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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Bound the $\infty$-norm of the eigenvector of the second minimum eigenvalue of normalized Laplacian from below

I meet the above problem while reading a paper. The problem can be stated as below. Consider an undirected graph $G$. Let $\mathbf{v}$ be a vector such that $\mathbf{D}^{1/2}\mathbf{v}$ is the ...
Lasting Howling's user avatar
3 votes
2 answers
356 views

Monotonicity of matrix conjugation

Let $A$ and $B$ be positive-definite matrices such that $A \le B.$ By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$ I am now curious under ...
António Borges Santos's user avatar
2 votes
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Bourgain-Gamburd-like theorems in the non-algebraic case

For $\mu$ a Borel probability measure on the compact group $G=\operatorname{SU}(d)$, Bourgain-Gamburd prove that the spectral radius of the associated operator on $L^2(G)$ is strictly less than one, ...
John Rached's user avatar
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Do the eigenvalues of a thin strip around a semicircle converge to the respective eigenvalues of a semicircle?

Background: Consider the sphere $S^2$ with the round metric $g$ and let $\gamma$ be one half of a great circle of length $\pi.$ Let $T_\epsilon$ denote a geodesically convex tube around $\gamma$ of ...
Student's user avatar
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Computation of Brown measure of the shift operator on $\ell^2(\mathbb N)$?

This looks an extremely simple question - I am just trying to give an example of Brown measure, https://en.wikipedia.org/wiki/Brown_measure, so I try to compute it for the left/right-shift operator on ...
Ma Joad's user avatar
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Discrete spectrum of $A \otimes 1+ 1 \otimes B$ [migrated]

Let $A, B$ be unbounded self-adjoint operators on Hilbert spaces $\mathcal{H_1}, \mathcal{H_2}$, with both non-empty discrete spectra. Let us say, for instance, $\inf \, \sigma(A) = \lambda_1^A$ and $...
Hugo's user avatar
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Difference in essential spectrum between Schrodinger operators

I am considering two Schrodinger operators on $\mathbb{Z}^2$ and compare their essential spectrum. The operators are both of the form $H=A+V$ where $A$ is the adjacency operator on the $\mathbb{Z}^2$-...
Keen-ameteur's user avatar
3 votes
1 answer
99 views

Dimension of spectral projection subspaces under local convergence

I'm interested in estimates on dimension of spectral projection subspaces of some limit operator. I recently asked a related question in the thread Dimension of spectral projection subspaces under ...
Keen-ameteur's user avatar
3 votes
2 answers
148 views

Dimension of spectral projection subspaces under strong convergence of operators

I have a possibly simple question regarding estimating bounds on spectral projection subspace. Let $H_n$ be a sequence of bounded self-adjoint operators on $\ell^2(\mathbb{Z}^2)$ converging in the ...
Keen-ameteur's user avatar
2 votes
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Spectrum of an almost Hamiltonian matrix

I have a complex-valued block matrix $N=\begin{bmatrix} A & B \\ C & -A^* \end{bmatrix}$, where $A$ is diagonal, $B=B^*$, and $C$ is rank-1 but not Hermitian. If $C$ were Hermitian, $N$ would ...
mathamphetamine's user avatar
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Spectral gap of a Markov operator on $L^2$ with a symmetric $L^\infty$ kernel

Let $I$ be a compact interval, say $I:=(0,1)$, and $k\in L^\infty(I\times I)$ a symmetric Markov kernel, i.e. $k(x,y)=k(y,x)$ and $$\int_I k(x,y) d y = 1\qquad\mbox{for almost all } x\in I.$$ Let $K:L^...
Johny B's user avatar
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7 votes
1 answer
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Why is the length spectrum called a spectrum?

Given a hyperbolic surface $X$, one considers the multiset of lengths of closed primitive geodesics. This multiset is called the length spectrum $\mathcal{L}(X)$. Question: is $\mathcal{L}(X)$ a ...
Andrey Ryabichev's user avatar
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2 answers
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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
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Induced higher Gershgorin estimate

I have a problem which I suspect appears in literature under a name I haven't found yet. Let $H:\ell^2(\mathbb{Z}^2)\to \ell^2(\mathbb{Z}^2)$ given by $H=\Delta + D$, where $\Delta$ is the graph ...
Keen-ameteur's user avatar
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In what sense does the Laplacian on compact intervals converge to one on all of $\mathbb{R}$?

I guess this topic may have been addressed somewhere but I cannot really find a reference myself, so I ask here. For each $N \in \mathbb{N}$, consider the Laplacian $\Delta$ on the interval $[-N,N]$ ...
Isaac's user avatar
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Eigenvalues of a Schrödinger operator

I'm interested in the existence of eigenfunctions and finding eigenvalues of the following operator $$L(\varphi) = \varphi_{rr} - \frac{1}{r} \varphi_r - [V + \frac{m}{r^2}] \varphi$$ $$\varphi(0) = \...
JMK's user avatar
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Gradient estimate of the eigenfunction of Laplacian on hyperbolic space

I am trying to understand the asymptotic behaviors of the gradient of the eigenfunction function of the Laplace-Beltrami operator on the hyperbolic plane $\mathbb{H}^2$. Specifically, my focus lies on ...
Atlas Tasilli's user avatar
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If we have $\mu_{xy}$, why can we only construct the spectral measure if $\| \mu_{xy} \| \le \| x \| \|y \|$?

Definitions Representation Let $X \subset \mathbb{C}^N$ and $\mathcal{A}$ be an algebra in $\mathcal{C}(X)$. Also, we denote $L(H)$ as the set of all linear operators on HIlbert space $H$. We call $\...
S-F's user avatar
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2 votes
1 answer
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Estimate for the operator $A A_D^{-1}$

Let $O\subset\mathbb{R}^d$ be a bounded domain of the class $C^{1,1}$ (or $C^2$ for simplicity). Let the operator $A_D$ be formally given by the differential expression $A=-\operatorname{div}g(x)\...
Yulia Meshkova's user avatar
1 vote
0 answers
174 views

Connection of eigenspace of finite Hilbert matrix and its continuous operator counterpart

I am trying to understand the connection between the eigenspace of the continuous operator $$ H(x,y) = \frac{1}{x+y} $$ which is nothing but the square of the Laplace operator, and its discrete ...
knuth's user avatar
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Self-adjoint operator with pure point spectrum

Suppose that A is a self-adjoint (possible unbounded) operator from a separable Hilbert space H to itself. I would like to know if the following statement is true: A has pure point spectrum (i.e., the ...
user3476591's user avatar
3 votes
2 answers
171 views

How to diagonalize this tridiagonal difference operator with unbounded coefficients?

Problem: I have a self-adjoint operator in $\ell^2(\mathbb{Z})$ which acts as $$T g(x)=q^{-2 x -3/2} g(x+1)+(1+q) q^{-2 x-1} g(x)+q^{-2 x +1/2} g(x-1),$$ and I am looking to diagonalize it. The ...
Leonid Petrov's user avatar
3 votes
1 answer
114 views

Spectra of products variously permutated

Let $x,y$ be elements of a Banach algebra $A$ and $\lambda\in\mathbb C\setminus\{0\}$. If $\lambda-xy $ is invertible, then $\frac1{\lambda}\big[1+y(\lambda-xy)^{-1}x \big]$ is clearly an inverse of $...
Pietro Majer's user avatar
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3 votes
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Operator Semigroup: Resolvent estimates and stabilization, a detail in the paper of Nicoulas Burq and Patrick Gerard

In Appendix A of the paper Stabilization of wave equations on the torus with rough dampings https://msp.org/paa/2020/2-3/p04.xhtml or https://arxiv.org/abs/1801.00983 by Nicoulas Burq and Patrick ...
monotone operator's user avatar
2 votes
1 answer
169 views

On spectral calculus and commutation of operators

Let $\mathcal{H}$ be a Hilbert space, $B\in\mathcal{B}(\mathcal{H})$ be bounded and self-adjoint and $A:\mathcal{D}(A)\to\mathcal{H}$ closed (but not necessarily self-adjoint or bounded). The ...
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Operator identity

Let $T:\mathcal{D}(A)\to\mathcal{H}$ be a unbounded, self-adjoint, operator with positive spectrum $\sigma(T)\subset [\varepsilon,\infty)$ for $\varepsilon>0$. Hence $T$ is bijective with bounded ...
B.Hueber's user avatar
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Dimension of Laplacian eigenspaces along a smooth 1-parameter family of metrics

Let $(M^n,g)$ be a closed Riemannian manifold, $n \geq 2$. For a smooth 1-parameter family $g_t$, $t \in (-\varepsilon, \varepsilon)$, of Riemannian metrics on $M$ with $g_0 = g$, let $\lambda_k(t)$, $...
Eduardo Longa's user avatar
4 votes
1 answer
224 views

Spectral density of symmetrized Haar matrix

Let $O$ be a random orthogonal matrix (according to Haar measure) of size $n$. I found by simulations that the spectral density of $O+O^\top$ is the arcsin law rescaled to the interval $[-2,2]$. I can'...
Pluviophile's user avatar
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Rotational invariance of Laplace-Beltrami eigenvalue problem on smooth manifolds

I am currently looking at the eigenvalue problems of the Laplace-Beltrami operator. Let $(M,g$) be a smooth and oriented Riemann manifold. I am investigating the eigenvalue problem of the Laplace-...
SebastianP's user avatar
1 vote
0 answers
141 views

Generalization of Borel functional calculus

[Repost from https://math.stackexchange.com/questions/4802593/generalization-of-borel-functional-calculus] Let $A$ be a normal operator on a Hilbert space $V$. The continuous functional calculus gives ...
oggius's user avatar
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1 vote
1 answer
72 views

Spectrum below zero for $-\beta(x) \partial^2_x : L^2(\mathbb{R}) \to L^2(\mathbb{R})$

Let $\beta \in L^\infty(\mathbb{R} ; (0, \infty))$ be bounded from above and below by positive constants. Consider the self-adjoint operator $ -\beta^{-1} \partial^2_x : L^2(\mathbb{R}; \beta dx) \to ...
JZS's user avatar
  • 469
3 votes
2 answers
197 views

Domain of spectral fractional Laplacian

Let $(M,g)$ be a complete Riemannian manifold with Laplacian $\Delta:C^{\infty}_{c}(M)\to C^{\infty}_{c}(M)$ (think of $\mathbb{R}^{d}$ if you wish). This operator is essentially self-adjoint in $L^{2}...
B.Hueber's user avatar
  • 987
6 votes
0 answers
165 views

Dependence of Neumann eigenvalues on the domain

I have the following problem in hands, in the context of a broader investigation: Let $V\in L^{n/2}$ compactly supported, where $n\geq 3$ is the dimension. I want to prove the following: For any $\...
Manuel Cañizares's user avatar
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0 answers
95 views

Reducing subspaces of unitary operators

Let $U$ be a unitary operator on a Hilbert space $\mathcal{H}$. We can assume $\mathcal{H}$ is an $L^2$ space and $U$ acts as multiplication by a function $u$ with $|u(x)| = 1$ a.e (by the spectral ...
bm3253's user avatar
  • 23
1 vote
1 answer
111 views

Sobolev-type estimate for irrational winding on a torus

Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. Let $\mathscr{C}^{\infty}_0(\mathbb{T})$ be the subset of $\mathscr{C}^{\infty}(\mathbb{T})$ of ...
user197284's user avatar
5 votes
2 answers
434 views

Reconstruction of second-order elliptic operator from spectrum

Let $M$ be a compact smooth manifold, $(\lambda_n)_{n=1}^{\infty}$ be a square-summable monotonically increasing sequence of non-negative numbers, and let $(f_k)_{k=1}^{\infty}$ be continuous ...
Math_Newbie's user avatar
3 votes
1 answer
198 views

Convergence of spectrum

Let $M$ be a compact manifold and $g_k$ be a sequence of Riemannain metrics smoothly converging to another Riemannian metric $g$. Let $\{\lambda^k_j\}$ be the spectrum of the Laplacian of the ...
Hammerhead's user avatar
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2 votes
1 answer
66 views

Spectral threshold effect: examples

I know that the effect of homogenization can be treated as a spectral threshold effect. I want to know more examples of spectral threshold effects in mathematical physics.
Yulia Meshkova's user avatar
2 votes
1 answer
114 views

Disturbance of self-adjoint operator

Assume that $ A $ is self-adjoint operator and $ B $ is a bounded self-adjoint operator. The definite domain of $ A,B $, denoted by $ D(A) $ and $ D(B) $ satisfies $ D(A)\subset D(B) $. Show that \...
Luis Yanka Annalisc's user avatar
0 votes
0 answers
90 views

Applications of finite speed of propagation property

Consider the Laplace operator $\Delta:=\sum_{j=1}^{n}\partial_{x_{j}}^{2}$ on $\mathbb{R}^n$. Let $E_{\lambda}$ be the spectral resolution of $\Delta$, and $$ H_{t}[f]:=\cos{(t\sqrt{-\Delta})}f=\int_{...
pxchg1200's user avatar
  • 265
2 votes
0 answers
133 views

Lp eigenfuntion bounds for the hermite operator on domain (or manifolds) with boundary

Let's define the harmonic oscillator $H = -\Delta+x^2$ in a domain $\Omega$ of $\mathbb R^d$. Thus, we consider the Dirichlet eigenvalue problem $$ (H - \lambda^2)u (x) = 0, \ x \in \Omega ; \ \text{...
L19's user avatar
  • 41
4 votes
0 answers
95 views

Eigenvalues of the Hodge-Laplace for 2-forms on $S^3$

Consider the Laplace operator $\Delta = d^{*} d + d d^{*}$ on $\Omega^2(S^3)$. What is the minimal eigenvalue of $\Delta$? (My computations showed that the answer is 4; the eigenforms correspond to ...
Swino's user avatar
  • 41
3 votes
0 answers
126 views

the second largest eigenvalue of transfer operators

A Gauss map $T$ is mixing and satisfies Lasota-York inequalities. By Henon's theorem, we know that the transfer operator $\hat{T}$ associated with $T$ has a spectral gap. This means there exists a ...
user avatar
0 votes
1 answer
147 views

Spectrum of a product of a symmetric positive definite matrix and a positive definite operator

Let $\mathbf H$ be an infinite dimensional Hilbert space. I want to find an example of a $2\times 2$ real symmetric positive definite matrix $M$ and a positive definite bounded operator $A : \mathbf H ...
SAKLY's user avatar
  • 63
3 votes
0 answers
113 views

Leibniz rule bound for the inverse of the Laplacian?

Let $f, g \in L^2[\mathbb{T}^2]$ be real-valued functions without zero modes. That is, $\int_{\mathbb{T}^2}f=\int_{\mathbb{T}^2}g=0$. Here, ${\mathbb{T}^2}$ is the $2$-dimensional torus $[\mathbb{R}/\...
Isaac's user avatar
  • 2,727
0 votes
0 answers
269 views

Spectral theorem for commuting operators

Let $A_{1},...,A_{n}$ be densely defined self-adjoint operators on a separable Hilbert space $\mathscr{H}$. Suppose these have a common dense domain $D\subset \mathscr{H}$ and satisfy commutation ...
IamWill's user avatar
  • 3,151
1 vote
0 answers
140 views

Reference on spectral theory of self-adjoint operators

I am reading this paper on comparing different moments of independent random variables. A initial step in their approach is designing an operator $L$ over smooth functions (and extended to an self ...
listener's user avatar
3 votes
1 answer
94 views

First Dirichlet eigenvalue of the harmonic oscillator on a bounded interval $(-a,a)$?

Let $a>0$ be a fixed number and consider the Hermite operator (or harmonic oscillator) defined by \begin{equation} Hf(x)=x^2f(x)-f''(x) \end{equation} for any smooth function $f$ compactly ...
Isaac's user avatar
  • 2,727
2 votes
1 answer
173 views

Leibniz rule for square root of Laplacian

Let $(M,g)$ be a compact Riemannian manifold (e.g. $M=S^3$ the 3-sphere) and let $\Delta$ be the metric Laplacian on $M$. Then $\Delta$ has an $L^2(M)$ basis of eigenfunctions $\pi_m$, $$ \Delta \pi_m ...
onamoonlessnight's user avatar
3 votes
0 answers
115 views

Reference request: trace norm estimate

In a paper I am currently reading, the author uses that if $T$ is an operator given by the kernel $$T(x,y) = \int_{\mathbb R} p(x,z) q(z,y) dz,$$ then $$\lvert \operatorname{tr} T \rvert \leq \lVert T ...
Staki42's user avatar
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