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10 votes
2 answers
1k views

Harmonic oscillator discrete spectrum

Let us act intentionally stupid and assume we do not know that we can solve for the spectrum of the harmonic oscillator $$-\frac{d^2}{dx^2}+x^2$$ explicitly. Is there an abstract argument why the ...
Zinkin's user avatar
  • 501
3 votes
2 answers
968 views

Can one estimate the distribution of eigenvalues of a matrix by its Cauchy/Stieltje transform?

Given a real symmetric $n$ dimensional matrix $A$, with eigenvalues $\lambda_i$ I am defining its Cauchy transform as the function, $f_A(z) = \sum_i \frac{1}{z-\lambda_i}\,$ Is there any information ...
Student's user avatar
  • 617
4 votes
1 answer
213 views

Mapping properties of backward and forward heat equation

In a previous question on mathoverflow, I asked about the following: Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions. The ...
Sascha's user avatar
  • 536
27 votes
0 answers
1k views

Unital $C^{*}$ algebras whose all elements have path connected spectrum

A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$. What is an example of a non commutative ...
Ali Taghavi's user avatar
22 votes
5 answers
1k views

Rigorous justification for this formal solution to $f(x+1)+f(x)=g(x)$

Let $g\in C(\Bbb R)$ be given, we want to find a solution $f\in C(\Bbb R)$ of the equation $$ f(x+1) + f(x) = g(x). $$ We may rewrite the equation using the right-shift operator $(Tf)(x) = f(x+1)$...
BigbearZzz's user avatar
  • 1,245
18 votes
4 answers
1k views

Who first used the multiplication operator version of spectral theory

This is another history question. Hilbert phrased the spectral theorem in terms of resolutions of the identity. While this remained the form of Stone and von Neumann, they did also have the ...
Barry Simon's user avatar
13 votes
7 answers
10k views

What is the best reference for Spectral theory?

I'm studying Bernard Aupetit: A Primer on Spectral Theory but the textbook we are using is a little bit heavy going for me. Is there a best book to learn about these things? Thank you.
user62498's user avatar
  • 823
11 votes
2 answers
2k views

Spectrum of $L^\infty(X,\mu)$

Suppose that $(X,\Sigma,\mu)$ is a measured set with respect to $\sigma$-algebra $\Sigma$. Suppose that $L^\infty(X,\mu)$ is the set of all $\mu$-equal bounded $\Sigma$-measurable functions on $X$. ...
unknown is my last name's user avatar
11 votes
1 answer
2k views

Bounded operator on a normed space with empty spectrum

A bounded operator acting on a complex Banach space has non-empty spectrum, and the proof of this fact uses the completeness of the space. Is there any example of bounded operator acting on a ...
M.González's user avatar
  • 4,461
8 votes
1 answer
421 views

$C^k$ one-parameter family of metrics

Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...
SMS's user avatar
  • 1,407
7 votes
3 answers
6k views

Integral kernel for the resolvent of the laplace operator

Consider the Laplace operator defined in the biggest possible subset of $L^2(\mathbb{R}^2)$ and let $z \in \mathbb{C}\backslash\mathbb{R}$. Therefore $z \notin \sigma (\Delta)$ the spectrum of $\Delta$...
Mike's user avatar
  • 225
7 votes
1 answer
413 views

Criteria for operators to have infinitely many eigenvalues

Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem. For non-normal operators this no longer has to be true. There ...
Sascha's user avatar
  • 536
7 votes
2 answers
920 views

Exotic spectrum of Laplace operator

Given a closed Riemannian manifold and a generalized Laplace $\Delta$ operator, it is well known that $\Delta$ has discrete spectrum $(\lambda_n)_n$ (arranged in a increasing way, not counting ...
Alex's user avatar
  • 101
6 votes
2 answers
539 views

Is there a reasonable notion of spectral theorem on a pre-Hilbert space?

I'm trying to understand how bad things could possibly get without Cauchy completeness as a criterion for Hilbert spaces in quantum mechanics. Obviously, doing calculus on a pre-Hilbert space would be ...
Sanchayan Dutta's user avatar
6 votes
3 answers
3k views

Non-empty resolvent set, then operator closed?

On Hilbert spaces, the following is true: Let $T$ be a densely-defined linear operator with non-empty resolvent set, then $T$ is closed. The obvious proof I see to show this uses explicitly the ...
gipom's user avatar
  • 115
6 votes
2 answers
529 views

Schrödinger eigenfunctions are bounded

Let $V:\mathbb{R}\rightarrow \mathbb{R}^{+ *}$ a real positive function such that $\displaystyle \lim_{ x \to \pm\infty} V(x)= +\infty $. Then the Schrödinger operator $H=-\frac{d^2}{dx^2}+V(x)$ has ...
M. Veruete's user avatar
5 votes
3 answers
986 views

Boundedness of Laplacian eigenfunctions

Let $A$ be a bounded domain in $\mathbb R^d$, $d>1$, and $\{u_k\}$ is the set of all $L^2$-normalized Laplacian eigenfunctions on $A$ with Dirichlet boundary condition (i.e., $\|u_k\|_2 = 1$). Is ...
Denis Grebenkov's user avatar
5 votes
1 answer
487 views

Fokker-Planck: equivalence between linear spectral gap and nonlinear displacement convexity?

In a smooth, bounded and convex domain $\Omega\subset \mathbb R^d$ consider the usual linear Fokker-Planck equation with Neumann (some would say Robin) boundary conditions \begin{equation} \label{FP} \...
leo monsaingeon's user avatar
5 votes
1 answer
564 views

Convergence of discrete Laplacian to continuous one

I make the following observation: Let $\Delta^{(n)}$ be the discrete Laplacian on $\mathbb{C}^n$ (ie the $n\times n $ matrix with diagonal $-2$ and upper/lower diagonal $1$.) This one has eigenvalues ...
Sascha's user avatar
  • 536
5 votes
1 answer
573 views

Analytic perturbation of eigenfunctions

Consider a domain $\Omega_0 \subset \mathbb{R}^n$, and deformations of $\Omega_0$, called $\Omega_t$, obtained by a one-to-one mapping $x \mapsto x + t\varphi (x)$, where $\varphi$ is smooth. It is ...
Noname's user avatar
  • 53
5 votes
1 answer
2k views

Commuting with self-adjoint operator

Let $T$ be an (unbounded) self-adjoint operator. Assume that there is a bounded operator $S$ such that $TS=ST.$ For which kind of $f$ do we have that $f(T)S=Sf(T)?$ My thought was that using a ...
Zinkin's user avatar
  • 501
3 votes
2 answers
280 views

Question about the Bessel operator

For $\nu>-1$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the succesive positive zeros of the Bessel function of the first kind $J_{\nu}$. The Bessel operator is given by \begin{equation*} L_\...
Tony419's user avatar
  • 421
3 votes
0 answers
160 views

Non-emptiness of spectrum $\sigma(a)$ in non-Archimedean Banach algebras

I'm trying to determine whether or not the standard proof that the spectrum of a point in a unital Banach algebra is non-empty can be adapted to prove the same thing over certain non-Archimedean ...
Very Forgetful Functor's user avatar
2 votes
1 answer
315 views

Parabolic PDE Long Time Asymptotics and Elliptic Operator Spectrum

How does one show directly that the solution following parabolic partial differential equation (PDE) of $p(t,v)$ approaches its stationary solution which is a solution of an elliptic partial ...
Hans's user avatar
  • 2,239
1 vote
1 answer
178 views

Parabolic PDE Long Time Asymptotics and Elliptic Operator Spectrum II

This is a follow-up on a previous question. Now the parabolic PDE of $P(t,x,v)$ has two spatial dimensions. $$ \partial_t P = L^* P \tag1 $$ $$L^*P = \frac12\left(\kappa^2\frac{\partial^2}{\partial v^...
Hans's user avatar
  • 2,239
1 vote
0 answers
151 views

Continuity of the spectrum under weaker notions of convergence

Let $T:X\to X$ be a linear operator on a Banach space $X$. We know that the spectrum of $T$ is an upper semicontinuous function of $T$ for the uniform convergence: that is, if $T_n:X\to X$ is a ...
Eduardo's user avatar
  • 757
1 vote
2 answers
274 views

$\mathcal P(K)=\mathcal R(K)$ iff $\Bbb C\backslash K$ is connected

Let $K$ be a compact set in $\Bbb C$. Let $\mathcal P(K)$ be the closed algebra generated by polynomials on $K$ and $\mathcal R(K)$ the closed algebra generated by rational functions without poles in $...
BigbearZzz's user avatar
  • 1,245
1 vote
0 answers
99 views

Minimize $\langle(1-\kappa)^{-1}f,f\rangle$ for a parameter-dependent integral operator $\kappa$

I've got a contractive self-adjoint linear integral operator $\kappa$ of the form $$(\kappa g)(x):=g(x)+\int\lambda({\rm d}y)k(x,y)(g(y)-g(x))\;\;\;\text{for }g\in L^2(\mu),$$ where $k$ depends on the ...
0xbadf00d's user avatar
  • 167
0 votes
0 answers
83 views

$ 0 $ locates in the continuous spectrum of Schrodinger operators?

This is question is motivated by Non-closed range space of Laplace operators?. We aim to determine what kind of potential will make corresponding schrodinger operators possess non-closed range. For ...
Yidong Luo's user avatar