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We consider the Dirac operators on R n , n ≥ 2 with singular potentials 1 D A , Φ , m , Γ δ Σ = D A , Φ , m + Γ δ Σ where 2 D A , Φ , m = ∑ j = 1 n α j - i ∂ x j + A j + α n + 1 m + Φ I N is a Dirac operator on R n with the variable magnetic and electrostatic potentials A = (A 1 ,... , A n) and Φ , and the variable mass m. In formula (2) α j are the N × N Dirac matrices, that is α j α k + α k α j = 2 δ jk I N , I N is the unit N × N matrix, N = 2 n + 1 / 2 , Γ δ Σ is a singular delta-type potential supported on a uniformly regular unbounded C 2 - hypersurface Σ ⊂ R n being the common boundary of the open sets Ω ± . Let H 1 (Ω ± , C N) be the Sobolev spaces of N - dimensional vector-valued distributions u on Ω ± , and H 1 (R n ╲ Σ , C N) = H 1 (Ω + , C N) ⊕ H 1 (Ω - , C N). We associate with the formal Dirac operator D A , Φ , m , Γ δ Σ the interaction (transmission) operator D A , Φ , m , B Σ = D A , Φ , m , B Σ defined by the Dirac operator D A , Φ , m on H 1 (R n ╲ Σ , C N) and the interaction condition B Σ : H 1 (R n B A , m , Φ , B Σ , C N) → H 1 / 2 (Σ , C N) associated with the singular potential. The main goal of the paper is to study the Fredholm property of the operators D A , Φ , m , B Σ for some non-compact C 2 -hypersurfaces. [ABSTRACT FROM AUTHOR]

We consider the Dirac operators with singular potentials 1 D A , Φ , m , Γ δ Σ = D A , Φ , m + Γ δ Σ where 2 D A , Φ , m = ∑ j = 1 n α j - i ∂ x j + A j + α n + 1 m + Φ I N is a Dirac operator on R n with variable magnetic and electrostatic potentials A = (A 1 ,... , A n) , Φ , and the variable mass m. In formula (2), α j are the N × N Dirac matrices, that is α j α k + α k α j = 2 δ jk I N , I N is the unit N × N matrix, N = 2 n + 1 / 2 , Γ δ Σ is a singular delta-potential supported on C 2 - hypersurface Σ ⊂ R n periodic with respect to the action of a lattice G on R n. We consider the self-adjointnes and discretness of the spectrum of unbounded in L 2 (T , C N) operators associated with the formal Dirac operator (1) on the torus T = R N ∕ G . We study the band-gap structure of the spectrum of self-adjoint operators D in L 2 (R n , C N) associated with the formal Dirac operator (1) on R n with G -periodic regular and singular potentials. We also consider the Fredholm property and the essential spectrum of unbounded operators associated with non-periodic regular and singular potentials supported on G -periodic smooth hypersurfaces in R n. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ R 3 be an open set. We study the spectral properties of the free Dirac operator H : = - i α · ∇ + m β coupled with the singular potential V κ = (ϵ I 4 + μ β + η (α · N)) δ ∂ Ω , where κ = (ϵ , μ , η) ∈ R 3 . The open set Ω can be either a C 2 -bounded domain or a locally deformed half-space. In both cases, self-adjointness is proved and several spectral properties are given. In particular, we give a complete description of the essential spectrum of H + V κ in the case of a locally deformed half-space, for the so-called critical combinations of coupling constants. Finally, we introduce a new model of Dirac operators with δ -interactions and deal with its spectral properties. More precisely, we study the coupling H ζ , υ = H + - i ζ α 1 α 2 α 3 + i υ β α · N δ ∂ Ω , with ζ , υ ∈ R . In particular, we show that H 0 , ± 2 is essentially self-adjoint and generates confinement. [ABSTRACT FROM AUTHOR]

The N -radial Schrödinger equation is analytically solved. The Cornell potential is extended to finite temperature. The energy eigenvalues and the wave functions are calculated in the N -dimensional form using the Nikiforov–Uvarov (NV) method. At zero temperature, the energy eigenvalues and the wave functions are obtained in good agreement with other works. The present results are applied on the charmonium and bottomonium masses at finite temperature. The effect of dimensionality number is investigated on the quarkonium masses. A comparison is discussed with other works, which use the QCD sum rules and lattice QCD. The present approach successfully generalizes the energy eigenvalues and corresponding wave functions at finite temperature in the N -dimensional representation. In addition, the present approach can successfully be applied to the quarkonium systems at finite temperature. [ABSTRACT FROM AUTHOR]

The possible existence of η′-nucleus bound states has been put forward through theoretical and experimental studies. It is strongly related to the η′ mass at finite density, which is expected to be reduced because of the interplay between the U(1) anomaly and partial restoration of chiral symmetry. The investigation of the C( p, d) reaction at GSI and FAIR, as well as an overview of the experimental program at GSI and future plans at FAIR are discussed. [ABSTRACT FROM AUTHOR]