We construct a class of matrix-valued Schrödinger operators with prescribed finite-band spectra of maximum spectral multiplicity. The corresponding matrix potentials are shown to be stationary solutions of the KdV hierarchy. The methods employed in this paper rely on matrix-valued Herglotz functions, Weyl–Titchmarsh theory, pencils of matrices, and basic inverse spectral theory for matrix-value...

We provide a new short proof of the following fact, first proved by one of us in 1998: If two Weyl-Titchmarsh m-functions, m j (z), of two Schrödinger operators H j = − d 2 dx 2 + q j , j = 1, 2 in L 2 ((0, R)), 0 < R ≤ ∞, are exponentially close, that is, |m 1 (z) − m 2 (z)| = |z|→∞ O(e −2 Im(z 1/2)a), 0 < a < R, then q 1 = q 2 a.e. on [0, a]. The result applies to any boundary conditions at x...

We develop a spectrally accurate numerical method to compute solutions of a model PDE used in plasma physics to describe diffusion in velocity space due to Fokker–Planck collisions. The solution is represented as a discrete and continuous superposition of normalizable and nonnormalizable eigenfunctions via the spectral transform associated with a singular Sturm–Liouville operator. We present a ...

We continue the study of the A-amplitude associated to a half-line d2 Schrodinger operator, -=t 4 in L2((0, b ) ) , b 5 oo.A is related to the iieylTitchmarsh m-function via m(-fi2) = A(a)e-2ff" d c x + ~ ( e ( ~ ~ & ) " ) -6-J: for all E > 0. We discuss five issues here. First, we extend the theory to general q in L1((O, a ) ) for all a , including q's which are limit circle at infinity. Secon...

We continue the study of the A-amplitude associated to a half-line Schrödinger operator, − d dx + q in L((0, b)), b ≤ ∞. A is related to the Weyl-Titchmarsh m-function via m(−κ2) = −κ− ∫ a 0 A(α)e −2ακ dα+O(e) for all ε > 0. We discuss five issues here. First, we extend the theory to general q in L((0, a)) for all a, including q’s which are limit circle at infinity. Second, we prove the followi...

We consider canonical systems (with 2 p × Hamiltonians H ( x ) ≥ 0 ), which correspond to matrix string equations. Direct and inverse problems are solved in terms of Titchmarsh–Weyl spectral functions related S -nodes. Procedures for solving given.

We provide an abstract framework for singular one-dimensional Schrödinger operators with purely discrete spectra to show when the spectrum plus norming constants determine such an operator completely. As an example we apply our findings to prove a new uniqueness results for perturbed quantum mechanical harmonic oscillators. In addition, we also show how to establish a Hochstadt–Liebermann type ...

We provide an abstract framework for singular one-dimensional Schrödinger operators with purely discrete spectra to show when the spectrum plus norming constants determine such an operator completely. As an example we apply our findings to prove new uniqueness results for perturbed quantum mechanical harmonic oscillators. In addition, we also show how to establish a Hochstadt–Lieberman type res...