Semiclassical eigenvalues and shape problems on surfaces of revolution

Abstract

Mark Kac [‘‘Can one hear the shape of drums?’’ Am. Math. Monthly 73, 1–23 (1966)] asked if the shape of a region Ω⊆Rn could be determined from its sound (spectrum of the Laplacian ΔΩ). He proved the conjecture for special classes of domains, including polygons and balls in Rn. Similar problems could be raised in other geometric contexts, including ‘‘shape of metric’’ for Laplacians on manifolds or ‘‘shape of potential’’ for Schrödinger operators HV=Δ+V. The latter two problems for surfaces of revolution Σ are addressed. An explicit reconstruction procedure will be outlined that leads from the joint spectrum of H=Δ or Δ+V and the angular momentum algebra so(n) to the ‘‘shape of Σ’’ and ‘‘V,’’ respectively. The metric result applies to generic surfaces of revolution, while the Schrödinger result allows all zonal (axisymmetric) potentials V on Σ. So the work extends the Kac’s ‘‘n-ball’’ result as well as the n-sphere ‘‘zonal Schrödinger theory’’ [D. Gurarie, (a) ‘‘Averaging methods in spectral theory of Schrödinger operators,’’ Maximal Principles and Eigenvalues Problems in Differential Equations, Pitman Research Notes Vol. 175, edited by P. W. Schaefer (Pitman, New York, 1980), pp. 167–77; (b) ‘‘Inverse spectral problem for the two-sphere Schrödinger operators with zonal potential,’’ Lett. Math. Phys. 16, 313–323 (1990); (c) ‘‘Zonal Schrödinger operators on the n-sphere: Inverse spectral problem and rigidity,’’ Commun. Math. Phys. 131, 571–603 (1990)].