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Spectrum, resolvent, numerical range, functional calculus, operator semigroups. Special classes of operators: compact, Fredholm, dissipative, differential, integral, pseudodifferential, etc.

4 votes
1 answer
178 views

Bound in terms of harmonic oscillator

I wonder if the following is true: Let $\alpha >0$ be a positive real number, do we have $$\Vert H^{\alpha} \psi''\Vert \le \Vert H^{\alpha+1} \psi\Vert,$$ where $H = -\frac{d^2}{dx^2} + x^2$ is the q …
António Borges Santos's user avatar
4 votes
1 answer
271 views

Schroedinger operator in 2 dimensions with singular potential

Consider the Schroedinger operator $$H = -\Delta + \frac{c}{\vert x \vert^2}$$ in two dimensions with $c >0$ This operator has a self-adjoint realization, since it is a positive symmetric operator on …
António Borges Santos's user avatar
6 votes
2 answers
456 views

Spectrum of operator involving ladder operators

The ladder operator in quantum mechanics are the operators $$a^\dagger \ = \ \frac{1}{\sqrt{2}} \left(-\frac{d}{dq} + q\right)$$ and $$a \ \ = \ \frac{1}{\sqrt{2}} \left(\ \ \ \!\frac{d}{dq} + q\rig …
António Borges Santos's user avatar