All Questions
22
questions
2
votes
1
answer
92
views
About the spectrum family of the multiplication operator
Let $<,>$ be the inner product of $L^2(\mathbb{R})$.
For a measurable function $F:\mathbb{R}^d\to\mathbb{R}$, we define
a multiplication operator $M_F:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$, ...
- 61
2
votes
1
answer
202
views
Spectrum of a sum of self-adjoint operators
This is a "sequel" to that question where I explain why I need the spectrum of an operator given as the sum of a convolution and a function multiplication. Here, I am considering the ...
- 1,139
2
votes
1
answer
77
views
If $0\leq A\leq B$ then the corresponding densities satisfy $0\leq\rho_A\leq \rho_B$
Let $A,B$ on $L^2(\mathbb R^d,\mathbb C^d)$ such that $0\leq A\leq B$ AND $A,B$ are trace-class. Then their densities are given by $$\rho_A(x):=\sum_jA\varphi_j(x)\overline{\varphi_j(x)},\qquad \rho_A(...
- 461
1
vote
1
answer
65
views
An inequality for Schatten norms with a compact self-adjoint operator
Let $Q$ be a compact self-adjoint operator on $L^2(\mathbb R,\mathbb C)$. Notice that $(1-\Delta)^{s}$ is a positive (and hence self-adjoint) operator on $H$ for any $s>0$.
We denote by $\mathfrak ...
- 461
5
votes
1
answer
191
views
Problem on a rank 1 perturbation of an self-adjoint operator
With my teacher we have been trying to solve for a while a problem that consists of two parts, and each one in three sections. The first part, which has to do with the problems that I was able to ...
- 160
2
votes
1
answer
139
views
Convergence of $e^{itH}$ as a sum for $H$ and unbounded operator
Let $H$ be an unbounded operator on a Hilbert space $\mathcal{H}$. We may define the propagator $U(t)=e^{itH}$ via the functional calculus. However, if $H$ were bounded, the series $\sum_{k=0}^\infty \...
- 3,738
0
votes
0
answers
558
views
Spectrum of momentum operator of quantum physics
I am stuck with the following problem:
Given the momentum operator of quantum mechanics $Af = - if'$ where the domain $D(A)$ consists of functions $f \in L^2(\mathbb{R})$ that are absolutely ...
- 51
1
vote
1
answer
78
views
Where precisely is topology required to prove the existence of non-zero eigenstates in the proof of the spectral theorem?
My professor today remarked that the proof of the spectral theorem (even for the discrete spectrum case) uses not just algebra but also topology to prove the existence of eigenstates. However, I'm not ...
user568976
4
votes
1
answer
903
views
Stone's theorem and the spectral theorem
I am struggling to formally derive the expression found in the Stone's theorem for one-parameter unitary groups. I am aware that this can be done by using the spectral theorem. I am mostly interested ...
- 43
1
vote
1
answer
520
views
Is the resolution of the identity 'unique'?
I want to know how many ways the identity operator $I$ on a (finite) Hilbert space $\mathcal{H}$ can be written as sum of outer products of states like $|\psi_i\rangle\langle \psi_i|$.
For example, $...
- 21
1
vote
3
answers
3k
views
Proving that every diagonalizable operator is normal
This question is insipired by the proof of the Spectral Theorem in Nielsen and Chuang's book (page 72). It says:
Theorem: Any normal operator $M$ on a vector space $V$ is diagonal with respect to ...
- 1,862
1
vote
0
answers
74
views
No embedded point spectra for (discrete) Schrodinger operators with compactly supported potential
Consider the lattice $\mathbb{Z}^d$ and let $H_0$ denote the (negative) Laplacian on $l^2(\mathbb{Z}^d)$ with spectrum $[-2d,2d]$. Suppose that I add a potential $q:\mathbb{Z}^d\rightarrow\mathbb{R}$ ...
- 1,035
4
votes
2
answers
349
views
Why is the magnetic Schrödinger operator positive?
In the book Schrödinger Operators by Cycon et al. they prove that the magnetic Schrödinger operator (as well as the Pauli operator) have essential spectrum $\sigma_{ess} = [0,\infty)$ if $B$ has decay ...
- 1,405
0
votes
0
answers
145
views
Simple proof that density operator has only pure point spectrum
Is there any simple proof that self-adjoint, nonnegative and trace one operators have only point spectrum? Namely, we want to show that if a bounded operator $\rho$ on $\mathcal{H}$ is such that
$$
\...
- 1,017
1
vote
0
answers
64
views
Let $\mathcal{H}$ be the Schrödinger operator. Why is $u(x,t)\xrightarrow{t\to\pm\infty} 0\iff\sigma_p(\mathcal{H})=\emptyset$?
Let $\mathcal{H}:=V-\Delta$ be the Schrödinger operator, $V\in C^\infty_0(\mathbb{R})$ and let $u\in L^2(\mathbb{R})$ be a solution of the equation
$$\left(\Delta-\frac{\partial^2}{\partial t^2}\...
- 1,038