Related papers: Inverse problems for symmetric doubly stochastic m…
Given an infinite graph $G$ on countably many vertices, and a closed, infinite set $\Lambda$ of real numbers, we prove the existence of an unbounded self-adjoint operator whose graph is $G$ and whose spectrum is $\Lambda$.
Polynomial $n\times n$ matrices $A(\lambda)$ and $B(\lambda)$ over a field $\mathbb F $ are called semi-scalar equivalent if there exist a nonsingular $n\times n$ matrix $P$ over the field $\mathbb F $ and an invertible $n\times n$ matrix…
This paper is concerned with the inverse problem of constructing a symmetric nonnegative matrix from realizable spectrum. We reformulate the inverse problem as an underdetermined nonlinear matrix equation over a Riemannian product manifold.…
A hollow matrix is a square matrix whose diagonal entries are all equal to zero. Define $\lambda^* = \rho^{1/2} + \rho^{-1/2} \approx 2.01980$, where $\rho$ is the unique real root of $x^3 = x + 1$. We show that for every $\lambda <…
The inverse eigenvalue problem for real symmetric matrices of the form 0 0 0 . 0 0 * 0 0 0 . 0 * * 0 0 0 . * * 0 . . . . . . . 0 0 * . 0 0 0 0 * * . 0 0 0 * * 0 . 0 0 0 is solved. The solution is shown to be unique. The problem is also…
Inverse problems constrained by partial differential equations are often ill-conditioned due to noisy and incomplete data or inherent non-uniqueness. A prominent example is full waveform inversion, which estimates Earth's subsurface…
We apply the approach developed in our previous papers to obtain examples of solutions to the inverse spectral problem (ISP) for the canonical Hamiltonian system. One of our goals is to illustrate connections of ISP with classical tools of…
In this work we study the existence of solutions to the following critical fractional problem with concave-convex nonlinearities, \begin{equation*} \left \{ \begin{array}{l} (-\Delta)^su=\lambda u^q+u^{2_s^*-1},\ u>0\quad\text{in…
This paper concerns some inverse eigenvalue problems of the quadratic $\star$-(anti)-palindromic system $Q(\lambda)=\lambda^2 A_1^{\star}+\lambda A_0 + \epsilon A_1$, where $\epsilon=\pm 1$, $A_1, A_0 \in \mathbb{C}^{n\times n}$,…
For a symmetric system, we want to study the problem of crossing an hypersurface in the neighborhood of a given point, when we suppose that all of the available vector fields are tangent to the hypersurface at the point. Classically one…
We say that a square real matrix $M$ is \emph{off-diagonal nonnegative} if and only if all entries outside its diagonal are nonnegative real numbers. In this note we show that for any off-diagonal nonnegative symmetric matrix $M$, there…
Simple semitoric systems were classified about ten years ago in terms of a collection of invariants, essentially given by a convex polygon with some marked points corresponding to focus-focus singularities. Each marked point is endowed with…
Let $\Omega_n$ denote the class of $n \times n$ doubly stochastic matrices (each such matrix is entrywise nonnegative and every row and column sum is 1). We study the diagonals of matrices in $\Omega_n$. The main question is: which $A \in…
Our main result is the proof of an inequality between the spectral numbers of a Lagrangian and the spectral numbers of its reductions, in the opposite direction to the classical inequality (see e.g [Vit92]). This has applications to the…
Necessary and sufficient conditions are presented for a measure to be the spectral measure of a finite range or exponentially decaying perturbation of a periodic Jacobi operator. As a corollary we can fully solve the inverse resonance…
An $n \times m$ non-negative matrix with row sum $m$ and column sum $n$ is called doubly stochastic. We answer the problem of finding doubly stochastic matrices of smallest posible support for every $1 <n \leq m$. Any matrix of minimum…
The specific violated mirror symmetry model is capable of generating the observed lepton weak mixing matrix with a structure similar to the observed one that almost lacks any visible regularities (the "flavor riddle"). The peculiarities of…
In this paper, we prove the existence of positive solutions $(\lambda_1,\lambda_2, u,v)\in \R^2\times H^1(\R^N, \R^2)$ to the following coupled Schr\"odinger system $$\begin{cases} -\Delta u + \lambda_1 u= \mu_1|u|^{p-2}u+\beta v \quad…
We consider spectra of $n$-by-$n$ irreducible tridiagonal matrices over a field and of their $n-1$-by-$n-1$ trailing principal submatrices. The real symmetric and complex Hermitian cases have been fully understood: it is necessary and…
Given a quantum semitoric system composed of pseudodifferential operators, Berezin-Toeplitz operators, or a combination of both, we obtain explicit formulas for recovering, from the semiclassical asymptotics of the joint spectrum, all…