Spectral theory is the study of generalized notions of eigenvalues and eigenvectors for linear operators in Banach spaces.

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How to find the Laplacian Eigen Values of the given graph

Find the Laplacian eigen-values of the of the graph on $N$ vertices whose edge set is given by $\{(i,i+1),1\le i<n\}$ and the edge $(1,n)$ . The answer is given to be $2-2\cos{\frac{2k\pi}{n}}$ . ...
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Max-Min Principle with an example on intervals

I have a few difficulties to understand the max-min principle (Intuitive understanding of Maximin Principle). Is there anyone could explain this theorem in using $[a,b] \subset [a',b']$? I know that ...
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1answer
31 views

Non square matrices and spectrum of $AB$

There is a known theorem (true in the general context of Banach algebras) that if $a,b \in A$ (where $A$ is a Banach algebra) then $\sigma(ab) \cup \{0\}=\sigma(ba) \cup \{0\}$ where $\sigma$ is the ...
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24 views

Any additional constraint will increase the value of the maximin

In the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $325$, there is the comment "any additional constraint will increase the value of the maximin",...
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42 views

Intuitive understanding of Maximin Principle

From the the book page $324$, does someone could explain to me the Theorem $2$. Maximin principle? I have a bit of difficulties to well understand how works this theorem. A simple example would be ...
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33 views

Spectral theorem for matrices…

From the spectral theorem we know that if $A$ is a symmetric matrix then there exists an orthogonal matrix $M$ such that $A=M^{-1}DM=M^TAM$ My question is: if I have the matrix $A$ how do I find ...
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Distribution $<tr(e^{it \sqrt{\Delta}},\varphi)>=<-i \ln(1-e^{it}), \varphi '>$ in term of Dirac-$\delta$-function

Personal question : Could it possible for the distribution distribution $<tr(e^{it \sqrt{\Delta}},\varphi)>=<-i \ln(1-e^{it}), \varphi'>$ to be expressed in term of the Dirac-$\delta$-...
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39 views

Why are functional calculi interesting?

In my class on spectral theory we have defined the continuous functional calculus for normal elements of a C*-algebra. We were told that this is one of the most important results in spectral theory on ...
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finite spectrum eigenvalue

Let $T:X \to X$ be a linear bounded operator where X is Banach space ,and $\sigma (T)$ is a finite set.Then does the spectrum consist of eigenvalues only? Any hint or counterexample is ok. thanks in ...
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33 views

Strauss's book - Weyl's asymptotic law for eigenvalues

Someone told me that in the book Partial Differential Equations by Strauss you can find a proof of Weyl's asymptotic law for eigenvalues (one can hear the volume and dimension of a domain). Is there ...
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33 views

Interval bounds for symmetric doubly-stochastic matrices (designed with Metropolis weights).

I'm facing an unusual problem with doubly-stochastic matrices, in the context of some undirected graph. I assume that it is connected, but this is not so important for this problem. Let me introduce ...
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25 views

Find spectral theorem of $A$ and find its singular eigenvalues.

The rotation matrix $$A=\pmatrix{ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta}$$ has complex eigenvalues $\{e^{\pm i\theta}\}$ corresponding to eigenvectors $\pmatrix{1 \\i}$ and $...
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33 views

The reason of $\int_{-\infty}^{\infty}\mu_k^2(x)dx=1$

Is there anyone could tell me why if $$\sum_{k \geq 0} e^{it \sqrt{-\lambda_k}}=\int_{-\infty}^{\infty} (\sum_{k \geq 0} e^{it \sqrt{-\lambda_k}} \mu_k^2(x))dx= \sum_{k \geq 0} e^{it \sqrt{-\lambda_k}}...
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24 views

Compact resolvent proof

I want to prove the following proposition: We consider the operator $$A=A_V=-\Delta+\frac{1}{4}|\nabla V|^2-\frac{1}{2}\Delta V$$(where $V$ is a polynomial) with domain $D(A)=C_c^{\infty}(R^...
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1answer
33 views

Expansion theorem or Poisson Summation Formula? - Basis of eigenfunctions gives rise to a Fourier series

Does anyone could explain to me why in the Semiclassical's answer on the question Wave kernel for the circle $\mathbb{S}^1$ - Poisson Summation Formula, the basis gives a series of the form $\...
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14 views

Set of normalized eigenfunctions - Expand some given function in the eigenfunction basis [closed]

The spectrum$(\mathbb{S}^1)=\{\lambda_k=k^2\ : k \in \mathbb{N}\}$, and the eigenfunctions $\mu_k(t)$ associated to the eigenvalues $\lambda_k$ are $a_k \cos kt + b_k \sin kt$ under the Laplacian ...
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126 views

Wave kernel for the circle $\mathbb{S}^1$ - Poisson Summation Formula

Question : How could I compute the (wave) kernel from the fact I have already found (wave) trace on unit circle? The definitions are related to the page $25$ of the following pdf. As the Spectrum$(S^...
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How to prove $r(T_1 + T_2) \leq r(T_1) + r(T_2)$ when $T_1T_2 = T_2T_1$ for bounded linear operators?

Suppose $T_1$ and $T_2$ are bounded linear operators in a complex banach space and $r(A)$ is the spectral radius of $A$, satisfying $$ r(A) = \inf_{n>0} \|A^n\|^{1/n} = \lim_{n\rightarrow \infty} \...
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1answer
24 views

Coefficients of the eigenfunction

Related to the question : Eigenvalues of the circle over the Laplacian operator, how is it possible to find $c_1$ and $c_2$ related the explicit function $g(x)=c_1 \cos (\mu x)+ c_2 \sin (\mu x)$? ...
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1answer
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Laplace-Beltrami operator on a circle - Explanation

I would like to find the spectrum of a circle. I know that the Laplace-Beltrami eigenvalue problem for unit circle is equivalent to the regular Laplacian eigenvalue problem for the interval of the ...
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1answer
35 views

The particularity of $k$ being an integer in the solution of a DE

Related to the Thomas's comment in the question : Eigenvalues of the circle over the Laplacian operator, is there anyone could tell me why the periodic function $g$ has a fundamental set of solutions ...
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Spectrum of linear operator, essential spectral radius

Consider the operator $L:L^1(S^1)\to L^1(S^1)$ given by $$ (Tf)(x)=\dfrac{1}{2}\left( f\left( \dfrac{x}{2}\mod 1\right)+f\left( \dfrac{x+1}{2} \mod 1 \right) \right) $$ where we identified $S^1$ with ...
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Periodic function - Differential equation on an existing question

Related to the question : Eigenvalues of the circle over the Laplacian operator, how could I get an explicit formula for the differential equation $g''=\lambda g$ with $g$ a $2 \pi$-periodic function. ...
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Periodic function with the capacity of being $g'' = \lambda g$

Related to the question : Eigenvalues of the circle over the Laplacian operator, what kind of periodic chart $c:(-\pi,\pi)\rightarrow S^1$ has the property that for a continuous function $f$, $g :=f \...
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50 views

Spectrum of the circle - Clarification of certain points

I have a bit of difficulty as doing the problem Eigenvalues of the circle over the Laplacian operator. Since $g$ is a periodic function, do I have to use the Fourier series? If so, how could I do that?...
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1answer
130 views

Eigenvalues of the circle over the Laplacian operator

I would like to find the spectrum of a circle. It seems that there are no boundary conditions, but I'm not quite certain. How could I find the spectrum of a circle over the Laplacian operator?
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Wave trace on $1$-dimensional circle - How about the spectrum of this circle?

I have to find the wave trace for the Laplacian on the $1$-dimensional circle. Generally, the wave trace is defined (see this website) as $$W(t)= \int_{M} K_t(x,y)dy=\sum_j \cos(t \lambda_j)=\Re \...
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How to prove that $ e_{\lambda}$ can be written in the following form?

Let $e_{\lambda}$ be the spectral density associated to the spectral function $E_{\lambda}$ for a self-adjoint operator $A$ on a complex Hilbert space $(H,\left<., .\right>)$. Haw to prove that ...
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range of weighted shift operator

Consider the weighted shift operator on $\ell^2$ space defined by $T(x_0, x_1, x_2, ...) = (0, x_0, 2x_1, 3x_2, 4x_3, ...)$ with domain $$\mathcal{D}(T) = \{(x_n) \in \ell^2 : \sum_{n=0}^{\infty}|(n+...
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1answer
25 views

Element with maximum magnitude in $A \leq \max(\sigma_{i})$, where $\sigma_{i}$ is singular values of A

Let $A$ be a matrix with real values. Is it true that element with maximum magnitude in $A$ is less than $\max(\sigma_{i})$, where $\sigma_{i}$ is singular values of A? That is, is $$ \max_{ij} |A_{ij}...
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1answer
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proof of an equality norm

Let the mapping $T:\ell^{2}\rightarrow \ell^{2}$ is defined as follow. $$T(x_1,x_2,\ldots,x_n,\ldots)=(x_1,\dfrac{1}{2}x_2,\ldots,\dfrac{1}{n}x_n,\ldots)$$ In this case, i've easily earned: $$\sigma(T)...
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1answer
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Green's function on the unit interval $[0,1]$

I would like to find the Green's function on the unit interval $[0,1]$. We have to solve the linear equation $$\frac{d^2f(x)}{dx}=g(x)$$ with boundary condition $f(0)=a$, $f(1)=b$. So the Green's ...
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What does 'mode' mean in this context?

The spectrum of an operator $\mathcal{L}$ is the disjoint union of two sets: the point spectrum that consists of all isolated eigenvalues with finite multiplicity and its complement, which we call the ...
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1answer
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Laplacian on the straight circle

I would like to well understand the notion of Green's function. I know that a Green's function, $G(x,s)$, of a linear differential operator $L = L(x)$ acting on distributions over a subset of the ...
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Trace of the heat operator $Z(t)=\sum_{m,n=1}^{\infty}e^{-\frac{\alpha_{m,n}^2}{r_0^2}t}$

I know that the spectrum of the disk of radius $r_0$ is $\lambda_{m,n}=\frac{\alpha_{m,n}^2}{r_0^2}$, where $\alpha_{m,n}$ is the n-th root of the Bessel's function of order $m$. I have to find the ...
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$U(r, \theta) = T_s(\theta) J_s(\sqrt{\lambda}r)$ - Extract the eigenvalues of the eigenfunction $U$?

On a certain problem, I obtain $U(r, \theta) = T_s(\theta) J_s(\sqrt{\lambda}r)$, where $J_s$ is the Bessel's function of order $s$, and we know by the Dirichlet boundary conditions that $U(r_0, \...
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Compute $\sum_{n=1}^\infty e^{\pi^2(\frac{n^2}{a^2})t}$ [duplicate]

I have to compute $$Z(t)=\sum_{n=1}^\infty e^{-\lambda_nt}$$ with $\lambda_n=\pi^2\left(\frac{m^2}{a^2}+\frac{n^2}{b^2}\right)$. So $$\sum_{m,n=1}^\infty \exp\left(-\pi^2\left(\frac{m^2}{a^2}+\frac{n^...
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Normal operators are unitarily equivalent iff they have the same spectrum

Here is a more general claim that implies the argument in the title: Show that if $S$ and $T$ are two normal operators, then there is a $*$-isomorphism $\pi: C^*(S)\to C^*(T)$ s.t. $\pi(S)=T$ iff $\...
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$g(y)=k_1 \cos (\sqrt{\alpha}y)+k_2 \sin (\sqrt{\alpha}y) $ - Value of $k_1$?

I have to find the constants $k_1$ and $k_2$ of the function $g(y)=k_1 \cos (\sqrt{\alpha}y)+k_2 \sin (\sqrt{\alpha}y) $ in using $g(0)=g(b)=0$. So I found that $k_1=0$. Furthermore, I think $\alpha = ...
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EDO by separation of variable - When are we certain that $\lambda \geq 0$?

If we use the separation of variables to solve $u_t-u_{xx}=-u$, $u=u(x,t)$, then we obtain $f(x)g'(t)-f''(x)g(t)=-f(x)g(t) \iff \frac{g'(t)}{g(t)}+1=\frac{f''(x)}{f(x)}$. The only way that a function ...
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1answer
25 views

Why is the Union of the Orthonormal Basis of $W$ and of $W^{\perp}$ an orthonormal Basis of $V$?

Let $V$ be a $\mathbb{K}$-vector space with a dot product. Let $T \in L(V)$ (endomorphism) be a self-adjoint operator ($\forall x, y \in V, T(x) \cdot y = x \cdot T(y)$) Let $W$ be an invariant ...
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relatively compact subspace

Can someone please help me to finf the answer for the following question: For $r\in \mathbb{N}$ ,we note $E_r=\{P\in \mathbb{R}[x_1,x_2,\ldots,x_d] \mid \deg P\le r \}$. and for $P\in E_r$ we ...
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Spectrum of a rectangle - Question

Question : Find the spectrum of a rectangle I know that the spectrum is the set of eigenvalues of the Laplacian within the region, with Dirichlet boundary conditions. For each $\lambda$, the ...
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Spectrum - rectangle, disk and the sphere

My supervisor asked me to find the spectrum of the rectangle, the disk and the sphere. Is anyone could give a bit of detail of what he means exactly? en.wikipedia.org/wiki/Spectral_shape_analysis. ...
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compacity of the resolvent

can someone please help me to solve the following problem: Let $A=-\Delta_V^{(0)}$ be an operator with domain $D(A)=C_0^{\infty}(\mathbb R^n)$(Witten Laplacian) which verifies : $\exists c\ge 0$ such ...
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20 views

Relation between the eigenvalues of $\Delta$ and counting lattice points

I was reading a paper with the following information: "Let $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$ be the flat torus, let $\varphi$ be the eigenfunctions and $\lambda$ the eigenvalues of the ...
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19 views

Essentially maximal accretive operator

Can someone please help me to prove the following equivalence: Let $A$ be an operator in $H$ (Hilbert space) with domain $D(A)$. $1)A$ is essentially maximally accretive. $2) A$ is maximally ...
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Spectrum of Laplace operator with potential acting on $L^2(\mathbb R)$ is discrete

Consider an operator $H=-\Delta +U(x)$ on $L^2(\mathbb R)$ for a function $U(x): \mathbb R \to \mathbb R$ that tends to $+\infty$ as $|x|$ grows. These kinds of operators appear all over non-...
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21 views

Witten Laplacian problem-show that $\lim\limits_{n \rightarrow +\infty} ||f_nu||=0$

Can someone help me please to correct my answer of this problem: We consider the Witten Laplacian with domain $C^∞_0(R^d)$. We know that this operator is essentially self adjoint,positive and with ...
2
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0answers
22 views

Estimating pseudo-periodicity of signals

I have pressure data which are measured at a given point in a standing wave. These data(signals) are 'almost' sinusoidal in nature. Each cycle may slightly vary from the original signal i.e the ...