TY - JOUR

T1 - STURM–LIOUVILLE OPERATORS WITH W-1,1-MATRIX POTENTIALS

AU - Granovskiy, Y.I.

AU - Malamud, M.M.

N1 - Export Date: 03 March 2026; Cited By: 0; Correspondence Address: Y.I. Granovskiy; Donetsk National Technical University, Institute of Applied Mathematics and Mechanics, Drohobych, Russian Federation; email: yarvodoley@mail.ru

PY - 2025

Y1 - 2025

N2 - The spectral structure of realizations for a matrix three-term Sturm–Liouville operator (Formula presented.) with singular potential Q(·)=Q(·)∗ on both the half-line and the line is investigated. It is shown that under certain conditions on the coefficients P(·) and R(·), the Dirichlet realization LD (as well as other selfadjoint realizations) in the case of Q(·)∈W-1,1(R+;Cm×m) has Lebesgue nonnegative spectrum with constant multiplicity m. In particular, the Schrödinger operator with matrix potential Q(·)∈W-1,1(R+;Cm×m) on the half-line R+ has Lebesgue spectrum with constant multiplicity m. This result is applied to the Sturm–Liouville expression L(P,Q,R) with delta-interactions on the line R. It is shown that if the minimal operator L:=Lmin in L2(R;R;Cm) is selfadjoint, then the nonnegative spectrum of L is Lebesgue of constant multiplicity 2m whenever Q(·)1R+(·)∈W-1,1(R+;Cm×m). In particular, if the minimal Schrödinger operator H on the line with potential matrix Q(·)=Q1(·)+∑k∈Zαkδ(·-xk) is selfadjoint, H=H∗, then its nonnegative spectrum is Lebesgue with constant multiplicity 2m whenever Q1(·)1R+∈L1(R+;Cm×m) and ∑k=1∞|αk|

AB - The spectral structure of realizations for a matrix three-term Sturm–Liouville operator (Formula presented.) with singular potential Q(·)=Q(·)∗ on both the half-line and the line is investigated. It is shown that under certain conditions on the coefficients P(·) and R(·), the Dirichlet realization LD (as well as other selfadjoint realizations) in the case of Q(·)∈W-1,1(R+;Cm×m) has Lebesgue nonnegative spectrum with constant multiplicity m. In particular, the Schrödinger operator with matrix potential Q(·)∈W-1,1(R+;Cm×m) on the half-line R+ has Lebesgue spectrum with constant multiplicity m. This result is applied to the Sturm–Liouville expression L(P,Q,R) with delta-interactions on the line R. It is shown that if the minimal operator L:=Lmin in L2(R;R;Cm) is selfadjoint, then the nonnegative spectrum of L is Lebesgue of constant multiplicity 2m whenever Q(·)1R+(·)∈W-1,1(R+;Cm×m). In particular, if the minimal Schrödinger operator H on the line with potential matrix Q(·)=Q1(·)+∑k∈Zαkδ(·-xk) is selfadjoint, H=H∗, then its nonnegative spectrum is Lebesgue with constant multiplicity 2m whenever Q1(·)1R+∈L1(R+;Cm×m) and ∑k=1∞|αk|

U2 - 10.1007/s10958-025-07993-w

DO - 10.1007/s10958-025-07993-w

M3 - статья

VL - 293

SP - 171

EP - 183

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 2

ER -