Elements of spectral theory without the spectral theorem

Krejčiřı́k, David; Siegl, Petr

doi:10.1002/9781118855300.ch5

Cited by 24 publications

(36 citation statements)

References 50 publications

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“…Using our purely mathematical tools we are able to arrive at a better understanding of certain purely formal connections between various structural aspects of the spectra, with the main emphasis put on its unboundedness from below (which could result into instabilities under small perturbations) in an interplay with the emergence of accumulation points in the point spectrum (in the latter case it makes sense to keep in mind the existing terminological ambiguities [12]). …”

Section: Discussion and Remarksmentioning

confidence: 99%

Markov constant and quantum instabilities

Pelantová

Starosta

Znojil

2016

J. Phys. A: Math. Theor.

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For a qualitative analysis of spectra of certain two-dimensional rectangular-well quantum systems several rigorous methods of number theory are shown productive and useful. These methods (and, in particular, a generalization of the concept of Markov constant known in Diophantine approximation theory) are shown to provide a new mathematical insight in the phenomenologically relevant occurrence of anomalies in the spectra. Our results may inspire methodical innovations ranging from the description of the stability properties of metamaterials and of certain hiddenly unitary quantum evolution models up to the clarification of the mechanisms of occurrence of ghosts in quantum cosmology.

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Section: Discussion and Remarksmentioning

confidence: 99%

Markov constant and quantum instabilities

Pelantová

Starosta

Znojil

2016

J. Phys. A: Math. Theor.

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“…Conversely, if P a satisfies the relation (8) with some positive, bounded and boundedly invertible operator Θ, then it is clearly similar to the self-adjoint operator Θ 1/2 P a Θ −1/2 . Hence, the spectrum of P a is necessarily real, which is equivalent to the condition (12) due to the explicit formula (18). So we are back in the situation where P a is self-adjoint.…”

Section: Hidden Symmetriesmentioning

confidence: 99%

“…The notion of quasi-selfadjoint (then called quasi-Hermitian) operators in quantum mechanics was first used by nuclear physicists Scholtz, Geyer and Hahne in 1992 [26], but it was actually considered previously by the mathematician Dieudonné as early as in 1961 [6]. We refer to the review article [19] and the book chapter [18] for mathematical aspects of quasi-self-adjoint quantum mechanics. Given a non-self-adjoint operator P with real spectrum, it is usually not easy to decide whether it is quasi-self-adjoint.…”

Section: Quasi-self-adjointness Of Momenta With Complex Magnetic Fieldsmentioning

confidence: 99%

Complex Magnetic Fields: An Improved Hardy--Laptev--Weidl Inequality and Quasi-Self-Adjointness

Krejčiřı́k¹

2019

SIAM J. Math. Anal.

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We show that allowing magnetic fields to be complex-valued leads to an improvement in the magnetic Hardy-type inequality due to Laptev and Weidl. The proof is based on the study of momenta on the circle with complex magnetic fields, which is of independent interest in the context of PT-symmetric and quasi-Hermitian quantum mechanics. We study basis properties of the non-self-adjoint momenta and derive closed formulae for the similarity transforms relating them to self-adjoint operators.

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“…Also, different scalar products can play a role, and this different products produce different adjoints of the same operators. Moreover, the role of pseudospectra in connection with unbounded operators becomes relevant, [5]. Then, in a sense, loosing self-adjointness makes the mathematical structure reacher.…”

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confidence: 99%

Finite-dimensional pseudo-bosons: A non-Hermitian version of the truncated harmonic oscillator

Багарелло

2018

Physics Letters A

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We propose a deformed version of the commutation rule introduced in 1967 by Buchdahl to describe a particular model of the truncated harmonic oscillator. The rule we consider is defined on a N -dimensional Hilbert space H N , and produces two biorhogonal bases of H N which are eigenstates of the Hamiltonians h = 1 2 (q 2 +p 2 ), and of its adjoint h † . Here q and p are non-Hermitian operators obeying [q, p] = i(1 1−N k), where k is a suitable orthogonal projection operator. These eigenstates are connected by ladder operators constructed out of q, p, q † and p † . Some examples are discussed.Quantum mechanics is often thought to be naturally associated to self-adjoint (or Hermitian 1 ) operators. In particular, the dynamics is deduced out of a self-adjoint Hamiltonian, and the observables of the system are almost always assumed to be self-adjoint as well.In recent years, mainly since the seminal work by Bender and Boettcher, [1], it was understood that self-adjointness is not an essential requirement, since other operators exist, not self-adjoint, having purely real (and discrete) spectra. We refer to [2,3,4] for some reviews on this alternative approach. What is interesting, from a mathematical point of view, is that orthonormal (o.n.) bases of eigenstates are replaced by biorthonormal sets which can be, or not, bases of the Hilbert space where the physical system lives. Also, different scalar products can play a role, and this different products produce different adjoints of the same operators. Moreover, the role of pseudospectra in connection with unbounded operators becomes relevant, [5]. Then, in a sense, loosing self-adjointness makes the mathematical structure reacher. Not only that: from a physical point of view the situation is also rather interesting since, for instance, some so-called PT-symmetric Hamiltonians can be naturally used to describe quantum systems with gain and loss phenomena, see [6,7] and references therein.In recent years, in connection with this kind of operators, we have developed a rather general formalism based on some suitable deformations of the canonical commutation and anticommutation relations (CCR and CAR). These deformations produce what we have called D-pseudo bosons and pseudo-fermions. A rather complete review on both these topics can be found in [8], to which we refer for several details and for some physical applications. Later on a similar framework was proposed for quons and for generalized Heisenberg algebra, [9,10].Here we consider a deformation of a different commutation rule, originally considered in [11], and later analyzed in [12], in connection with a truncated version of the harmonic oscillator. The operator c considered in these papers obeys the following rule 1) in which N = 2, 3, 4, . . . is a fixed natural number, while K is a self-adjoint projection operator, K = K 2 = K † , satisfying the equality Kc = 0. The presence of the term N K in (1.1) makes it possible to find a representation of K and c in terms of N × N matrices. In fact, in absence of this term ...

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Elements of spectral theory without the spectral theorem

Cited by 24 publications

References 50 publications

Markov constant and quantum instabilities

Markov constant and quantum instabilities

Complex Magnetic Fields: An Improved Hardy--Laptev--Weidl Inequality and Quasi-Self-Adjointness

Finite-dimensional pseudo-bosons: A non-Hermitian version of the truncated harmonic oscillator

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