Narrow Results

In this work, we solve the time-independent Schrödinger equation for a Landau system modulated by a non-Hermitian Hamiltonian. The system consists of a spinless particle in a uniform magnetic field submitted to action of a non-${𝒫}{𝒯}$ symmetric complex potential. Although the Hamiltonian is neither Hermitian nor ${𝒫}{𝒯}$-symmetric, we find that the Landau problem under study exhibits an entirely real energy spectrum.

We discuss the quantum theory of an harmonic oscillator with time-dependent mass and frequency submitted to action of a complex time-dependent linear potential with ${𝒫}{𝒯}$ symmetry. Combining the Lewis and Riesenfeld approach to time-dependent non-Hermitian Hamiltonians having ${𝒫}{𝒯}$ symmetry and linear invariants, we solve the time-dependent Schrödinger equation for this problem and use the corresponding quantum states to construct a Gaussian wave packet solution. We show that the shape of this wave packet does not depend on the driving force. Afterwards, using this wave packet state, we calculate the expectation values of the position and momentum, their fluctuations and the associated uncertainty product. We find that these expectation values are complex numbers and as a consequence the position and momentum operators are not physical observables and the uncertainty product is physically unacceptable.

Let $G$ be an $N\times N$ real matrix whose entries are independent identically distributed standard normal random variables $G_{ij}\sim {𝒩}(0,1)$. The eigenvalues of such matrices are known to form a two-component system consisting of purely real and complex conjugated points. The purpose of this paper is to show that by appropriately adapting the methods of [E. Kanzieper, M. Poplavskyi, C. Timm, R. Tribe and O. Zaboronski, Annals of Applied Probability26(5) (2016) 2733–2753], we can prove a central limit theorem of the following form: if ${\lambda }_1,\dots ,{\lambda }_{N_{ℝ}}$ are the real eigenvalues of $G$, then for any even polynomial function $P(x)$ and even $N=2n$, we have the convergence in distribution to a normal random variable

Let $\zeta =\xi +i{\xi }^{\prime }$ where $\xi ,{\xi }^{\prime }$ are iid copies of a mean zero, variance one, subgaussian random variable. Let $N_n$ be an $n\times n$ random matrix with entries that are iid copies of $\zeta$. We prove that there exists a $c\in (0,1)$ such that the probability that $N_n$ has any real eigenvalues is less than $c^n$ where $c$ only depends on the subgaussian moment of $\xi$. The bound is optimal up to the value of the constant $c$. The principal component of the proof is an optimal tail bound on the least singular value of matrices of the form $M_n:=M+N_n$ where $M$ is a deterministic complex matrix with the condition that $\parallel M\parallel \le K{n}^{1/2}$ for some constant $K$ depending on the subgaussian moment of $\xi$. For this class of random variables, this result improves on the results of Pan–Zhou [Circular law, extreme singular values and potential theory, J. Multivariate Anal.101(3) (2010) 645–656] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math.218(2) (2008) 600–633]. In the proof of the tail bound, we develop an optimal small-ball probability bound for complex random variables that generalizes the Littlewood–Offord theory developed by Tao–Vu [From the Littlewood–Offord problem to the circular law: Universality of the spectral distribution of random matrices, Bull. Amer. Math. Soc.$($N.S.$)$46(3) (2009) 377–396; Inverse Littlewood–Offord theorems and the condition number of random discrete matrices, Ann. of Math.$(2)$169(2) (2009) 595–632] and Rudelson–Vershynin [The Littlewood–Offord problem and invertibility of random matrices, Adv. Math.218(2) (2008) 600–633; Smallest singular value of a random rectangular matrix, Comm. Pure Appl. Math.62(12) (2009) 1707–1739].

In this paper, we consider the elliptic Ginibre matrices in the orthogonal symmetry class that interpolates between the real Ginibre ensemble and the Gaussian orthogonal ensemble. We obtain the finite size corrections of the real eigenvalue densities in both the global and edge scaling regimes, as well as in both the strong and weak non-Hermiticity regimes. Our results extend and provide the rate of convergence to the previous recent findings in the aforementioned limits. In particular, in the Hermitian limit, our results recover the finite size corrections of the Gaussian orthogonal ensemble established by Forrester, Frankel and Garoni.