Spectral analysis of the spin-boson Hamiltonian with two photons for arbitrary coupling

We study the spectrum of the spin-boson model with two photons in $\mathbb{R}^d$ for arbitrary coupling $\alpha>0$. It is shown that the discrete spectrum is finite and the essential spectrum consists of a half-line the bottom of which is a unique zero of a simple Nevanlinna function. Besides the simplicity and more abstract nature of our approach, the main novelty is the achievement of these results under "minimal" regularity conditions on the photon dispersion and the coupling function.Comment: 16 page

Sharp spectral bounds for complex perturbations of the indefinite Laplacian

We derive quantitative bounds for eigenvalues of complex perturbations of the indefinite Laplacian on the real line. Our results substantially improve existing results even for real-valued potentials. For $L^1$-potentials, we obtain optimal spectral enclosures which accommodate also embedded eigenvalues, while our result for $L^p$-potentials yield sharp spectral bounds on the imaginary parts of eigenvalues of the perturbed operator for all $p\in[1,\infty)$. The sharpness of the results are demonstrated by means of explicit examples.Comment: References added before Theorem 2 and

Analysis of the essential spectrum of singular matrix differential operators

A complete analysis of the essential spectrum of matrix-differential operators $\mathcal A$ of the form \begin{align} \begin{pmatrix} -\displaystyle{\frac{\rm d}{\rm d t}} p \displaystyle{\frac{\rm d}{\rm d t}} + q & -\displaystyle{\frac{\rm d}{\rm d t}} b^* \! + c^* \\[2mm] \hspace{6mm} b \displaystyle{\frac{\rm d}{\rm d t}} + c & \hspace{4mm} D \end{pmatrix} \quad \text{in } \ L^2((\alpha, \beta)) \oplus \bigl(L^2((\alpha, \beta))\bigr)^n \label{mo} \end{align} singular at $\beta\in\mathbb R\cup\{\infty\}$ is given; the coefficient functions $p$, $q$ are scalar real-valued with $p>0$, $b$, $c$ are vector-valued, and $D$ is Hermitian matrix-valued. The so-called "singular part of the essential spectrum" $\sigma_{\rm ess}^{\rm \,s}(\mathcal A)$ is investigated systematically. Our main results include an explicit description of $\sigma_{\rm ess}^{\rm \,s}(\mathcal A)$, criteria for its absence and presence; an analysis of its topological structure and of the essential spectral radius. Our key tools are: the asymptotics of the leading coefficient $\pi(\cdot,\lambda)=p-b^*(D-\lambda)^{-1}b$ of the first Schur complement of $\mathcal A$, a scalar differential operator but non-linear in $\lambda$; the Nevanlinna behaviour in $\lambda$ of certain limits $t\!\nearrow\!\beta$ of functions formed out of the coefficients in $\mathcal A$. The efficacy of our results is demonstrated by several applications; in particular, we prove a conjecture on the essential spectrum of some symmetric stellar equilibrium models.Comment: 32 pages, 1 figur

Essential spectrum of non-self-adjoint singular matrix differential operators

The purpose of this paper is to study the essential spectrum of non-self-adjoint singular matrix differential operators in the Hilbert space $L^2(\mathbb{R})\oplus L^2(\mathbb{R})$ induced by matrix differential expressions of the form \begin{align}\label{abstract:mdo} \left(\begin{array}{cc} \tau_{11}(\,\cdot\,,D) & \tau_{12}(\,\cdot\,,D)\\[3.5ex] \tau_{21}(\,\cdot\,,D) & \tau_{22}(\,\cdot\,,D) \end{array}\right), \end{align} where $\tau_{11}$, $\tau_{12}$, $\tau_{21}$, $\tau_{22}$ are respectively $m$-th, $n$-th, $k$-th and 0 order ordinary differential expressions with $m=n+k$ being even. Under suitable assumptions on their coefficients, we establish an analytic description of the essential spectrum. It turns out that the points of the essential spectrum either have a local origin, which can be traced to points where the ellipticity in the sense of Douglis and Nirenberg breaks down, or they are caused by singularity at infinity.Comment: 25 pages, 1 figure, a few typos correcte