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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 …

I would like to know how one solves Sturm-Liouville problems on $(0,\infty)$ NUMERICALLY for the eigenvalues that are of the form $$-f''(x)+\frac{1}{\sinh(x)^2}f(x)=\lambda f(x).$$ So even if there …

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 strat …

An operator $A$ is called dissipative if for all $x \in D(A)$ and $\lambda >0$ $$ \left\lVert (A-\lambda)x \right\rVert \ge \lambda \left\lVert x \right\rVert.$$ On a Hilbert space this is equivalen …

Let $A \in L(X,Y)$ be a bounded operator between Banach spaces. Then its dual operator $A' \in L(Y',X')$ has the same spectrum as $A$ by the closed range theorem. Now, if we have an unbounded operato …

I just saw the following question that was asked yesterday on math overflow on meromorphic solutions to ODEs Although, I understand the answers and comments to the questions, I did not understand how …