Author
Tosis Kato
Bio: Tosis Kato is an academic researcher from University of California, Berkeley. The author has an hindex of 1, co-authored 1 publications receiving 640 citations.
Papers
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TL;DR: In this paper, the authors extend Simon's theorem to a more general case, where the Schrodinger operator is essentially self-adjoint onC(R)m), if 0≦q ∈L>>\s 2(R>>\s m), andq>>\s 1(x)≧−q*(|x|) withq *(r) monotone nondecreasing inr ando(r petertodd 2) asr → ∞.
Abstract: Recently B. Simon proved a remarkable theorem to the effect that the Schrodinger operatorT=−Δ+q(x) is essentially selfadjoint onC
0
∞
(R
m
if 0≦q ∈L
2(R
m). Here we extend the theorem to a more general case,T=−Σ
=1/
(∂/∂x
j −ib
j(x))2 +q
1(x) +q
2(x), whereb
j, q1,q
2 are real-valued,b
j ∈C(R
m),q
1 ∈L
loc
2
(R
m),q
1(x)≧−q*(|x|) withq*(r) monotone nondecreasing inr ando(r
2) asr → ∞, andq
2 satisfies a mild Stummel-type condition. The point is that the assumption on the local behavior ofq
1 is the weakest possible. The proof, unlike Simon’s original one, is of local nature and depends on a distributional inequality and elliptic estimates.
677 citations
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TL;DR: In this paper, it was shown that the Strong Maximum Principle is true for weak solutions of − Δu + β(u) = f with β a non-negative superharmonic continuous function in a domain Ω ⊂ ℝ� n�,n ⁽ 1,n ↽ 1.
Abstract: In its simplest form the Strong Maximum Principle says that a nonnegative superharmonic continuous function in a domain Ω ⊂ ℝ
n
,n ⩾ 1, is in fact positive everywhere. Here we prove that the same conclusion is true for the weak solutions of − Δu + β(u) = f withβ a nondecreasing function ℝ → ℝ,β(0)=0, andf⩾0 a.e. in Ω if and only if the integral∫(β(s)s)
−1/2
ds diverges ats=0+. We extend the result to more general equations, in particular to − Δ
p
u + β(u) =f where Δ
p
(u) = div(|Du|
p-2
Du), 1
1,137 citations
TL;DR: In this article, uniform estimates and blow-up behavior for solutions of −δ(u) = v(x)eu in two dimensions are presented, with a focus on partial differential equations.
Abstract: (1991). Uniform estimates and blow–up behavior for solutions of −δ(u)=v(x)eu in two dimensions. Communications in Partial Differential Equations: Vol. 16, No. 8-9, pp. 1223-1253.
679 citations
514 citations
TL;DR: In this paper, it was shown that vector-like global symmetries (like isospin or baryon number) are not spontaneously broken in vectorlike gauge theories with θ = 0 (like QCD), and massless bound states do not form from massive constituents.
502 citations
TL;DR: In this paper, the strong maximum principle was used to show that if such a solution exists, then u > 0 in in, then n > 3 with 0 0, p not identically zero.
Abstract: (1) - Au = p(x)u a in A n , n > 3 with 0 0, p not identically zero. We shall assume throughout the paper that p E L m We look for a solution u > 0, u not identically zero, so that, by the loc" strong maximum principle, if such a solution exists then u > 0 in in
340 citations