Self-adjoint operator ≈ Self-adjoint operator
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Pseudospectra in non-Hermitian quantum mechanics Open
We propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-Hermitian operators. We relate pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint operators, and basis pr…
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Hamiltonian for the Zeros of the Riemann Zeta Function Open
A Hamiltonian operator H[over ^] is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. The classical…
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Measuring non-Hermitian operators via weak values Open
In quantum theory, a physical observable is represented by a Hermitian\noperator as it admits real eigenvalues. This stems from the fact that any\nmeasuring apparatus that is supposed to measure a physical observable will\nalways yield a r…
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Energy observable for a quantum system with a dynamical Hilbert space and a global geometric extension of quantum theory Open
A non-Hermitian operator may serve as the Hamiltonian for a unitary quantum\nsystem, if we can modify the Hilbert space of state vectors of the system so\nthat it turns into a Hermitian operator. If this operator is time-dependent,\nthe mo…
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Localization for a one-dimensional split-step quantum walk with bound states robust against perturbations Open
For given two unitary and self-adjoint operators on a Hilbert space, a spectral mapping theorem was proved in the work of Higuchi et al. (e-print arXiv:1506.06457) [see also E. Segawa and A. Suzuki, Quantum Stud.: Math. Found. 3, 11 (2016)…
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Schrödinger Operators on Graphs and Geometry II. Spectral Estimates for $${\varvec{L}}_\mathbf{1}$$ L 1 -potentials and an Ambartsumian Theorem Open
In this paper we study Schrödinger operators with absolutely integrable potentials on metric graphs. Uniform bounds—i.e. depending only on the graph and the potential—on the difference between the $$n^\mathrm{th}$$ eigenvalues of the Lapla…
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Atomic and molecular decomposition of homogeneous spaces of\n distributions associated to non-negative self-adjoint operators Open
We deal with homogeneous Besov and Triebel-Lizorkin spaces in the setting of\na doubling metric measure space in the presence of a non-negative self-adjoint\noperator whose heat kernel has Gaussian localization and the Markov property.\nTh…
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Functions of self-adjoint operators in ideals of compact operators Open
For self-adjoint operators $A, B$, a bounded operator $J$, and a function\n$f:\\mathbb R\\to\\mathbb C$ we obtain bounds in quasi-normed ideals of compact\noperators for the difference $f(A)J-Jf(B)$ in terms of the operator $AJ-JB$.\nThe f…
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Quantitative inductive estimates for Green’sfunctions of non-self-adjoint matrices Open
We provide quantitative inductive estimates for Green's functions of matrices\nwith (sub)expoentially decaying off diagonal entries in higher dimensions.\nTogether with Cartan's estimates and discrepancy estimates, we establish\nexplicit b…
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Posterior consistency and convergence rates for Bayesian inversion with hypoelliptic operators Open
Bayesian approach to inverse problems is studied in the case where the\nforward map is a linear hypoelliptic pseudodifferential operator and\nmeasurement error is additive white Gaussian noise. The measurement model for\nan unknown Gaussia…
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Spectral representations of normal operators in quaternionic Hilbert spaces via intertwining quaternionic PVMs Open
The possibility of formulating quantum mechanics over quaternionic Hilbert spaces can be traced back to von Neumann’s foundational works in the thirties. The absence of a suitable quaternionic version of spectrum prevented the full develop…
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Survey Article: Self-adjoint ordinary differential operators and their spectrum Open
We survey the theory of ordinary self-adjoint differential operators in\nHilbert space and their spectrum. Such an operator is generated by a\nsymmetric differential expression and a boundary condition. We discuss the\nvery general modern …
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Brown measures of free circular and multiplicative Brownian motions with self-adjoint and unitary initial conditions Open
Let Z_N be a Ginibre ensemble and let A_N be a Hermitian random matrix independent of Z_N such that A_N converges in distribution to a self-adjoint random variable x_0 in a W^* -probability space (\mathscr{A},\tau) . For each t>0 , the ran…
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Quasi boundary triples and semi-bounded self-adjoint extensions Open
In this note semi-bounded self-adjoint extensions of symmetric operators are investigated with the help of the abstract notion of quasi boundary triples and their Weyl functions. The main purpose is to provide new sufficient conditions on …
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Characterization of domains of self-adjoint ordinary differential operators of any order, even or odd Open
We characterize the domains of very general ordinary differential operators of any order, even or odd, with complex coefficients and arbitrary deficiency index.
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Repeatability of measurements: Non-Hermitian observables and quantum Coriolis force Open
A noncommuting measurement transfers, via the apparatus, information encoded in a system's state to the external “observer.” Classical measurements determine properties of physical objects. In the quantum realm, the very same notion restri…
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Fourth-Order Comprehensive Adjoint Sensitivity Analysis (4th-CASAM) of Response-Coupled Linear Forward/Adjoint Systems: I. Theoretical Framework Open
The most general quantities of interest (called “responses”) produced by the computational model of a linear physical system can depend on both the forward and adjoint state functions that describe the respective system. This work presents…
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Spectral enclosures for non-self-adjoint extensions of symmetric operators Open
The spectral properties of non-self-adjoint extensions $A_{[B]}$ of a\nsymmetric operator in a Hilbert space are studied with the help of ordinary and\nquasi boundary triples and the corresponding Weyl functions. These extensions\nare give…
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On the theory of self-adjoint extensions of symmetric operators and its applications to quantum physics Open
This is a series of five lectures around the common subject of the construction of self-adjoint extensions of symmetric operators and its applications to Quantum Physics. We will try to offer a brief account of some recent ideas in the the…
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Spectral analysis of Dirac operators with delta interactions supported on the boundaries of rough domains Open
Given an open set Ω⊂R3, we deal with the spectral study of Dirac operators of the form Ha,τ = H + Aa,τδ∂Ω, where H is the free Dirac operator in R3 and Aa,τ is a bounded, invertible, and self-adjoint operator in L2(∂Ω)4, depending on param…
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The nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (nth-CASAM-L): I. Mathematical Framework Open
This work presents the mathematical framework of the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (abbreviated as “nth-CASAM-L”), which is conceived for obtaining the …
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Self-Adjointness of Dirac Operators with Infinite Mass Boundary Conditions in Sectors Open
This paper deals with the study of the two-dimensional Dirac operatorwith infinite mass boundary condition in a sector. We investigate the question ofself-adjointness depending on the aperture of the sector: when the sector is convexit is …
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Unbounded operators having self‐adjoint, subnormal, or hyponormal powers Open
We show that if a densely defined closable operator A is such that the resolvent set of A 2 is nonempty, then A is necessarily closed. This result is then extended to the case of a polynomial . We also generalize a recent result by Sebesty…
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On Lieb-Thirring inequalities for one-dimensional non-self-adjoint Jacobi and Schrödinger operators Open
We study to what extent Lieb–Thirring inequalities are extendable from self-adjoint to general (possibly non-self-adjoint) Jacobi and Schrödinger operators. Namely, we prove the conjecture of Hansmann and Katriel from [12] and answer anoth…
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Two-dimensional Dirac operators with general δ-shell interactions supported on a straight line Open
In this paper the two-dimensional Dirac operator with a general hermitian δ -shell interaction supported on a straight line is introduced as a self-adjoint operator and its spectral properties are investigated in detail. In particular, it …
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Observables in Quantum Mechanics and the Importance of Self-Adjointness Open
We are focused on the idea that observables in quantum physics are a bit more then just hermitian operators and that this is, in general, a “tricky business”. The origin of this idea comes from the fact that there is a subtle difference be…
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Winding in non-Hermitian systems Open
This paper extends the property of interlacing of the zeros of eigenfunctions\nin Hermitian systems to the topological property of winding number in\nnon-Hermitian systems. Just as the number of nodes of each eigenfunction in a\nself-adjoi…
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Tridiagonality, supersymmetry and non self-adjoint Hamiltonians Open
In this paper we consider some aspects of tridiagonal, non self-adjoint,\nHamiltonians and of their supersymmetric counterparts. In particular, the\nproblem of factorization is discussed, and it is shown how the analysis of the\neigenstate…
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Quasinormal modes and self-adjoint extensions of the Schrödinger operator Open
We revisit here the analytical continuation approach usually employed to\ncompute quasinormal modes (QNM) and frequencies of a given potential barrier\n$V$ starting from the bounded states and respective eigenvalues of the\nSchroedinger op…
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Adjoint shadowing directions in hyperbolic systems for sensitivity analysis Open
For hyperbolic diffeomorphisms, we define adjoint shadowing directions as a bounded inhomogeneous adjoint solution whose initial condition has zero component in the unstable adjoint direction. For hyperbolic flows, we define adjoint shadow…