the half-spacey>0, x
n
, whereP(D
x) is a constant coefficient operator. Under suitable conditions on limy0u(x, y), b(y), f(x, y) and , it is shown that suppu = suppf. This generalizes a result due to Walter Littman.Research partially supported by USNSF Grant 79-02538-A02. 相似文献
2.
Consider the Schrödinger equation {fx25-1}. The following estimates are proved: (A) IfV≡0 then for any 0≤α<1/2, {fx25-2}, and for α=1/2,s>1/2, {fx25-3} (B) If |V(x)|≤C(1+|x|2)?1?δ, δ>0, then (if 0 is neither an eigenvalue nor a resonance of ?Δ+V), {fx25-4}. 相似文献
3.
The notions of norm and spectral radius of a matrix are generalized to spaces with an indefinite metric. A spectral radius formula is established. 相似文献
4.
Matania Ben-Artzi 《Journal of Differential Equations》1980,38(1):51-60
Let H = ?Δ + V, where the potential V is spherically symmetric and can be decomposed as a sum of a short-range and a long-range term, V(r) = VS(r) + VL. Let λ = lim supr→∞VL(r) < ∞ (we allow λ = ? ∞) and set λ+ = max(λ, 0). Assume that for some r0, VL(r) ?C2k(r0, ∞) and that there exists δ > 0 such that . Assume further that and that 2kδ > 1. It is shown that: (a) The restriction of H to C∞(Rn) is essentially self-adjoint, (b) The essential spectrum of H contains the closure of (λ, ∞). (c) The part of H over (λ, ∞) is absolutely continuous. 相似文献
5.
The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is now well-known and can be formulated
as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution
at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation
of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence
the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it
can be “diagonalized” and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled
systems (WCS). Such systems have only “partial set” of Riemann invariants, but these sets are weakly coupled in a way which
enables a “diagonalized” treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic
fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A “propagation of singularities”
argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to “rotate” initial spatial slopes
into “time derivative”. In particular, the case of a “sonic point” is incorporated easily into the general treatment. A GRP
scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the
“acoustic approximation” of the analytical solution. It can be viewed as a proper linearization (different from the approach
of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization
of the Godunov scheme. 相似文献
6.
Matania Ben-Artzi 《Israel Journal of Mathematics》1981,40(3-4):259-274
LetH=?Δ+V(r) be a Schrödinger operator with a spherically symmetric exploding potential, namely,V(r)=V S(r)+V L(r), whereV S(r) is short-range and the exploding partV L(r) satisfies the following assumptions: (a) Λ=lim sup r→∞ V L(r)<∞ (but Λ=?∞ is possible). Denote Λ+= max(Λ,0). (b)V L(r)∈C 2k (r 0, ∞) and, with someδ>0 such that 2kδ>1: (d/dr) j V L(r) · (Λ+?V L(r))?1=O(r jδ) asr → ∞,j=1, ..., 2k. (c) ∫ r0 ∞ dr|V L(r|1/2 dr|V L(r)|1/2=∞. (d) (d/dr)V L(r)≦0. Under these assumptions a limiting absorption principle forR(z)=(H?z)?1 is established. More specifically, ifK ?C +={zImz≧0} is compact andK ∩ (?∞, Λ]=Ø thenR (z) can be extended as a continuous map ofK intoB (Y, Y*) (with the uniform operator topology), whereY ?L 2(R n) is a weighted-L 2 space. To ensure uniqueness of solutions of (H?z)u=f, z ∈K, a suitable radiation condition is introduced. 相似文献
7.
A theorem of Yamamoto on singular numbers of matrices is extended to G-bounded matrices. 相似文献
8.
9.
Matania Ben Artzi 《Journal of Differential Equations》1984,52(3):327-341
Let be a Schrödinger operator in Rn. Here is an “exploding” radially symmetric potential which is at least C2 monotone nonincreasing and O(r2) as r → ∞. V is a general potential which is short range with respect to VE. In particular, VE 0 leads to the “classical” short-range case (V being an Agmon potential). Let Λ = limr → ∞VE(r) and R(z) = (H ? z)?1, 0 < Im z, Λ < Re z < ∞. It is shown that R(z) can be extended continuously to Im z = 0, except possibly for a discrete subset ?(Λ, ∞), in a suitable operator topology . And L ? L2(Rn) is a weighted L2-space; H is then absolutely continuous over (Λ, ∞), except possibly for a discrete set of eigenvalues. The corresponding eigenfunctions are shown to be rapidly decreasing. 相似文献
10.
In this paper we obtain general infinite dimensional inertia theorems for linear pencils in Hilbert space which cover previously known results for the finite dimensional case and for block weighted shifts. Connections with definite subspaces for contractions in spaces with indefinite metric are discussed. 相似文献