TY - JOUR

T1 - Absence of reflection as a function of the coupling constant

AU - Killip, Rowan

AU - Sims, Robert

N1 - Funding Information:
The first author was supported in part by NSF Grant No. DMS-0401277 and a Sloan Foundation Fellowship.

PY - 2006

Y1 - 2006

N2 - We consider solutions of the one-dimensional equation - u″+(Q+λV)u=0 where Q:ℝ→ℝ is locally integrable, V:ℝ→ℝ is integrable with supp(V) ⊂[0, 1], and λ ∈ ℝ is a coupling constant. Given a family of solutions {u λ}λ∈ℝ which satisfy u λ(x) = u0(x) for all x<0, we prove that the zeros of b(λ) := W[u0, uλ], the Wronskian of u0 and uλ, form a discrete set unless V≡0. Setting Q(x) :=-E, one sees that a particular consequence of this result may be stated as: if the fixed energy scattering experiment -u″+λVu=Eu gives rise to a reflection coefficient which vanishes on a set of couplings with an accumulation point, then V≡0.

AB - We consider solutions of the one-dimensional equation - u″+(Q+λV)u=0 where Q:ℝ→ℝ is locally integrable, V:ℝ→ℝ is integrable with supp(V) ⊂[0, 1], and λ ∈ ℝ is a coupling constant. Given a family of solutions {u λ}λ∈ℝ which satisfy u λ(x) = u0(x) for all x<0, we prove that the zeros of b(λ) := W[u0, uλ], the Wronskian of u0 and uλ, form a discrete set unless V≡0. Setting Q(x) :=-E, one sees that a particular consequence of this result may be stated as: if the fixed energy scattering experiment -u″+λVu=Eu gives rise to a reflection coefficient which vanishes on a set of couplings with an accumulation point, then V≡0.

UR - http://www.scopus.com/inward/record.url?scp=33745683253&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33745683253&partnerID=8YFLogxK

U2 - 10.1063/1.2206691

DO - 10.1063/1.2206691

M3 - Article

AN - SCOPUS:33745683253

VL - 47

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 6

M1 - 062102

ER -