Spectral-Theory on Chen Kai Blog

Functional Analysis (9): Unbounded Operators — When Boundedness Fails

Sun, 17 Oct 2021 09:00:00 +0000

Two articles ago I was talking about how spectral theory is the linear-algebraic infrastructure of quantum mechanics. The trouble is that nearly every operator a physicist actually cares about – the position operator, the momentum operator, the Laplacian, the Schrodinger Hamiltonian – is not bounded. They are not defined on the whole Hilbert space. They are densely defined, with domains that depend on the regularity or decay of the input function. None of the previous spectral apparatus applies directly. We need to extend it.

Functional Analysis (8): Spectral Theory — Decomposing Operators

Fri, 15 Oct 2021 09:00:00 +0000

When I first saw the word “spectrum” used for an operator I assumed it was a fancy synonym for “set of eigenvalues.” That is the right intuition for matrices and for compact operators, and it is exactly what one wants in introductory linear algebra. The trouble is that it is wrong as soon as the operator is not compact. The position operator $$(Mf)(x) = x f(x)$$ on $$L^2[0, 1]$$ has no eigenvalues: any eigenfunction would have to satisfy $x f(x) = \lambda f(x)$ a.e., which forces $$f = 0$$ everywhere away from a single point, hence $$f = 0$$ in $$L^2$$ . And yet the operator is clearly not invertible, since $\lambda I - M$ is multiplication by $x - \lambda$ , which fails to be boundedly invertible whenever $\lambda \in [0, 1]$ .

Functional Analysis (7): Compact Operators — The Bridge to Finite Dimensions

Wed, 13 Oct 2021 09:00:00 +0000

I owe my fondness for compact operators to a small embarrassment. As an undergraduate I assumed that infinite-dimensional linear algebra would feel exotic everywhere. It does not. There is a wide and well-mapped suburb of operator theory in which everything one learned about symmetric matrices – eigenvalues, orthogonal eigenvectors, the spectral decomposition – comes back almost unchanged, just with eigenvalues tailing off to zero instead of a finite list. That suburb is the world of compact operators, and the price of admission is a single condition: the operator must squeeze the unit ball into a relatively compact set. Once that condition is met, nearly everything follows: the spectrum is countable, nonzero eigenvalues are isolated with finite-dimensional eigenspaces, the Fredholm alternative holds, and integral equations of the second kind become as tractable as linear systems. The line between matrices and infinite-dimensional operators ceases to be a wall and becomes a permeable membrane.