# 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

833 questions
Filter by
Sorted by
Tagged with
108 views

202 views

### First eigenvalue of the Laplacian on the traceless-transverse 2-forms

Let $(S^3/\Gamma, g)$ be a spherical space form with constant sectional curvature $1$, where $\Gamma$ is a finite subgroup of $SO(4)$ acting freely on $S^3$. Consider the first nonzero eigenvalue ...
50 views

### The non empty set of accumulation points of a bounded linear operator is the spectrum of another operator

Let $X$ be an infinite dimensional Banach space, and let $T \in L(X)$ such that the set of accumulation points of $T$ is non empty, i-e $\mbox{acc}\,\sigma(T)\neq 0.$\ Is there a Banach space $Y$ ...
69 views

### Existence of solution of a difference equation

Consider the operator $T$ acting from $L^2(0,1+r)$ to it self defined by $$Tu(x)=a1_{(0,1)}(x)u(x+r)+b1_{(1,1+r)}(x)u(x-1)$$ where $r\in (0,1)$ and $a,b$ are real, and $1_A(x)$ is the characteristic ...
165 views

41 views

### Convergence in the resolvent sense and spectral properties

Let $\{T_k\}_k, T$ be unbounded selfadjoint operators on a Hilbert space $H$. If $T_k\to T$ in the norm-resolvent sense, then for any $(a,b)\subset \mathbb R$ with $\{a,b\}\cap \sigma(T)=\emptyset$, ...
73 views

### Expressing the singular values of a 2-by-2 real-valued matrix by the norm of the two columns and the angle between them

I'm looking for an elegant way to show the following claim. Claim: Let $m_1, m_2 \in \mathbb{R}^2$ be the two columns of matrix $M \in \mathbb{R}^{(2 \times 2)}$. The singular values of the matrix are ...
161 views

### Eigenvalues of operator

In the question here the author asks for the eigenvalues of an operator $$A = \begin{pmatrix} x & -\partial_x \\ \partial_x & -x \end{pmatrix}.$$ Here I would like to ask if one can extend ...
140 views

### Explicit eigenvalues of matrix?

Consider the matrix-valued operator $$A = \begin{pmatrix} x & -\partial_x \\ \partial_x & -x \end{pmatrix}.$$ I am wondering if one can explicitly compute the eigenfunctions of that object on ...
141 views

### Relation between the spectrum of 𝐷𝐴 and 𝐴 where 𝐴 is a bounded linear self-adjoint positive operator and 𝐷 is a constant diagonal positive matrix

Let $D$ be a $3\times 3$ constant real positive-definite diagonal matrix and $\Omega\subset\mathbb{R}^3$ be a bounded lipschitz domain. Denote by $(L^2(\Omega))^3$ the set of square integrable ...
69 views

### Essential spectrum of constant invertible diagonal matrix acting on a product of Hilbert spaces [closed]

Let $M$ be a $3\times 3$ real invertible diagonal matrix and $H$ a Hilbert space of infinite dimension (for example, we can take $H$ as the space of square integrable functions over a bounded ...
90 views

89 views

### Fourier transform without characters (Eigenfunctions of an operator)

Let's consider a very simple problem in quantum mechanics: We have, in $\mathbb R,$ a potential barrier of the form $$V(x) = V_0 \mathbf 1_{[-a,a]}(x),$$ where $\mathbf 1_{[-a,a]}$ denotes the ...
15 views

### 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 (...
122 views

### 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 ...
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 ...
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 (...