Spectral Theory of Laplace–Beltrami Operators with Periodic Metrics

The spectra of Laplace–Beltrami operators with periodic metrics has been less investigated than that of Schrödinger operators with a periodic potentials, and there are many differences between these two cases. It has been established that the spectrum of a Laplace–Beltrami operator with periodic metric is the union of closed intervals and that spectral gaps are possible, but whether an infinite number of spectral gaps is possible is an open question. Utilizing special transformations it is shown that there are two-dimensional Laplace–Beltrami operators having an arbitrarily large number of spectral gaps by proving that a particular two-dimensional operator has an infinite number of spectral gaps. In the case of a periodic conformal metric relationships between the scalar curvature and the number of gaps are investigated.