All Questions

Let us act intentionally stupid and assume we do not know that we can solve for the spectrum of the harmonic oscillator $$-\frac{d^2}{dx^2}+x^2$$ explicitly. Is there an abstract argument why the ...

In a previous question on mathoverflow, I asked about the following: Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions. The ...

Let $g\in C(\Bbb R)$ be given, we want to find a solution $f\in C(\Bbb R)$ of the equation $$ f(x+1) + f(x) = g(x). $$ We may rewrite the equation using the right-shift operator $(Tf)(x) = f(x+1)$...

Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem. For non-normal operators this no longer has to be true. There ...

I make the following observation: Let $\Delta^{(n)}$ be the discrete Laplacian on $\mathbb{C}^n$ (ie the $n\times n $ matrix with diagonal $-2$ and upper/lower diagonal $1$.) This one has eigenvalues ...

Let $T$ be an (unbounded) self-adjoint operator. Assume that there is a bounded operator $S$ such that $TS=ST.$ For which kind of $f$ do we have that $f(T)S=Sf(T)?$ My thought was that using a ...