Definition of discrete and continuous spectrum

Question

I am reading about automorphic forms for $GL(2)$ and I am having trouble understand the definition of "spectrum." For instance, in Goldfeld and Hundley's book on automorphic representations they write, "The continuous spectrum of $\Delta_k$ (the Laplace operator) is spanned by the Eisenstein series $E_\frak{a}$. However, in the references I have consulted (e.g. these notes by Garrett: http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/06_ops_hsp.pdf), the spectrum of an operator is defined to be a certain set of the complex numbers. In Bump's book on automorphic forms, he says that the Laplacian on $L^2(\mathbb{R})$ has a continuous spectrum, but doesn't define the term "continuous spectrum." It seems from context that an operator has a discrete spectrum if its eigenvalues form a discrete subset of the complex numbers, while it has a continuous spectrum if its eigenfunctions can be parametrized by a continuous subset of $\mathbb{R}$, but I am almost certain that this is wrong. So, my question is "What is the definition of spectrum, discrete spectrum, and continuous spectrum" in this context?

Answer 1

Yes, the use of "spectrum" is ambiguous, but not "in a bad way". Certainly warranting this question, yes.

As in the standard remark that the Laplacian on $L^2(\mathbb R)$ "has continuous spectrum", what is meant is partly the obvious negative, that there are no $L^2$ eigenvalues at all, but, nevertheless, Fourier inversion does express nice functions as integrals of non-$L^2$ eigenfunctions (the exponentials $x\to e^{i\xi x}$). Operationally, the point is that there are no $L^2$ eigenfunctions (no "discrete spectrum"), but there is still a (useful!) spectral decomposition-and-synthesis in terms of eigenfunctions from a (slightly) larger space.

In greatest generality, I don't think there is any terribly useful-practical generalization to abstract scenarios... despite "spectral theorems" for various sorts of operators on Hilbert spaces.

One slightly-useful incarnation is in the idea of "Hilbert integrals" of Hilbert spaces. Perhaps this is most compelling in the "integral of representations" version, so that we are expressing a representation of some reductive group, maybe a Lie group like $SL_2(\mathbb R)$, as a sum-and-integral of irreducibles. The same idea applies "even" to representations of just a single operator such as the Laplacian, but certain starknesses of the story seem (to me) to generate considerable cognitive dissonance.

Ignoring that route for a moment, the powerful operational sense of "presence of continuous spectrum" consists of: denial that there is a basis of (genuine) eigenfunctions, and, in cases of interest, explicit description of "generalized eigenfunctions", and explicit description of how vectors in the Hilbert space are expressible as sums of genuine eigenfunctions, and integrals of "generalized" eigenfunctions. In practice, having a Plancherel theorem for such expression is essential, or it's nearly useless.

Gelfand's school did some work abstracting "generalized eigenfunctions", and, indeed, looking at tangible Hilbert spaces of functions of various sorts, there is intuitive suggestion of larger spaces in which "missing" eigenfunctions visibly lie. In practice, I think that the abstract forms of this are not particularly helpful in understanding specific situations. Indeed, one might reasonably claim that the substance and subtlety of many specific situation where "there is continuous spectrum" resides in alternative, explicit identification of those "generalized eigenfunctions". The abstractions for this give no useful prescription!

In the various cases of automorphic forms here-and-there, the fact that integrals of Eisenstein series span the orthogonal complement to cuspforms is not obvious, not easy, and the corresponding Plancherel is certainly not obvious. But these things are provable, justifying vague or ambiguous, but descriptive, terminology.

That is, in practice, especially for automorphic forms, "continuous spectrum" means that beyond discrete spectrum (not merely cuspforms, in general, but also all Speh forms, namely, square-integrable residues of Eisenstein series with cuspidal data associated to Levi components... for $GL_2$, these are essentially just constants, which are prosaic-enough to allow us to overlook the highly non-trivial larger issue) integrals of natural-enough Eisenstein series, not in $L^2$, fill out the rest of $L^2$.