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We investigate the representation of the geometrical information of theuniverse in terms of the eigenvalues of the Laplacian defined on the universe.We concentrate only on one specific problem along this line: To introduce aconcept of distance between universes in terms of the difference in thespectra.  We can find out such a measure of closeness from a general discussion. Thebasic properties of this `spectral distance' are then investigated. It can berelated to a reduced density matrix element in quantum cosmology. Thus,calculating the spectral distance gives us an insight for the quantumtheoretical decoherence between two universes. The spectral distance does notin general satisfy the triangular inequality, illustrating that it is notequivalent to the distance defined by the DeWitt metric on the superspace.  We then pose a question: Whether two universes with different topologiesinterfere with each other quantum mechanically? We concentrate on thedifference in the orientabilities. Several concrete models in 2-dimension areset up, and the spectral distances between them are investigated: Tori andKlein's bottles, spheres and real projective spaces. Quite surprisingly, wefind many cases of spaces with different orientabilities in which the spectraldistance turns out to be very short. It may suggest that, without any otherspecial mechanism, two such universes interfere with each other quite strongly.Comment: 47 page

Spectral representation of the spacetime structure: The ‘‘distance’’ between universes with different topologies