A review is given on the foundations and applications of non-Hermitian classical and quantum physics. First, key theorems and central concepts in non-Hermitian linear algebra, including Jordan normal form, biorthogonality, exceptional points, pseudo-Hermiticity and parity-time symmetry, are delineated in a pedagogical and mathematically coherent manner. Building on these, we provide an overview of how diverse classical systems, ranging from photonics, mechanics, electrical circuits, acoustics to active matter, can be used to simulate non-Hermitian wave physics. In particular, we discuss rich and unique phenomena found therein, such as unidirectional invisibility, enhanced sensitivity, topological energy transfer, coherent perfect absorption, single-mode lasing, and robust biological transport. We then explain in detail how non-Hermitian operators emerge as an effective description of open quantum systems on the basis of the Feshbach projection approach and the quantum trajectory approach. We discuss their applications to physical systems relevant to a variety of fields, including atomic, molecular and optical physics, mesoscopic physics, and nuclear physics with emphasis on prominent phenomena/subjects in quantum regimes, such as quantum resonances, superradiance, continuous quantum Zeno effect, quantum critical phenomena, Dirac spectra in quantum chromodynamics, and nonunitary conformal field theories. Finally, we introduce the notion of band topology in complex spectra of non-Hermitian systems and present their classifications by providing the proof, firstly given by this review in a complete manner, as well as a number of instructive examples. Other topics related to non-Hermitian physics, including nonreciprocal transport, speed limits, nonunitary quantum walk, are also reviewed.

Non-Hermitian Physics

In this paper we show that many kernel methods can be adapted to deal with indefinite kernels, that is, kernels which are not positive semidefinite. They do not satisfy Mercer's condition and they induce associated functional spaces called Reproducing Kernel Kreĭn Spaces (RKKS), a generalization of Reproducing Kernel Hilbert Spaces (RKHS).Machine learning in RKKS shares many "nice" properties of learning in RKHS, such as orthogonality and projection. However, since the kernels are indefinite, we can no longer minimize the loss, instead we stabilize it. We show a general representer theorem for constrained stabilization and prove generalization bounds by computing the Rademacher averages of the kernel class. We list several examples of indefinite kernels and investigate regularization methods to solve spline interpolation. Some preliminary experiments with indefinite kernels for spline smoothing are reported for truncated spectral factorization, Landweber-Fridman iterations, and MR-II.

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Learning with non-positive kernels

Preface. 1. Introduction. Part I: Realization. 2. Notation and Properties of Non-Uniform Spaces. 3. Time-Varying State Space Realizations. 4. Diagonal Algebra. 5. Operator Realization Theory. 6. Isometric and Inner Operators. 7. Inner-Outer Factorization and Operator Inversion. Part II: Interpolation and Approximation. 8. J-Unitary Operators. 9. Algebraic Interpolation. 10. Hankel-Norm Model Reduction. 11. Low-Rank Matrix Approximation and Subspace Tracking. Part III: Factorization. 12. Orthogonal Embedding. 13. Spectral Factorization. 14. Lossless Cascade Factorizations. 15. Conclusions. Appendices: A. Hilbert Space Definitions and Properties. References. Glossary of Notation. Index.

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Time-Varying Systems and Computations

We study a non-Hermitian PT symmetric generalization of an N-particle, two-mode Bose–Hubbard system, modeling for example a Bose–Einstein condensate in a double well potential coupled to a continuum via a sink in one of the wells and a source in the other. The effect of the interplay between the particle interaction and the non-Hermiticity on characteristic features of the spectrum is analyzed drawing special attention to the occurrence and unfolding of exceptional points (EPs). We find that for vanishing particle interaction there are only two EPs of order N + 1 which under perturbation unfold either into [(N +1 )/2] eigenvalue pairs (and in the case of N + 1 odd, into an additional zero-eigenvalue) or into eigenvalue triplets (third-order eigenvalue rings) and (N +1) mod 3 single eigenvalues, depending on the direction of the perturbation in parameter space. This behavior is described analytically using perturbational techniques. More general EP unfoldings into eigenvalue rings up to (N +1 )th order are indicated.

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A non-Hermitian \mathcal{P}\mathcal{T} symmetric Bose–Hubbard model: eigenvalue rings from unfolding higher-order exceptional points

The concepts of boundary relations and the corresponding Weyl families are introduced. Let S be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space h, let H be an auxiliary Hilbert space, let [GRAPHICS] and let JH be defined analogously. A unitary relation G from the Krein space (h(2), J(h)) to the Kre. in space (H-2, J(H)) is called a boundary relation for the adjoint S* if ker Gamma = S. The corresponding Weyl family M(lambda) is de. ned as the family of images of the defect subspaces (n) over cap (lambda), lambda is an element of C \ R under Gamma. Here Gamma need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space H and the class of unitary relations Gamma : ( H-2, J(H)) -> (H-2, J(H)), it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every H-valued maximal dissipative (for lambda is an element of C+) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.

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Boundary relations and their Weyl families

The geometry of spaces with an indefinite metric fundamental classes of operators in spaces with an indefinite metric invariant semi-definite subspaces spectral topics and some applications theory of extensions of isometric and symmetric operators in spaces with an indefinite metric.

I. S. Iokhvidov

Papers

Linear Operators in Spaces with an Indefinite Metric