# Questions tagged [spectral-theory]

Spectral theory is the study of generalized notions of eigenvalues and eigenvectors for linear operators in Banach spaces.

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### eigenvalues of contraction

Assume $A - B$ is a contraction, i.e., its spectral radius is smaller than $1$. We also assume $A$ is diagonalizable. I am trying to show that if $x^TB=0$, then $x^T$ can be written as a sum of ...
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### Spectrum of the laplacian outside of a compact

Let us consider $A$ a translation invariant lower semi-bounded operator on $L^2(\mathbb{R}^n)$ with domain $D(A)$ and with empty discrete spectrum (I exclude bound states). I have the following ...
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### Quantum Ergodic Theorem: why $\sqrt{-\Delta}$ is used instead of $-\Delta$?

I'm studying the proof of Quantum Ergodic Theorems in the book Partial Differential Equations II: Qualitative Studies of Linear Equations (3rd edition) by Michael E. Taylor. The book includes the ...
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### Spectrum of operator $T:l_1\to l_1, Tx = (0,0,x_5,2x_1,0,x_1+x_3,0,0,...)$

What is the spectrum of an operator $T: l_1 \to l_1$, $x = (x_1,x_2,...,x_n,...)$, $Tx = (0,0,x_5,2x_1,0,x_1+x_3,0,0,...)$? For $\lambda \ne 0$, equation $Tx=\lambda x$ doesn't have non-zero solutions....
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### Let $H$ be a Hilbert space and let $A \in \mathcal{B}(H)$ be such an operator that $A^3 = A^*$.

Let $H$ be a Hilbert space and let $A \in \mathcal{B}(H)$ be such an operator that $A^3 = A^*$. (a) Show that from $A^4 = 0$ it follows $A = 0$. (b) Find the eigenvalues of the operator $A$. (c) ...
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### Restrictions on a set to be the spectrum of a 1D (discrete) Schrödinger operator.

What restrictions are there on a compact set $E\subset\mathbb{R}$ for $E$ to be the spectrum of a bounded (discrete) Schrödinger operator on $l^2(\mathbb{Z})$? Is there a known necessary and ...
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### Thus $MV - VM = V^2$. So the spectrum of $V^2$ is $\sigma(V^2) = (\sigma(V))^2=0$. Why??

I am curious if this statement holds (it doesn't make much sense to me, but it was written in solutions in this form): $\sigma(A)=0\implies\sigma(A^2)=(\sigma(A))^2=0.$ Can anybody explain to me why ...
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### Spectral analysis for harmonic oscillator operator?

Let $L=-\frac{d^2}{dx^2}+x^2, x\in\mathbb R$, the one-dimensional harmonic oscillator; this is an unbounded self-adjoint operator acting in $L^2(\mathbb R)$. I am looking for a reference that deals ...
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### Determine the spectrum of the operator $A$. Is the spectrum composed only of eigenvalues of the operator $A$?

On the space $l^2$, define the linear operator $A$ with the prescription $A(x_1,x_2,x_3,x_4,...)=((x_1+x_2)/2,(x_2+x_3)/2,(x_3+x_4)/2,...)$. (a) Prove that $A$ is a bounded operator on $l^2$ and ...
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### Let H be a Hilbert space and $A \in B(H)$. Prove or disprove the following statements.

Let H be a Hilbert space and $A \in B(H)$. Prove or disprove the following statements: (a) If $||A|| = 1$, then $0 \in \sigma(A)$. (b) $\sigma(A^*) = \overline{\sigma(A)}$. Attempt: I actually have no ...
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