# Questions tagged [eigenvalue]

A linear operator (including a matrix) acting on a non-zero *eigenvector* preserves its direction but, in general, scales its magnitude by a scalar quantity *λ* called the *eigenvalue* or characteristic value associated with that eigenvector. Even though it is normally used for linear operators, it may also extend to nonlinear operations, such as Schroeder functional composition, which evoke linear operations.

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### Eigenvectors of a 4D rotation, and their interpretation

Let us define a 4D rotation by using two unit quaternions: $$\mathring{q}_l=\frac{a+ib+jc+kd}{\left|a+ib+jc+kd\right|}$$ and $$\mathring{q}_r=\frac{e+ib+jc+kd}{\left|e+ib+jc+kd\right|}.$$ They differ ...
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### Discreteness of set of energy eigenvalues

Given some potential $V$, we have the eigenvalue problem $$-\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi$$ with the boundary condition $$\lim_{|x|\rightarrow \infty} \psi(x) = 0$$ If we ...
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### Eigenvalue $a_n$

Q1: In Zetilli's book page 166 (ch. "Postulates of QM", eq. 3.1) i encountered an expression $\hat{A}|\psi\rangle = a_n|\psi_n\rangle$. I know this is an eigenvalue equation, but i have seen another ...
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### NP-completeness of non-planar Ising model versus polynomial time eigenvalue algorithms

From the papers by Barahona and Istrail I understand that a combinatorial approach is followed to prove the NP-completeness of non-planar Ising models. Basic idea is non-planarity here. On the other ...
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### How do we know that $\psi$ is the eigenfunction of an operator $\hat{H}$ with eigenvalue $W$?

I am kind of new to this eigenvalue, eigenfunction and operator things, but I have come across this quote many times: $\psi$ is the eigenfunction of an operator $\hat{H}$ with eigenvalue $W$. ...
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### Geometrical interpretation of complex eigenvectors in a system of differential equations

Let's consider a system of differential equations in the form $$\dot{X} = M X$$ in two dimensions ($X = (x(t), y(t))$). In the case that $M$ has real values, it is easy to give a geometric ...
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### Quantum graph theory: complex spectra

In quantum graph theory, what are the properties of a given graph to own complex conjugated complex eigenvalues, either finite or infinite? Spectral graph theory is as far as I know a not completely ...
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### Spectral properties of CFT

What are the general spectral properties of CFT? I mean what is the "spectrum"/eigenvalues of CFT in 2d and d>2 spacetime dimensions? I understand the "spectrum" and "Fock space" realization of Dirac ...
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### Mysterious spectra?

In my blog post Why riemannium? , I introduced the following idea. The infinite potential well in quantum mechanics, the harmonic oscillator and the Kepler (hydrogen-like) problem have energy spectra, ...
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### Physical meaning of Laplace-Beltrami eigenfunctions?

The eigenfunctions of Laplace-Beltrami operator are often used as the basis of functions defined on some manifolds. It seems that there is some kind of connection between eigen analysis of Laplace-...
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### Lower bounds on spectral gaps of ferromagnetic spin-1/2 XXX Hamiltonians?

Question. Are there any references or techniques which can be applied to obtain energy gaps for ferromagnetic XXX spin-1/2 Hamitlonians, on general interaction graphs, or tree-graphs? I'm interested ...
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Typically one writes simultaneous eigenstates of the angular momentum operators $J_3$ and $J^2$ as $|j,m\rangle$, where $$J^2|j,m\rangle = \hbar^2 j(j+1)|j,m\rangle$$ $$J_3 |j,m\rangle = \hbar m|j,m\... 2answers 2k views ### Non-Degeneracy of Eigenvalues of Number Operator for Simple Harmonic Oscillator [duplicate] Possible Duplicate: Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate? I'm trying to convince myself that the eigenvalues n of the number operator N=a^{\dagger}a for ... 1answer 403 views ### Perturbation method & eigenvalues I have a problem but I don't understand the question. It says: "Show that, to first order in energy, the eigenvalues ​​are unchanged." What does it mean? It means that if the Hamiltonian has the ... 1answer 52 views ### growth condition for the potential what growth conditions should the potential inside the Hamiltonian  H=p^{2}+ V(x)  has in order to get ALWAYS a discrete spectrum ?? for example how can we know for teh cases  |x|^{a}  ,  exp(... 2answers 6k views ### Using eigenvalues to determine the stability/behaviour of the system first time I've been on physics.se but have used the math and cs before... Anyway, here's my question: If we have a damped pendulum described by the equation$$y'' + ay' + b = 0 , a>0$$Using the ... 4answers 12k views ### Why do we use Hermitian operators in QM? Position, momentum, energy and other observables yield real-valued measurements. The Hilbert-space formalism accounts for this physical fact by associating observables with Hermitian ('self-adjoint') ... 0answers 62 views ### Random quantum systems with asymmetric Lifshitz tails? For a quantum mechanical system with a periodic Hamiltonian (Schrödinger operator) H, let N(E) be its integrated density of states, i.e. the fraction of eigenvalues in the spectrum \sigma(H) ... 1answer 405 views ### Random Hankel matrix and eigenvalues distribution I would like to know if there are any theoretical results on the distribution of the eigenvalues of Hankel matrices. I seek a result like the Marchenko–Pastur distribution for random matrices. 4answers 1k views ### Why are the inner products of the eigenfunctions of an operator with a discrete eigenvalue spectrum guaranteed to exist? I was reading through a textbook, and the statement was made that the inner products are guaranteed to exist if the eigenvalue spectrum of the operator is discrete. I have come across no support for ... 1answer 123 views ### When does the “norm of quasi-eigenvectors” matter in calculations? For which physical results are these even used? Which physical system in nonrelativistic quantum mechanics is actually described by a model, where the norm of the "position eigenstate" (i.e. the delta distribution as limit of vectors in the Hilbert ... 1answer 146 views ### Are Quantum Physics and statistical theory always the same as semiclassical approximations? Quantum Mechanics and Statistical physics is a bit hard , could we then study only the WKB approximation ? In the form: replace  \sum_{n=0}^{\infty}exp(- \beta E_{n})=Z(\beta)\sim\iint dxdpexp(-p^{... 1answer 260 views ### Inverse of a sum of two easy matrices Let A be a symmetric positive semidefinite matrix and I the identity matrix. Given the linear equation$$ y = A(A + \sigma^2I)^{-1} x $$Write A in terms of its eigenvectors |u_i\rangle,$$...
Let two Hamiltonians $H_{1}$ and $H_{2}$ be defined in such a manner that their eigenvalue staircases satisfy $N_{1} (E) = N_{2} (E)+A +O(E^{-1})$ What can we say about their potentials \$ V_{1} (x)...