The Dirichlet/Neumann spectrum determines the domain among the class of analytic domains with some discrete symmetries by work of Zeldich.

The unit disk is determined by its Dirichlet spectrum (among any region, with sufficiently regular boundary; I'm not sure what the minimal assumptions are): First, by Weyl's law, "you can hear the area of a drum" (i.e., the area is a spectral invariant). Then, the claim follows from the rigidity statement in the Faber--Krahn inequality.

The Dirichlet/Neumann spectrum determines the domain among the class of analytic domains with some discrete symmetries by work of Zelditch.

The unit disk is determined by its Dirichlet spectrum (among any region, with sufficiently regular boundary; I'm not sure what the minimal assumptions are): First, by Weyl's law, "you can hear the area of a drum" (i.e., the area is a spectral invariant). Then, the claim follows from the rigidity statement in the Faber--Krahn inequality.

The spectrum determines the domain among the class of analytic domains with some discrete symmetries by work of Zeldich.

The unit disk is determined by its spectrum (among any region, with sufficiently regular boundary; I'm not sure what the minimal assumptions are): First, by Weyl's law, "you can hear the area of a drum" (i.e., the area is a spectral invariant). Then, the claim follows from the rigidity statement in the Faber--Krahn inequality.

The Dirichlet/Neumann spectrum determines the domain among the class of analytic domains with some discrete symmetries by work of Zeldich.

The unit disk is determined by its Dirichlet spectrum (among any region, with sufficiently regular boundary; I'm not sure what the minimal assumptions are): First, by Weyl's law, "you can hear the area of a drum" (i.e., the area is a spectral invariant). Then, the claim follows from the rigidity statement in the Faber--Krahn inequality.