# Questions tagged [sp.spectral-theory]

Schrodinger operators, operators on manifolds, general differential operators, numerical studies, integral operators, discrete models, resonances, non-self-adjoint operators, random operators/matrices

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### Singular values of a matrix that its rows have (1) a tightly bounded angle, and (2) at least some norm

We're looking for a connection between a matrix's singular values and some information we hold about its rows. We wish to find a tight bound on the singular values. We know on the rows that they have (...
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### Massive dirac operator symmetric spectrum

Consider the Dirac operator $$H = \begin{pmatrix} m & -i\partial_z \\ -i\partial_{\bar z} & -m \end{pmatrix},$$ where $\partial_{\bar z}$ is the Cauchy-Riemann operator and $m \ge 0.$ It is ...
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### Existence of periodic solution to ODE

We shall consider the matrix-valued differential operator $$(L u)(x) :=u'(x) - \begin{pmatrix} 0 & \sin(2\pi x-\frac{\pi}{6})\\ - 2\sin(2\pi x+\frac{\pi}{6}) & 0 \end{pmatrix} u(x).$$ This is ...
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### Eigenvalues of Laplacian and eigenvalues of curvature operator

Let $(M^n,g)$ be a compact Riemannian manifold (without boundary). The symmetries of the curvature $R$ of (the Levi-Civita connection associated to) $g$ allow one to realise $R$ as a self-adjoint (...
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### Isolated eigenvalues of “bipartite” operators

Please note: This is a reformulation of a previous question of mine. The old question has been already answered to, so I prefer asking a new one. However, it looked like the old formulation did not ...
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### Detecting isolated eigenvalues from local spectral measures

Please note: This question has been edited after it became clear from Christian Remling's answer that the original formulation was far from what I really meant to ask. Let $T\ne 0$ be a self-adjoint ...
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### A gap in the proof of uniqueness of functional calculus based on a spectral theorem

This question considers the proof of a fundamental theorem of functional calculus, given in the book Spectral Theory - Basic Concepts and Applications by David Borthwick (Theorem 5.9). Firstly we have ...
Let $A$ be a self-adjoint (unbounded) operator on a separable Hilbert space $H$. From the following form of spectral theorem, we may define a functional calculus by $f(A)=Q^{-1} M_{f\circ \alpha} Q$. (...