# 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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### Applications and motivations of resolvent for elliptic operator

Let $A(x)=(a_{ij}(x)):\mathbb{R}^d\to\mathbb{R}^{d\times d}$ be a matrix satisfying ellipticity condition, that is \begin{align} \mu^{-1}|\xi|^2\geq \sum_{i,j=1}^da_{ij}(x)\xi_i\xi_j\geq\mu|\xi|^2 \...
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### Closure of the point spectrum of an unbounded diagonalizable operator

Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to$H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...
1 vote
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### Are these $L_2$-spectral radii approximations strictly increasing?

Suppose that $V$ is a finite dimensional complex Hilbert space. Let $L(V)$ denote the collection of all linear mappings from $V$ to $V$. Let $A_1,\dots,A_r:V\rightarrow V$ be linear operators. Then ...
114 views

### Spectral theory of infinite volume hyperbolic manifolds

I have a question about the discrete spectrum of the Laplace operator on hyperbolic manifolds with infinite volume. I understand the case of infinite area surfaces: see Chapter 7 (Sections 1 and 2) of ...
338 views

### Positiveness of Banach limit [closed]

I‘m currently reading Arveson’s “A Short Course on Spectral Theory”, and I’m stuck at Exercise 3.1 (1). The question is: Let $l^{\infty}(\mathbb{N})$ be the set of all bounded sequences of complex ...
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### Deriving Sommerfeld radiation condition from limiting absorption principle

For the Helmholtz equation $$-(\Delta + k ^2) u = f, \label{1}\tag{1}$$ imposing the Sommerfeld radiation condition $$\lim_{r\to\infty} r ^{\frac{m-1}2} \left( u_r - i k u\right) = 0$$ on $u$ ...
1 vote
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1 vote
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### Class of spectral zeta functions whose analytic extension takes a particular form

In quantum field theory the one-loop effective action is expressed in terms of the functional determinant of the (elliptic and self-adjoint) operator of small disturbances. Since the real eigenvalues ...
64 views

### Mathematical reason for scatter states being special?

In infinite spectral theory, we have the discrete and continuous spectrum, which are called "bound" and "scatter states" in physics. My understanding is, if $O \in B(H)$ is a self-...
123 views

### The definition of essential spectrum for general closed operators

I've asked this problem in MSE several days ago, see here. But there is no reply up until now. Maybe I wrote things too complicated there and so I'll write a very clean problem here. For background ...
39 views

### Can we describe the transition affected by the measurement of a quantum mechanical observable in the language of probability theory?

Consider a quantum mechanical system $S$ with the state space being given by a $\mathbb C$-Hilbert space $H$. It is assumed each physically measurable quantity $\mathcal A$ is described by an ...
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### Proving an eigenvalue bound without resorting to Weyl's law

Suppose $(M,g)$ is a smooth compact Riemannian manifold of dimension $n\geq 2$ with smooth boundary and denote by $\{\phi_k,\lambda_k\}_{k\in \mathbb N}$ its Dirihclet spectral decomposition for the ...
112 views

### Reed-Simon Vol. IV: Question regarding convergence of eigenvalues

I am reading through Chapter XIII.16 of Reed and Simon's Methods of Modern Mathematical Physics IV: Analysis of Operators about Schrödinger operators with periodic potentials. Since the topic is kind ...
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### Weyl's law and eigenfunction bounds for weighted Laplace-Beltrami operator

I would appreciate any answers or even references for the following problem. Let $(M,g)$ be a complete smooth Riemannian manifold with an asymptotically Euclidean metric (let's even say that the ...
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### How to prove that a finite rank perturbation on an infinite matrix does not change its continuous spectrum?

I have the discrete Laplace operator on an infinite Hilbert space with an orthonormal basis $\psi_x$ ($\forall x \in \mathbb Z$), given by $\Delta \psi_x=\psi_{x-1}+\psi_{x+1}$. If I introduce a ...
1 vote
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### Why is spectral theory developed for $\mathbb C$

Spectral theory is a fundamental part of operator theory and the spectrum of many operators is investigated throughout the existing literature. And that is for a good reason: If $A$ is some closed ...
1 vote
Let $M$ be a compact smooth manifold with a smooth boundary. Given a smooth Riemannian metric $g$ on $M$, we denote by $\{\phi_k\}_{k=1}^{\infty}$ an $L^2(M)$--orthonormal basis consisting of ...
Let $M$ be a compact smooth manifold without boundary. Let $T>0$ and let $g$ be a smooth Riemannian metric on $M$. Given any $f \in L^2(M)$ let $u$ be the unique solution to the equation \...