# Tagged Questions

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### A vector space with countable and uncountable basis at the same time

Let $V$ be a vector space over $\mathbb{C}$. Two self-adjoint, commutable linear operators $\xi$ and $\eta$ act on it. Both of their eigenvectors form a complete set of $V$, but $\xi$'s eigenvalues ...
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### What is the smallest non-trivial Hilbert space?

I came to know without proof or explanation that smallest non-trivial Hilbart space is generated by two basis vectors. What is its proof? One example I know. Denote $a = (0 , 1)$ and $b = (1 , 0)$. ...
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### Does the shift operator on $\ell^2(\mathbb{Z})$ have a logarithm?

Consider the Hilbert space $\ell^2(\mathbb{Z})$, i.e., the space of all sequences $\ldots,a_{-2},a_{-1},a_0,a_1,a_2,\ldots$ of complex numbers such that $\sum_n |a_n|^2 < \infty$ with the usual ...
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### Basic Quantum Mechanics Concepts with Continuous Spectra

The following are a couple excerpts of the first chapter of Sakurai and Napolitano, Modern Quantum Mechanics, 2nd edition: Prior to these formulas, the text discusses the fundamental mathematics ...
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### Book on periodic Schrödinger operators

I am looking for good books about the spectral theory of periodic (1-dimensional) Schrödinger operators on a compact interval. A good reference I found was Reed/Simon Analysis of Operators (and a ...
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Let $H$ be a diagonalizable matrix (not necessarily Hermitian). Then, it induces a biorthogonal left and right vectors, such that $$... 0answers 18 views ### When can we get discrete spectrum? Suppose that T is a densely defined closed operator on a separable Hilbert space H. Form N = T^*T. Assume further that T has a finite dimensional kernel and satisfies the commutation relation ... 1answer 50 views ### Domain of the quantum free Hamiltonian in 1D Consider the quantum free Hamiltonian H_0 =-\frac{d^2}{dx^2} (the Laplacian on the real line). I want to show that it is (essentially) self-adjoint in its domain of definition. The usual approach ... 1answer 80 views ### Spectral theorem in Quantum Mechanics In Quantum Mechanics one often looks at self-adjoint(unbounded and closed) linear operators A,B that are defined dense on L^2. My question is: Is it true that if we have [A,B]=0, where [,] is ... 2answers 80 views ### Selfadjointness of Coulomb Hamiltonian in d>=3 dimensions I know that the Coulomb-Hamiltonian H=-\Delta - |\cdot|^{-1} is self-adjoint with dom(H)=H^2(\mathbb R^3). This follows by the Kato-Rellich-Theorem. Has the corresponding quadratic form a form ... 1answer 30 views ### Continous and Discrete basis, Multiplication of Density Matrix and Hamiltonian. Suppose I have a wave function \psi(x) in position basis. I can make a density function by simply multiplying \psi(x) and its conjugate \psi^*(x). If I operate the density matrix ... 4answers 340 views ### Correct spaces for quantum mechanics The general formulation of quantum mechanics is done by describing quantum mechanical states by vectors |\psi_t(x)\rangle in some Hilbert space \mathcal{H} and describes their time evolution by ... 2answers 181 views ### Can the 0-norm represent determinism? In Scott Aaronson's Quantum Computing since Democritus, he presents classical probability theory as based on the 1-norm, and QM as based on the 2-norm. Call \{v_1,\ldots,v_N\} a unit vector ... 0answers 47 views ### Doubt about the spectrum of an operator I consider the Laplacian operator$$A=-\Delta$$in the domain$$H^2(\mathbb{R}^3)$$where it is selfadjoint. We know that its spectrum is [0,+\infty). Now I want to consider the restriction of A ... 0answers 70 views ### Fourier transform of integral kernel of the free resolvent The free resolvent in \mathbb{R}^3 has this rapresentation$$(R_0(z)f)(x)=\int_{\mathbb{R}^3}\frac{e^{i\sqrt{z}|x-y|}}{4\pi|x-y|}f(y)dy$$with \Im \sqrt{z}>0. Then its integral kernel is ... 1answer 63 views ### Is multiplying by a measurable function V always self-adjoint? There are a handful of results establishing conditions on the measurable real-valued function V(x) under which the operator:$$-\Delta + V(x) Is (essentially) self-adjoint on ...
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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 ...
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### 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 ...