Optimization of quantum graph eigenvalues with preferred orientation vertex conditions
Pavel Exner, Jonathan Rohleder
TL;DR
The paper investigates Laplacian eigenvalue optimization on finite metric graphs with time-reversal breaking vertex couplings, introducing a nonstandard vertex condition that yields a tractable quadratic form and self-adjoint operator. It proves that, under a fixed total length, the ground state on a star graph is maximized by an equilateral configuration with 3 or 4 edges depending on parity, using surgery (transplantation) arguments to compare edge-length distributions. For general graphs, it establishes sharp universal eigenvalue bounds, notably $\lambda_1(\Gamma) \le -1$ and $\lambda_{2k+1}(\Gamma) \le \frac{4k^2\pi^2}{L(\Gamma)^2}$, with equality realized by the equilateral figure-8; the bounds are saturated and demonstrate that the operator does not scale simply with graph size. A Dirichlet-to-Neumann framework is developed to derive negative eigenvalue conditions on general graphs, and the results extend to Dirichlet-endpoint variants of star graphs. Overall, the work provides novel spectral optimization tools for nonstandard vertex couplings and reveals how graph surgery can yield sharp, size-invariant bounds.
Abstract
We discuss Laplacian spectrum on a finite metric graph with vertex couplings violating the time-reversal invariance. For the class of star graphs we determine, under the condition of a fixed total edge length, the configurations for which the ground state eigenvalue is maximized. Furthermore, for general finite metric graphs we provide upper bounds for all eigenvalues.