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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23
votes
3answers
2k views

Trapped rays bouncing between two convex bodies

At some point during my research I was confronted with this problem, but I did not dedicate serious time to it. Anyway it stayed in the back of my mind and I'm still interested in hints for it. ...
25
votes
10answers
4k views

real symmetric matrix has real eigenvalues - elementary proof

Every real symmetric matrix has at least one real eigenvalue. Does anyone know how to prove this elementary, that is without the notion of complex numbers?
10
votes
5answers
873 views

Can always a family of symmetric real matrices depending smoothly on a real parameter be diagonalized by smooth similarity transformations?

This question is related to another question, but it is definitely not the same. Is it always possible to diagonalize (at least locally around each point) a family of symmetric real matrices $A(t)$ ...
14
votes
2answers
605 views

Eigenvalues of an “oblique diagonal” matrix

I am looking for guidance about the behavior of powers of a particular matrix (call it $A_n$ for $n\ge2$), which has come up in a counting problem about quantum knot mosaics (a good reference for ...
7
votes
2answers
2k views

Conditions for smooth dependence of the eigenvalues and eigenvectors of a matrix on a set of parameters

Let $A\in\mathcal M_n$ be an $n\times n$ real [symmetric] matrix which depends smoothly on a [finite] set of parameters, $A=A(\xi_1,\ldots,\xi_k)$. We can view it as a smooth function $A:\mathbb ...
3
votes
3answers
2k views

eigenvalues of sum of a non-symmetric matrix and its transpose (A+A^T)

Suppose we have a matrix $M$ such that $M$ is non-symmetric real and has positive eigenvalues. Do we have a relation between eigenvalues/eigenvectors of $(M+M^T)$ and those of $M$? What if $M$ and ...
2
votes
2answers
564 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 ...
6
votes
3answers
669 views

Some explanation about Dynin's formalism

I have seen this claim on the Wikipedia page for the Yang-Mills Millenium problem by Alexander Dynin. He is a mathematician working at the Department of Mathematics of Ohio State University and so, I ...
5
votes
1answer
1k 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 ...
5
votes
1answer
230 views

Spectrum of this ODE

I noticed something interesting studying this Sturm-Liouville Problem: $$ \frac{d}{dx}\left(\sqrt{(1-x^{2})}\frac{df}{dx} \right)+\frac{\left(n \alpha x+\alpha^2 x^{2} + ...
2
votes
1answer
494 views

Global Lichnerowicz Formula Proof (in the Kahler case)?

For a Kahler manifold $M$, let us denote its Dirac operator $\overline{\partial} + \overline{\partial}^\ast$, with respect to a metric $g$, by $D$. Moreover, let us dentoe the Levi-Civita connection ...
7
votes
1answer
632 views

0 eigenvalue for a symmetric tridiagonal matrix

Let $T\in \mathbb{R}^{n\times n}$ be a symmetric tridiagonal matrix having the off--diagonal entries equal to -1. The diagonal entries are all positive, $a_i>0$, $i=\overline{1,n}$, and there ...
3
votes
0answers
133 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 ...
3
votes
1answer
277 views

The first eigenvalue of the Schrödinger operator is simple.

Hello, let $(M,g)$ be a compact and connected Riemannian manifold (possibly with $\partial M\neq \emptyset$). We consider the Friedrichs extension of $L=-\Delta +V: C^{\infty}(M,\mathbb{R})\subset ...
0
votes
0answers
62 views

Signed Laplacians and Ramanujan graphs

Given a signing/2-lift matrix $A_s$ of a $d-$regular graph one has the relationship that the ``Signed Laplacian" is $L = d + A_s$. This $L$ is still the same size as the base graph. But the lifted ...
0
votes
1answer
203 views

Spectrum of the operator PAP, with A self-adjoint and P strictly positive

Let $A$ be an unbounded self-adjoint operator with spectrum $\sigma(A)=\mathbb R$ in a Hilbert space $\mathcal H$. Let $P$ be a bounded operator in $\mathcal H$ satisfying $P\ge1$ and $$ {\rm ...