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We derive a counting formula for the eigenvalues of Schr\"odinger operators with self-adjoint boundary conditions on quantum star graphs. More specifically, we develop techniques using Evans functions to reduce full quantum graph eigenvalue problems into smaller subgraph eigenvalue problems. These methods provide a simple way to calculate the spectra of operators with localized potentials., Comment: 6 figures

The Evans function is a well known tool for locating spectra of differential operators in one spatial dimension. In this paper we construct a multidimensional analogue as the modified Fredholm determinant of a ratio of Dirichlet-to-Robin operators on the boundary. This gives a tool for studying the eigenvalue counting functions of second-order elliptic operators that need not be self-adjoint. To do this we use local representation theory for meromorphic operator-valued pencils, and relate the algebraic multiplicities of eigenvalues of elliptic operators to those of the Robin-to-Robin and Robin-to-Dirichlet operator pencils. In the self-adjoint case we relate our construction to the Maslov index, another well known tool in the spectral theory of differential operators. This gives new insight into the Maslov index and allows us to obtain crucial monotonicity results by complex analytic methods., Comment: 61 pages, comments welcome!

Extending work of Carty, we show that $H^1$ solutions of a simplified 1D BGK model decay exponentially in $L^2$ to a subclass of the class of grossly determined solutions as defined by Truesdell and Muncaster. In the process, we determine the spectrum and generalized eigenfunctions of the associated non-selfadjoint linearized operator and derive the associated generalized Fourier transform and Parseval's identity. Notably, our analysis makes use of rigged space techniques originating from quantum mechanics, as adapted by Ljance and others to the nonselfadjoint case.

Working with a general class of linear Hamiltonian systems with at least one singular boundary condition, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of $\mathbb{C}^{2n}$. This extends previous work by the authors for regular linear Hamiltonian systems., Comment: 55 pages, 2 figures

It was recently shown by the authors that a semilinear elliptic equation can be represented as an infinite-dimensional dynamical system in terms of boundary data on a shrinking one-parameter family of domains. The resulting system is ill-posed, in the sense that solutions do not typically exist forward or backward in time. In this paper we consider a radial family of domains and prove that the linearized system admits an exponential dichotomy, with the unstable subspace corresponding to the boundary data of weak solutions to the linear PDE. This generalizes the spatial dynamics approach, which applies to infinite cylindrical (channel) domains, and also generalizes previous work on radial domains as we impose no symmetry assumptions on the equation or its solutions.

A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in $\mathbb{R}^n$ is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for the PDE. This is a novel approach to elliptic problems that enables the use of dynamical systems tools in studying the corresponding PDE. The dynamical system is ill-posed, meaning solutions do not exist forwards or backwards in time for generic initial data. We offer a framework in which this ill-posed system can be analyzed. This can be viewed as generalizing the theory of spatial dynamics, which applies to the case of an infinite cylindrical domain., Comment: v2: minor corrections and notational changes

We establish a Sturm{Liouville theorem for quadratic operator pencils counting their unstable real roots, with applications to stability of waves. Such pencils arise, for example, in reduction of eigenvalue systems to higher-order scalar problems.

We show that for Sturm-Liouville Systems on the half-line $[0,\infty)$, the Morse index can be expressed in terms of the Maslov index and an additional term associated with the boundary conditions at $x = 0$. Relations are given both for the case in which the target Lagrangian subspace is associated with the space of $L^2 ((0,\infty), \mathbb{C}^{n})$ solutions to the Sturm-Liouville System, and the case when the target Lagrangian subspace is associated with the space of solutions satisfying the boundary conditions at $x = 0$. In the former case, a formula of H\"ormander's is used to show that the target space can be replaced with the Dirichlet space, along with additional explicit terms. We illustrate our theory by applying it to an eigenvalue problem that arises when the nonlinear Schr\"odinger equation on a star graph is linearized about a half-soliton solution., Comment: 34 pages

By reduction to a generalized Sturm Liouville problem, we establish spectral stability of hydraulic shock profiles of the Saint-Venant equations for inclined shallow-water flow, over the full parameter range of their existence, for both smooth-type profiles and discontinuous-type profiles containing subshocks. Together with work of Mascia-Zumbrun and Yang-Zumbrun, this yields linear and nonlinear $H^2\cap L^1 \to H^2$ stability with sharp rates of decay in $L^p$, $p\geq 2$, the first complete stability results for large-amplitude shock profiles of a hyperbolic relaxation system.

Working with a general class of linear Hamiltonian systems on $[0, 1]$, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of $\mathbb{C}^{2n}$. We verify that our applicability class includes Dirac and Sturm-Liouville systems, as well as a system arising from differential-algebraic equations for which the spectral parameter appears nonlinearly.

In a scalar reaction-diffusion equation, it is known that the stability of a steady state can be determined from the Maslov index, a topological invariant that counts the state's critical points. In particular, this implies that pulse solutions are unstable. We extend this picture to pulses in reaction-diffusion systems with gradient nonlinearity. In particular, we associate a Maslov index to any asymptotically constant state, generalizing existing definitions of the Maslov index for homoclinic orbits. It is shown that this index equals the number of unstable eigenvalues for the linearized evolution equation. Finally, we use a symmetry argument to show that any pulse solution must have nonzero Maslov index, and hence be unstable., Comment: 19 pages, 1 figure. To appear in Philos. Trans. Roy. Soc. A

Applying the Lyapunov-Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth-order scalar Swift-Hohenberg equation, we carry out a rigorous small-amplitude stability analysis of Turing patterns for the model introduced by Matthews and Cox for pattern formation with a conservation law. Our results confirm that stability is accurately predicted in the small-amplitude limit by the formal modified Ginzburg-Lanadau system (mGL) consisting of a coupled Ginzburg-Landau equation and mean mode equation derived by Matthews and Cox, rigorously validating the standard weakly unstable approximation., Comment: arXiv admin note: substantial text overlap with arXiv:1608.08476

Applying the Lyapunov-Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth-order scalar Swift-Hohenberg equation, we carry out a rigorous small-amplitude stability analysis of Turing patterns for the canonical second-order system of reaction diffusion equations given by the Brusselator model. Our results confirm that stability is accurately predicted in the small-amplitude limit by the formal Ginzburg Landau amplitude equations, rigorously validating the standard weakly unstable approximation and Eckhaus criterion.

Assuming a symmetric potential that approaches constant endstates with a sufficient asymptotic rate, we relate the Maslov and Morse indices for Schr\"odinger operators on $\mathbb{R}$. In particular, we show that with our choice of convention, the Morse index is precisely the negative of the Maslov index.

We discuss a definition of the Maslov index for Lagrangian pairs on $\mathbb{R}^{2n}$ based on spectral flow, and develop many of its salient properties. We provide two applications to illustrate how our approach leads to a straightforward analysis of the relationship between the Maslov index and the Morse index for Sch\"odinger operators on $[0,1]$ and $\mathbb{R}$.

Assuming a symmetric potential and separated self-adjoint boundary conditions, we relate the Maslov and Morse indices for Schr\"odinger operators on $[0, 1]$. We find that the Morse index can be computed in terms of the Maslov index and two associated matrix eigenvalue problems. This provides an efficient way to compute the Morse index for such operators.

Given a star-shaped bounded Lipschitz domain $\Omega\subset{\mathbb R}^d$, we consider the Schr\"odinger operator $L_{\mathcal G}=-\Delta+V$ on $\Omega$ and its restrictions $L^{\Omega_t}_{\mathcal G}$ on the subdomains $\Omega_t$, $t\in[0,1]$, obtained by shrinking $\Omega$ towards its center. We impose either the Dirichlet or quite general Robin-type boundary conditions determined by a subspace ${\mathcal G}$ of the boundary space $H^{1/2}(\partial\Omega)\times H^{-1/2}(\partial\Omega)$, and assume that the potential is smooth and takes values in the set of symmetric $(N\times N)$ matrices. Two main results are proved: First, for any $t_0\in(0,1]$ we give an asymptotic formula for the eigenvalues $\lambda(t)$ of the operator $L^{\Omega_t}_{\mathcal G}$ as $t\to t_0$ up to quadratic terms, that is, we explicitly compute the first and second $t$-derivatives of the eigenvalues. This includes the case of the eigenvalues with arbitrary multiplicities. Second, we compute the first derivative of the eigenvalues via the (Maslov) crossing form utilized in symplectic topology to define the Arnold-Maslov-Keller index of a path in the set of Lagrangian subspaces of the boundary space. The path is obtained by taking the Dirichlet and Neumann traces of the weak solutions of the eigenvalue problems for $L^{\Omega_t}_{\mathcal G}$., Comment: arXiv admin note: substantial text overlap with arXiv:1408.1103

We study the spectrum of Schr\"odinger operators with matrix valued potentials utilizing tools from infinite dimensional symplectic geometry. Using the spaces of abstract boundary values, we derive relations between the Morse and Maslov indices for a family of operators on a Hilbert space obtained by perturbing a given self-adjoint operator by a smooth family of bounded self-adjoint operators. The abstract results are applied to the Schr\"{o}dinger operators with quasi-periodic, Dirichlet and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for the multidimensional Schr\"odinger operators with periodic potentials. For quasi convex domains in $\mathbb{R}^n$ we recast the results connecting the Morse and Maslov indices using the Dirichlet and Neumann traces on the boundary of the domain., Comment: 36 pp

We study the Schr\"odinger operator $L=-\Delta+V$ on a star-shaped domain $\Omega$ in $\mathbb{R}^d$ with Lipschitz boundary $\partial\Omega$. The operator is equipped with quite general Dirichlet- or Robin-type boundary conditions induced by operators between $H^{1/2}(\partial\Omega)$ and $H^{-1/2}(\partial\Omega)$, and the potential takes values in the set of symmetric $N\times N$ matrices. By shrinking the domain and rescaling the operator we obtain a path in the Fredholm-Lagrangian-Grassmannian of the subspace of $H^{1/2}(\partial\Omega)\times H^{-1/2}(\partial\Omega)$ corresponding to the given boundary condition. The path is formed by computing the Dirichlet and Neumann traces of weak solutions to the rescaled eigenvalue equation. We prove a formula relating the number of negative eigenvalues of $L$ (the Morse index), the signed crossings of the path (the Maslov index), the number of negative eigenvalues of the potential matrix evaluated at the center of the domain, and the number of negative squares of a bilinear form related to the boundary operator., Comment: 62 pages, 1 figure. Second version: typos corrected; additional references added

We study the linear stability of localized modes in self-interacting spinor fields, analyzing the spectrum of the operator corresponding to linearization at solitary waves. Following the generalization of the Vakhitov--Kolokolov approach, we show that the bifurcation of real eigenvalues from the origin is completely characterized by the Vakhitov--Kolokolov condition $dQ/d\omega=0$ and by the vanishing of the energy functional. We give the numerical data on the linear stability in the generalized Gross--Neveu model and the generalized massive Thirring model in the charge-subcritical, critical, and supercritical cases, showing the agreement with the Vakhitov--Kolokolov and the energy vanishing conditions., Comment: 17 pages. Earliner application to Dirac--Maxwell was flawed and led to incorrect conclusions