TY - JOUR

T1 - Donoghue-type m-functions for Schrödinger operators with operator-valued potentials

AU - Gesztesy, Fritz

AU - Naboko, Sergey N.

AU - Weikard, Rudi

AU - Zinchenko, Maxim

N1 - Gesztesy, F., Naboko, S.N., Weikard, R. et al. JAMA (2019) 137: 373. https://doi.org/10.1007/s11854-018-0076-1

PY - 2019

Y1 - 2019

N2 - Given a complex, separable Hilbert space H, we consider differential expressions of the type τ = −(d 2 /dx 2 )I H + V(x), with x ∈ (x 0 ,∞) for some x 0 ∈ ℝ, or x ∈ ℝ (assuming the limit-point property of τ at ±∞). Here V denotes a bounded operator-valued potential V(·) ∈ B(H) such that V(·) is weakly measurable, the operator norm ||V(⋅)||B(H) is locally integrable, and V(x) = V(x)* a.e. on x ∈ [x 0 ,∞) or x ∈ ℝ. We focus on two major cases. First, on m-function theory for self-adjoint half-line L 2 -realizations H +,α in L 2 ((x 0 ,∞); dx;H) (with x 0 a regular endpoint for τ, associated with the self-adjoint boundary condition sin(α)u′(x 0 ) + cos(α)u(x 0 ) = 0, indexed by the selfadjoint operator α = α* ∈ B(H)), and second, on m-function theory for self-adjoint full-line L 2 -realizations H of τ in L 2 (ℝ; dx;H). In a nutshell, a Donoghue-type m-function MA,NiDo(⋅) associated with self-adjoint extensions A of a closed, symmetric operator A˙ in H with deficiency spaces N z = ker (A˙ * −zIH) and corresponding orthogonal projections PNz onto N z is given by MA,NiDo(z)=PNi(zA+IH)(A)−zIH)−1PNi|Ni;=zINi+(z2+1)PNi(A−zIH)−1PNi|Ni,z∈C∖R. In the concrete case of half-line and full-line Schrödinger operators, the role of A˙ is played by a suitably defined minimal Schrödinger operator H +,min in L 2 ((x 0 ,∞); dx;H) and H min in L 2 (ℝ; dx;H), both of which will be proven to be completely non-self-adjoint. The latter property is used to prove that if H +,α in L 2 ((x 0 ,∞); dx;H), respectively, H in L 2 (ℝ; dx;H), are self-adjoint extensions of H +,min , respectively, H min , then the corresponding operator-valued measures in the Herglotz–Nevanlinna representations of the Donoghue-type m-functions MH+,αDo,N+,i(⋅) and MH,NiDo(⋅) encode the entire spectral information of H +,α , respectively, H.

AB - Given a complex, separable Hilbert space H, we consider differential expressions of the type τ = −(d 2 /dx 2 )I H + V(x), with x ∈ (x 0 ,∞) for some x 0 ∈ ℝ, or x ∈ ℝ (assuming the limit-point property of τ at ±∞). Here V denotes a bounded operator-valued potential V(·) ∈ B(H) such that V(·) is weakly measurable, the operator norm ||V(⋅)||B(H) is locally integrable, and V(x) = V(x)* a.e. on x ∈ [x 0 ,∞) or x ∈ ℝ. We focus on two major cases. First, on m-function theory for self-adjoint half-line L 2 -realizations H +,α in L 2 ((x 0 ,∞); dx;H) (with x 0 a regular endpoint for τ, associated with the self-adjoint boundary condition sin(α)u′(x 0 ) + cos(α)u(x 0 ) = 0, indexed by the selfadjoint operator α = α* ∈ B(H)), and second, on m-function theory for self-adjoint full-line L 2 -realizations H of τ in L 2 (ℝ; dx;H). In a nutshell, a Donoghue-type m-function MA,NiDo(⋅) associated with self-adjoint extensions A of a closed, symmetric operator A˙ in H with deficiency spaces N z = ker (A˙ * −zIH) and corresponding orthogonal projections PNz onto N z is given by MA,NiDo(z)=PNi(zA+IH)(A)−zIH)−1PNi|Ni;=zINi+(z2+1)PNi(A−zIH)−1PNi|Ni,z∈C∖R. In the concrete case of half-line and full-line Schrödinger operators, the role of A˙ is played by a suitably defined minimal Schrödinger operator H +,min in L 2 ((x 0 ,∞); dx;H) and H min in L 2 (ℝ; dx;H), both of which will be proven to be completely non-self-adjoint. The latter property is used to prove that if H +,α in L 2 ((x 0 ,∞); dx;H), respectively, H in L 2 (ℝ; dx;H), are self-adjoint extensions of H +,min , respectively, H min , then the corresponding operator-valued measures in the Herglotz–Nevanlinna representations of the Donoghue-type m-functions MH+,αDo,N+,i(⋅) and MH,NiDo(⋅) encode the entire spectral information of H +,α , respectively, H.

UR - http://www.scopus.com/inward/record.url?scp=85058471065&partnerID=8YFLogxK

U2 - 10.1007/s11854-018-0076-1

DO - 10.1007/s11854-018-0076-1

M3 - Article

AN - SCOPUS:85058471065

VL - 137

SP - 373

EP - 427

JO - Journal d'Analyse Mathematique

JF - Journal d'Analyse Mathematique

SN - 0021-7670

IS - 1

ER -