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 $(D, H_1, H_2)$ 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 $A= I_m (d/dx) + \phi$ in $L^2(\mathbb{R})^m$, with a matrix-valued coefficient $\phi = \phi^* \in L^1_{\text{loc}}(\mathbb{R})^{m \times m}$, $m \in \mathbb{N}$, thus explicitly permitting distributional potential coefficients $V_j$ in $H_j$, $j=1,2$, 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 $H_j$, with (possibly, distributional) matrix-valued potentials $V_j$, we provide some spectral theoretic applications, including a derivation of the corresponding spectral representations for $H_j$, $j=1,2$. Finally, we derive a local Borg--Marchenko uniqueness theorem for $H_j$, $j=1,2$, by employing the underlying supersymmetric structure and reducing it to the known local Borg--Marchenko uniqueness theorem for $D$.Comment: 36 page

Spectro-temporal post-smoothing in NMF based single-channel source separation

In this paper, we propose a new, simple, fast, and effective method to enforce temporal smoothness on nonnegative matrix factorization (NMF) solutions by post-smoothing the NMF decomposition results. In NMF based single-channel source separation, NMF is used to decompose the magnitude spectra of the mixed signal as a weighted linear combination of the trained basis vectors. The decomposition results are used to build spectral masks. To get temporal smoothness of the estimated sources, we deal with the spectral masks as 2-D images, and we pass the masks through a smoothing filter. The smoothing direction of the filter is the time direction of the spectral masks. The smoothed masks are used to find estimates for the source signals. Experimental results show that, using the smoothed masks give better separation results than enforcing temporal smoothness prior using regularized NMF

J-spectral factorization and equalizing vectors

For the Wiener class of matrix-valued functions we provide necessary and sufficient conditions for the existence of a $J$-spectral factorization. One of these conditions is in terms of equalizing vectors. A second one states that the existence of a $J$-spectral factorization is equivalent to the invertibility of the Toeplitz operator associated to the matrix to be factorized. Our proofs are simple and only use standard results of general factorization theory. Note that we do not use a state space representation of the system. However, we make the connection with the known results for the Pritchard-Salamon class of systems where an equivalent condition with the solvability of an algebraic Riccati equation is given. For Riesz-spectral systems another necessary and sufficient conditions for the existence of a $J$-spectral factorization in terms of the Hamiltonian is added

Nonsquare Spectral Factorization for Nonlinear Control Systems

This paper considers nonsquare spectral factorization of nonlinear input affine state space systems in continuous time. More specifically, we obtain a parametrization of nonsquare spectral factors in terms of invariant Lagrangian submanifolds and associated solutions of Hamilton–Jacobi inequalities. This inequality is a nonlinear analogue of the bounded real lemma and the control algebraic Riccati inequality. By way of an application, we discuss an alternative characterization of minimum and maximum phase spectral factors and introduce the notion of a rigid nonlinear system.