On the M-function and Borg–Marchenko theorems for vector-valued Sturm–Liouville equations

Abstract

We will consider a vector-valued Sturm–Liouville equation of the form R[U]≔−(PU′)′+QU=λWU, x∈[0,b), with P−1, W, Q∈Lloc1([0,b))m×m being Hermitian and under some additional conditions on P−1 and W. We give an elementary deduction of the leading order term asymptotics for the Titchmarsh–Weyl M-function corresponding to this equation. In the special case of P=W=I, Q∈L1([0,∞))m×m and the Neumann boundary conditions at 0, we will also prove that M=(1/−λ)(I+R)(I−R)−1, where R=limn→∞ Rn=∑n=1∞Qn, for recursively defined sequences {Rn} and {Qn}. If Q∈Lloc1([0,b))m×m, 0<b⩽∞, the same formula is valid with an exponentially small error for large λ. It is clear that expansions of this type are helpful in finding representatives of the KdV invariants. For P=W=I, we prove that the spectral measure corresponding to the equation R[U]=λU uniquely determines Q as well as b and the boundary conditions at 0 and b. We finally give a new proof of a local form of the Borg–Marchenko theorem (cf. Gesztesy and Simon, “On local Borg–Marchenko uniqueness results,” Commun. Math. Phys. 211, 273–287 (2000), Chap. 3); a theorem which is due to Simon [see Simon, “A new approach to inverse spectral theory, I. fundamental formalism,” Ann. Math. 150, 1–29 (1999)] in the scalar case. For applications to physics, it is worth mentioning that vector-valued Sturm–Liouville equations appear in some problems in magneto-hydro-dynamics.