All Questions
Tagged with sp.spectral-theory nt.number-theory
18 questions
29
votes
3
answers
3k
views
Perron-Frobenius "inverse eigenvalue problem"
The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...
- 2,200
12
votes
0
answers
508
views
Weyl law for Maass forms with nontrivial character
The classical Weyl law for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$ counts the number of Maass cusp forms on $\Gamma \backslash \mathbb{H}$ with Laplace eigenvalue less than $T$. This is originally due to ...
- 8,374
10
votes
3
answers
3k
views
Number Theory and Geometry/Several Complex Variables
This is a question for all you number theorists out there...based on my skimming of number theory textbooks and survey articles, it seems like most of the applications of geometry and complex ...
- 1,665
9
votes
1
answer
313
views
The scope of correspondence principle in quantum chaos
My understanding of the so-called correspondence principle in quantum chaos, is that it is a connection between the behaviour of a classical Hamiltonian system (chaotic/completely integrable) and the ...
- 809
8
votes
3
answers
526
views
Lower bound for spectral radius on $\operatorname{GL}(n,\mathbb{Z})$
Consider the group of matrices $G =\operatorname{GL}(n,\mathbb{Z})$ with integer entries and determinant $\pm 1$. For each matrix $D \in G$, the product of the eigenvalues of $D$ is equal to $\det D =\...
- 345
8
votes
1
answer
3k
views
The Guinand-Weil explicit formula without entire function theory
I'll admit from the outset that this question is slightly vague. The actual question appears at the end of the post.
The explicit formula of Guinand and Weil can be written in the following way:
For ...
- 2,151
7
votes
4
answers
1k
views
Multiplicity one conjecture
I recently became interested in Maass cusp forms and heared people mentioning a "multiplicity one conjecture". As far as I understood it, it says that the dimension of the space of Maass cusp form for ...
- 123
7
votes
1
answer
639
views
History of spectral methods to the study of real analytic $GL_2$-Eisenstein series
I'm trying to sort out the history of spectral methods in the study of real analytic $GL_2$-Eisenstein series. From what I read so far, I would say that the subject was really kicked off by the ...
- 7,609
7
votes
0
answers
291
views
"Non-critical" zeros of $\zeta$ and the $\zeta$-cycles of Connes and Consani
In the recent preprint of Connes and Consani https://arxiv.org/abs/2106.01715 a new spectral realization of the critical zeros of $\zeta$ (edit: defined as being those on the critical line only, see ...
- 883
6
votes
0
answers
348
views
Recent work on Pseudo-Laplacian and Pseudo-cuspform in the spirit of Riemann Hypothesis after the work of Bombieri and Garrett
( This is my first MO question . I'm totally inexperienced on MO so, forgive me for my mistakes .)
Paul Garrett and Enrico Bombieri were (are?) Secretly Working on Pseudo-Laplacians and Pseudo-...
user153636
5
votes
2
answers
476
views
Weyl law for arithmetic Fuchsian groups known?
For congruence subgroups of $PSL(2,\mathbb{Z})$, the Weyl law for the eigenvalues of Maass cusp forms had been proven by Selberg. How is the status of such a Weyl law for eigenvalues of Maass cusp ...
- 123
5
votes
0
answers
262
views
p-adic analogue of self-adjoint operator
Consider the very well-known result that any Hermitian matrix over $\mathbb{C}$, say $T$, admits a decomposition $T = UDU^*$ where $U$ is unitary and $D$ is diagonal with real entries. I am looking ...
5
votes
0
answers
539
views
An inverse eigenvalue problem on Jacobi matrices
I am interested in trying to design a Hermitian Jacobi (tridiagonal) matrix $H$ that has specific properties. The basic property, which is simple enough to construct, is that for an $N\times N$ matrix ...
- 291
4
votes
1
answer
375
views
The exceptional eigenvalues and Weyl's law in level aspect
The Weyl law for Maass cusp forms for $SL_2(\mathbb Z)$ was obtained by Selberg firstly and has been generalized to various caces (Duistermaat-Guillemin, Lapid-Müller and others). For example, one of ...
- 143
4
votes
0
answers
624
views
The Bilu-Linial conjecture and Ramanujan graphs
The Bilu-Linial conjecture claims that every $d-$regular graph has a $2-$lift such that for the signing matrix has its eigenvalues between $[-2\sqrt{d-1},2\sqrt{d-1}]$ (the ``signing matrix" is the ...
- 1,893
3
votes
0
answers
186
views
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, ...
- 31
2
votes
0
answers
110
views
Spectral decomposition of idele class group $L^2(\mathbb{I}_F/ F^{\times})$
Let $F$ be a number field. Let $\mathbb{A}_F$ be the ring of adeles. The group of units of $\mathbb{A}_F$ is called the group of ideles $\mathbb{I}_F=\mathbb{A}_F^{\times}= GL_1(\mathbb{A}_F)$. The ...
- 421
1
vote
0
answers
239
views
Norm related to diophantine approximation?
I'm trying to read this paper: Bourgain, J.; Jitomirskaya, S., Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential, J. Stat. Phys. 108, No. 5-6, 1203-1218 (2002), ...
- 785