Spectral properties on a circle with a singularity

Scattering cross section resonance originating from a spectral singularity

Using techniques of supersymmetric quantum mechanics, scattering properties of Hermitian Hamiltonians, which are related to non-Hermitian ones by similarity transformations, are studied. We have found that the scattering matrix of the Hermitian Hamiltonian coincides with the phase factor of the non-unitary scattering matrix of the non-Hermitian Hamiltonian. The possible presence of a spectral singularity in a non-Hermitian Hamiltonian translates into a pronounced resonance in the scattering cross section of its Hermitian counterpart. This opens a way for detecting spectral singularities in sca…

Zero width resonance (spectral singularity) in a complex PT-symmetric potential

We show that the complex PT-Symmetric potential, $V(x)=-V_1 {sech}^2x + iV_2 {sech}x ~\tanh x, $, entails a single zero-width resonance (spectral singularity) when $V_1+|V_2|=4n^2+4n+{3\over 4}(n=1,2,3.., |V_2|>|V_1|+ {{sgn}(V_1) \over 4})$ and the positive resonant energy is given as $E_*={1 \over 4}[|V_2|-(1/4+V_1)]$.

Complete particle-pair annihilation as a dynamical signature of the spectral singularity

Motivated by the physical relevance of a spectral singularity of interacting many-particle system, we explore the dynamics of two bosons as well as fermions in one-dimensional system with imaginary delta interaction strength. Based on the exact solution, it shows that the two-particle collision leads to amplitude-reduction of the wave function. For fermion pair, the amplitude-reduction depends on the spin configuration of two particles. In both cases, the residual amplitude can vanish when the relative group velocity of two single-particle Gaussian wave packets with equal width reaches the mag…

Spectral singularities for Non-Hermitian one-dimensional Hamiltonians: puzzles with resolution of identity

We examine the completeness of bi-orthogonal sets of eigenfunctions for non-Hermitian Hamiltonians possessing a spectral singularity. The correct resolutions of identity are constructed for delta like and smooth potentials. Their form and the contribution of a spectral singularity depend on the class of functions employed for physical states. With this specification there is no obstruction to completeness originating from a spectral singularity.

Spectral singularity and deep multiple minima in the reflectivity in non-Hermitian (complex) Ginocchio potential

We bring out the existence of at most one spectral singularity (SS) and deep multiple minima in the reflectivity of the non-Hermitian (complex) Ginocchio potential. We find a parameter dependent single spectral singularity in this potential provided the imaginary part is emissive (not absorptive). The reflectionlessness of the real Hermitian Ginocchio's potential at discrete positive energies gives way to deep multiple minima in reflectivity when this potential is perturbed and made non-Hermitian (complex). A novel co-existence of a SS with deep minima in reflectivity is also revealed wherein …

Phase transition through the splitting of self-dual spectral singularity in optical potentials

We consider optical media which feature antilinear symmetries. We show that: (i) spectral singularities of such media (if any) are always self-dual, i.e., correspond to CPA-lasers; (ii) under the change of a system's parameter the self-dual spectral singularity may split into a pair of isolated complex conjugate eigenvalues, which corresponds to an unconventional and overlooked in the most of previous studies scenario of the phase transition (known as $PT$-symmetry breaking in systems obeying parity-time symmetry); (iii) if the antilinear symmetry is local, i.e., does not involve any spatial r…

随机自伴量子信道的极限谱分布

Limiting spectral distribution of random self-adjoint quantum channels

研究了Kraus算子为随机Hermitian矩阵的量子信道极限谱分布，发现Kraus秩随n趋于无穷时分布为半圆分布，秩固定时则为可用自由概率描述的显式律。

A Free Particle on a Circle with Point Interaction

The quantum dynamics of a free particle on a circle with point interaction is described by a U(2) family of self-adjoint Hamiltonians. We provide a classification of the family by introducing a number of subfamilies and thereby analyze the spectral structure in detail. We find that the spectrum depends on a subset of U(2) parameters rather than the entire U(2) needed for the Hamiltonians, and that in particular there exists a subfamily in U(2) where the spectrum becomes parameter-independent. We also show that, in some specific cases, the WKB semiclassical approximation becomes exact (modulo p…

半直线上离散随机游走的谱量子化及单位圆上的正交多项式

Spectral quantization of discrete random walks on half-line, and orthogonal polynomials on the unit circle

本研究提出半直线离散随机游走的谱量子化方案，建立其与单位圆正交多项式的关联，推广了现有方法并给出常转移概率情形下的谱几何构造及显式表达式。

Spectral singularity in composite systems and simulation of a resonant lasing cavity

We investigate herein the existence of spectral singularities (SSs) in composite systems that consist of two separate scattering centers A and B embedded in one-dimensional free space, with at least one scattering center being non-Hermitian. We show that such composite systems have an SS at $k_{c}$ if the reflection amplitudes $r^{A}\left( k_{c}\right) $ and $r^{B}\left( k_{c}\right) $ of the two scattering centers satisfy the condition $r_{\mathrm{R}% }^{A}\left( k_{c}\right) r_{\mathrm{L}}^{B}\left( k_{c}\right) e^{i2k_{c}\left( x_{B}-x_{A}\right) }=1$. We also extend the condition to the sy…

具有正则奇点的 Schrödinger 方程的谱解

Spectral solutions for the Schrödinger equation with a regular singularity

通过修改 Bethe-like ansatz 重现氢原子光谱，提出并验证了原点奇异势的精确量子化条件（EQC），为具有正则奇点的一维 Schrödinger 方程提供了谱解途径。

Particular spectral singularity in the continuum energies: a manifestation as resonances

We study the coalescence of two bound energy eigenstates embedded in the continuous spectrum of a real Hamiltonian $H[4]$ and the singular point produced by this coalescence. At the singular point, the two unnormalized Jost eigenfunctions are no longer linearly independent but coalesce to give rise to a bound state eigenfunction embedded in the continuum. We disturb the potential $V[4]$ by means of a truncation, this perturbation breaks the singular point in two resonances. The phase shift shows a jump of magnitude $2π$ and the shape of the cross section shows two inverted peaks, this behaviou…

Self-similar features around spectral singularity in complex barrier potential

The spectral singularity (SS) from a non-Hermitian potential is one of the most remarkable scattering feature of non-Hermitian quantum mechanics. At the spectral singular point, the scattering amplitudes diverge to infinite. This phenomena have been extensively studied over the last two decades. The previous studies have suggested the need of extremely fine control of the various system parameters for practical applications of SS. However no study have been carried out to understand the behavior of the scattering amplitude in the vicinity of SS. Such study would be important towards the practi…

A new type of spectral mapping theorem for quantum walks with a moving shift on graphs

The conventional spectral mapping theorem for quantum walks can only be applied for walks employing a shift operator whose square is the identity. This theorem gives most of the eigenvalues of the time evolution $U$ by lifting the eigenvalues of an induced self-adjoint matrix $T$ onto the unit circle on the complex plane. We acquire a new spectral mapping theorem for the Grover walk with a shift operator whose cube is the identity on finite graphs. Moreover, graphs we can consider for a quantum walk with such a shift operator is characterized by a triangulation. We call these graphs triangulab…