Related papers: Bounds on Eigenfunctions of Quantum Cat Maps

In this paper, we consider a generalized polyharmonic eigenvalue problem of the form $A(u)= \lambda h(u)$ in a bounded smooth domain with Dirichlet boundary conditions in the setting of higher-order Orlicz-Sobolev spaces. Here, $A$ is a…

We study quantum algorithms for testing bipartiteness and expansion of bounded-degree graphs. We give quantum algorithms that solve these problems in time O(N^(1/3)), beating the Omega(sqrt(N)) classical lower bound. For testing expansion,…

The subleading eigenvalues and associated eigenfunctions of the Perron-Frobenius operator for 2-dimensional area-preserving maps are numerically investigated. We closely examine the validity of the so-called Ulam method, a numerical scheme…

Quasi-normal-eigenvalue optimization is studied under constraints $b_1(x) \le B(x) \le b_2 (x)$ on structure functions $B$ of 2-side open optical or mechanical resonators. We prove existence of various optimizers and provide an example when…

We estimate the $L^2$ norm of the restriction to a totally geodesic submanifold of the eigenfunctions of the Laplace-Beltrami operator on the standard flat torus $\mathbb{T}^d$, $d\ge2$. We reduce getting correct bounds to counting lattice…

If $(M,g)$ is a compact Riemannian surface then the integrals of $L^2(M)$-normalized eigenfunctions $e_j$ over geodesic segments of fixed length are uniformly bounded. Also, if $(M,g)$ has negative curvature and $\gamma(t)$ is a geodesic…

Given a compact, $m$-dimensional Riemann manifold $(M,g)$ and a large positive constant $L$ we denote by $U_L$ the subspace of $C^\infty(M)$ spanned by the eigenfunctions of the Laplacian corresponding to eigenvalues $\leq L$. We equip…

We seek to improve the restriction bounds of Neumann data of Laplace eigenfunctions $u_h$ by studying the $L^2$ restriction bounds of Neumann data and their $L^2$ concentration as measured by defect measures. Let $\gamma$ be a closed smooth…

We prove the existence of nontrivial unbounded domains $\O$ in the Euclidean space $\R^d$ for which the Dirichlet eigenvalue problem for the Laplacian on $\Omega$ admits sign-changing eigenfunctions with constant Neumann values on $\partial…

We propose the notion of $E_{2}$-quasi-exact solvability and apply this idea to find explicit solutions to the eigenvalue problem for a non-Hermitian Hamiltonian system depending on two parameters. The model considered reduces to the…

We consider a class of singular Riemannian metrics on a compact Riemannian manifold with boundary and the eigenfunctions of the corresponding Laplace-Beltrami operator. In our setting, the average density of eigenfunctions with eigenvalue…

We analyze the quantum capacity of a unital quantum channel, using ideas from the proof of near-optimality of Petz recovery map [Barnum and Knill 2000] and give an upper bound on the quantum capacity in terms of regularized output $2$-norm…

Based on empirical evidence, quantum systems appear to be strictly linear and gauge invariant. This work uses concise mathematics to show that quantum eigenvalue equations on a one dimensional ring can either be gauge invariant or have a…

We consider half-harmonic maps from $\mathbb{R}$ (or $\mathbb{S}$) to $\mathbb{S}$. We prove that all (finite energy) half-harmonic maps are non-degenerate. In other words, they are integrable critical points of the energy functional. A…

Consider an $ N \times N$ Hermitian one-dimensional random band matrix with band width $W > N^{1 / 2 + \frak c} $ for any $ {\frak c} > 0$. In the bulk of the spectrum and in the large $ N $ limit, we obtain the following results: (i) The…

We consider periodic homogenization of boundary value problems for second-order semilinear elliptic systems in 2D of the type $$ \partial_{x_i}\left(a_{ij}^{\alpha…

Let M be a compact manifold of dimension n, P = P(h) a semiclassical pseudodifferential operator on M, and u = u(h) an L^2 normalised family of functions such that Pu is O(h) in L^2(M) as h goes to 0. Let H be a compact submanifold of M. In…

The paper addresses the doubly elliptic eigenvalue problem $$\begin{cases} -\Delta u=\lambda u \qquad &\text{in $\Omega$,}\\ u=0 &\text{on $\Gamma_0$,}\\ -\Delta_\Gamma u +\partial_\nu u =\lambda u\qquad &\text{on $\Gamma_1$,} \end{cases}…

Let $\mathbb C$ be the complex plane, $E$ be a measurable subset in a segment $[0, R]$ of the positive semiaxis $\mathbb R^+$, $u\not\equiv -\infty$ be a subharmonic function on $\mathbb C$. The main result of this article is an upper…

Nielsen \cite{Nielsen05} recently asked the following question: "What is the minimal size quantum circuit required to exactly implement a specified $% \mathit{n}$-qubit unitary operation $U$, without the use of ancilla qubits?" Nielsen was…