Tagged Questions

3
votes
2answers
115 views

Can the $0$-norm represent determinism?

In Scott Aaronson's Quantum Computing since Democritus, he presents classical probability theory as based on the $1$-norm, and QM as based on the $2$-norm. Call $\{v_1,\ldots,v_N\}$ a unit vector ...
2
votes
0answers
39 views

Doubt about the spectrum of an operator

I consider the Laplacian operator $$A=-\Delta$$ in the domain $$H^2(\mathbb{R}^3)$$ where it is selfadjoint. We know that its spectrum is $[0,+\infty)$. Now I want to consider the restriction of $A$ ...
1
vote
0answers
46 views

Fourier transform of integral kernel of the free resolvent

The free resolvent in $\mathbb{R}^3$ has this rapresentation $$(R_0(z)f)(x)=\int_{\mathbb{R}^3}\frac{e^{i\sqrt{z}|x-y|}}{4\pi|x-y|}f(y)dy$$with $\Im \sqrt{z}>0$. Then its integral kernel is ...
2
votes
1answer
36 views

Is multiplying by a measurable function $V$ always self-adjoint?

There are a handful of results establishing conditions on the measurable real-valued function $V(x)$ under which the operator: $$-\Delta + V(x)$$ Is (essentially) self-adjoint on ...
6
votes
1answer
90 views

Kernel of adjoint operator

This problem is puzzling me, even though it should be really simple. Let $L=-\partial_x^2 + \frac 1 2 x^{-2}$ be an operator defined on $D(L)=C^\infty_c(0,+\infty)\subset L^2(0,+\infty)$. Its adjoint ...
3
votes
1answer
28 views

Essential selfadjointness preserved under unitarily transfomration?

I am wondering if essential selfadjointness of an operator in a Hilbert space is preserved under unitarily transformations. In other terms: let $H,H'$ be two isomorphic Hilbert spaces, with an ...
0
votes
1answer
48 views

Inequality from Von Neumann entropy.

I am looking over some old course notes. First, Von Neumann entropy is defined. The Von Neumann entropy of a system described by a density matrix $\rho$ is defined by $S(\rho)\equiv ...
2
votes
0answers
72 views

Defining entanglement in subspaces of tensor product

Let $\mathcal{H}=\mathbb{C}^n$ be a Hilbert space. A state $\rho\in\mathcal{B(H)}$ is a positive semi-definite operator with unit trace. $\rho\in \mathcal{B(H)}$, where ...
1
vote
1answer
119 views

Under What Conditions Does the Action of the Dual Space Induce an Hermitian Inner Product?

I'm starting to learn about Dirac notation in Quantum Mechanics, and am coming from a pure background. The notes I'm reading states that we assume that the action of the dual space on the state space ...