1,135 research outputs found
Matrix representations of Sturm–Liouville problems with transmission conditions
AbstractWe identify a class of Sturm–Liouville equations with transmission conditions such that any Sturm–Liouville problem consisting of such an equation with transmission condition and an arbitrary separated or real coupled self-adjoint boundary condition has a representation as an equivalent finite dimensional matrix eigenvalue problem. Conversely, given any matrix eigenvalue problem of certain type and an arbitrary separated or real coupled self-adjoint boundary condition and transmission condition, we construct a class of Sturm–Liouville problems with this specified boundary condition and transmission condition, each of which is equivalent to the given matrix eigenvalue problem
On Discontinuous Dirac Operator with Eigenparameter Dependent Boundary and Two Transmission Conditions
In this paper, we consider a discontinuous Dirac operator with eigenparameter
dependent both boundary and two transmission conditions. We introduce a
suitable Hilbert space formulation and get some properties of eigenvalues and
eigenfunctions. Then, we investigate Green's function, resolvent operator and
some uniqueness theorems by using Weyl function and some spectral data
The Finite Spectrum of Sturm-Liouville Operator With δ-Interactions
The goal of this paper is to study the finite spectrum of Sturm-Liouville operator with δ- interactions. Such an equation gives us a Sturm-Liouville boundary value problem which has n transmission conditions. We show that for any positive numbers mj (j = 0; 1; :::; n) that are related to number of partition of the intervals between two successive interaction points, we can construct a Sturm-Liouville equations with δ-interactions, which have exactly d eigenvalues. Where d is the sum of mj’s
Supersymmetry and Schr\"odinger-type operators with distributional matrix-valued potentials
Building on work on Miura's transformation by Kappeler, Perry, Shubin, and
Topalov, we develop a detailed spectral theoretic treatment of Schr\"odinger
operators with matrix-valued potentials, with special emphasis on
distributional potential coefficients.
Our principal method relies on a supersymmetric (factorization) formalism
underlying Miura's transformation, which intimately connects the triple of
operators of the form [D= (0 & A^*, A & 0) \text{in}
L^2(\mathbb{R})^{2m} \text{and} H_1 = A^* A, H_2 = A A^* \text{in}
L^2(\mathbb{R})^m.] Here in , with a
matrix-valued coefficient , , thus explicitly permitting distributional
potential coefficients in , , where [H_j = - I_m
\frac{d^2}{dx^2} + V_j(x), \quad V_j(x) = \phi(x)^2 + (-1)^{j} \phi'(x),
j=1,2.] Upon developing Weyl--Titchmarsh theory for these generalized
Schr\"odinger operators , with (possibly, distributional) matrix-valued
potentials , we provide some spectral theoretic applications, including a
derivation of the corresponding spectral representations for , .
Finally, we derive a local Borg--Marchenko uniqueness theorem for ,
, by employing the underlying supersymmetric structure and reducing it
to the known local Borg--Marchenko uniqueness theorem for .Comment: 36 page
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