$$\mathcal{P}\mathcal{T}$$ -supersymmetric partner of a short-range square well

In a box of size L, a spatially antisymmetric square-well potential of a purely imaginary strength ig and size l < L is interpreted as an initial element of the SUSY hierarchy of solvable Hamiltonians, the energies of which are all real for g < g _c (l). The first partner potential is constructed in closed form and discussed. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

Pseudo-Hermiticity, $$\mathcal{P}\mathcal{T}$$ symmetry,and the metric operator

The main achievements of Pseudo-Hermitian Quantum Mechanics and its distinction from the indefinite-metric quantum theories are reviewed. The issue of the non-uniqueness of the metric operator and its consequences for defining the observables are discussed. A systematic perturbative expression for the most general metric operator is offered and its application for a toy model is outlined. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

Point Interactions: $$\mathcal{P}\mathcal{T}$$ -Hermiticity and Reality of the Spectrum

General point interactions for the second derivative operator in one dimension are studied. In particular, -self-adjoint point interactions with the support at the origin and at points ±l are considered. The spectrum of such non-Hermitian operators is investigated and conditions when the spectrum is pure real are presented. The results are compared with those for standard self-adjoint point interactions.     

Spontaneous breakdown of $$ \mathcal{P}\mathcal{T} $$ symmetry in the complex Coulomb potential

The $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} symmetry of the Coulomb potential and its solutions are studied along trajectories satisfying the $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} symmetry requirement. It is shown that with appropriate normalization constant the general solutions can be chosen $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric if the L parameter that corresponds to angular momentum in the Hermitian case is real. $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} symmetry is spontaneously broken, however, for complex L values of the form L = −1/2 + iλ. In this case the potential remains $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric, while the two independent solutions are transformed to each other by the $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} operation and at the same time, the two series of discrete energy eigenvalues turn into each other’s complex conjugate.     

Spectra of $$ \mathcal{P}\mathcal{T} $$-symmetric Hamiltonians on tobogganic contours

A non-standard generalization of the Bender potentials x ²(ix ^ɛ) is suggested. The spectra are obtained numerically and some of their particular properties are discussed.     

Particles versus fields in $$ \mathcal{P}\mathcal{T} $$-symmetrically deformed integrable systems

The elastic scattering and the ⁶He angular distributions were measured in ⁷Li + ⁷Li reaction at two energies, E _lab = 20 and 25 MeV. FRDWBA calculations have been performed to explain the measured ⁶He data. The calculations were very sensitive to the choice of the optical model potentials in entrance and exit channels. The one-step proton transfer was found to be the dominant reaction mechanism in ⁶He production.      

${ \mathcal P }{ \mathcal T }$-symmetry berry phases,topology and ${ \mathcal P }{ \mathcal T }$-symmetry breaking

We study the complex Berry phases in non-Hermitian systems with parity- and time-reversal $\left({ \mathcal P }{ \mathcal T }\right)$ symmetry. We investigate a kind of two-level system with ${ \mathcal P }{ \mathcal T }$ symmetry. We find that the real part of the the complex Berry phases have two quantized values and they are equal to either 0 or π, which originates from the topology of the Hermitian eigenstates. We also find that if we change the relative parameters of the Hamiltonian from the unbroken-${ \mathcal P }{ \mathcal T }$-symmetry phase to the broken-${ \mathcal P }{ \mathcal T }$-symmetry phase, the imaginary part of the complex Berry phases are divergent at the exceptional points. We exhibit two concrete examples in this work, one is a two-level toys model, which has nontrivial Berry phases; the other is the generalized Su–Schrieffer–Heeger (SSH) model that has physical loss and gain in every sublattice. Our results explicitly demonstrate the relation between complex Berry phases, topology and ${ \mathcal P }{ \mathcal T }$-symmetry breaking and enrich the field of the non-Hermitian physics.     

Krein-space formulation of \mathcal{P}\mathcal{T} symmetry, \mathcal{C}\mathcal{P}\mathcal{T}-inner products, and pseudo-Hermiticity

Emphasizing the physical constraints on the formulation of the quantum theory, based on the standard measurement axiom and the Schrödinger equation, we comment on some conceptual issues arising in the formulation of the $\mathcal{P}\mathcal{T}$ -symmetric quantum mechanics. In particular, we elaborate on the requirements of the boundedness of the metric operator and the diagonalizability of the Hamiltonian. We also provide an accessible account of a Krein-space derivation of the $\mathcal{C}\mathcal{P}\mathcal{T}$ -inner product, that was widely known to mathematicians since 1950’s. We show how this derivation is linked with the pseudo-Hermitian formulation of the $\mathcal{P}\mathcal{T}$ -symmetric quantum mechanics.     

Isospectrality of conventional and new extended potentials,second-order supersymmetry and role of $$ \mathcal{P}\mathcal{T} $$ symmetry

We develop a systematic approach to construct novel completely solvable rational potentials. Second-order supersymmetric quantum mechanics dictates the latter to be isospectral to some well-studied quantum systems. $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} symmetry may facilitate reconciling our approach to the requirement that the rationally extended potentials be singularity free. Some examples are shown.     

Scarcity of real discrete eigenvalues in non-analytic complex $$ \mathcal{P}\mathcal{T} $$-symmetric potentials

We find that a non-differentiability occurring whether in real or imaginary part of a complex $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric potential causes a scarcity of the real discrete eigenvalues despite the real part alone possessing an infinite spectrum. We demonstrate this by perturbing the real potentials x ² and |x| by imaginary $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -symmetric potentials ix/it|x| and ix, respectively.     

New quasi-exactly solvable Hermitian as well as non-Hermitian $$ \mathcal{P}\mathcal{T} $$-invariant potentials

We start with quasi-exactly solvable (QES) Hermitian (and hence real) as well as complex $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -invariant, double sinh-Gordon potential and show that even after adding perturbation terms, the resulting potentials, in both cases, are still QES potentials. Further, by using anti-isospectral transformations, we obtain Hermitian as well as $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} -invariant complex QES periodic potentials. We study in detail the various properties of the corresponding Bender-Dunne polynomials.     

Non-supersymmetric extremal RN-AdS black holes in $$ \mathcal{N} = 2 $$ gauged supergravity

We investigate extremal Reissner-Nordström-AdS black holes in fourdimensional \( \mathcal{N} = 2 \) abelian gauged supergravity. We find a new attractor equation which is not reduced to the one in the asymptotically flat spacetime. We also argue a formula which is available even in the presence of the scalar potential. We apply them to the T³-model and the STU-model in generic black hole charge distributions. In addition, focusing on the so-called T³-model with a single neutral vector multiplet, we obtain non-supersymmetric extremal Reissner-Nordström-AdS black hole solutions with regular event horizons in the D0-D4 and the D2-D6 black hole charge configurations. The negative cosmological constant emerges even without the Fayet-Iliopoulos parameters.     

Global Well-Posedness of the Energy-Critical Defocusing NLS on $${\mathbb{R} \times \mathbb{T}^3}$$

We prove global well-posedness in H ¹ for the energy-critical defocusing initial-value problem \({(i\partial_t+\Delta_x)u=u|u|^2,\quad u(0)=\phi,}\) in the semiperiodic setting \({x\in\mathbb{R} \times \mathbb{T}^3}\) .     

Strong generators of the subregular $$\mathcal {W}$$W-algebra $$\mathcal {W}^{K-N}(\mathfrak {sl}_N,f_{sub})$$WK-N(slN,fsub) and combinatorial description at critical level

Letters in Mathematical Physics - We construct explicitly strong generators of the affine $$\mathcal {W}$$-algebra $$\mathcal {W}^{K_0-N}(\mathfrak {sl}_N, f_{sub})$$ of subregular type A....     

Global symmetries and $$\mathcal{N}=2$$ SUSY

We prove that \(\mathcal{N}=2\) theories that arise by taking n free hypermultiplets and gauging a subgroup of \({\text {Sp}}(n)\), the non-R global symmetry of the free theory, have a remaining global symmetry, which is a direct sum of unitary, symplectic, and special orthogonal factors. This implies that theories that have \({\text {SU}}(N)\) but not \({\text {U}}(N)\) global symmetries, such as Gaiotto’s \(T_N\) theories, are not likely to arise as IR fixed points of RG flows from weakly coupled \({\mathcal{N}=2}\) gauge theories.     

Reality and non-reality of the spectrum of $$ \mathcal{P}\mathcal{T} $$-symmetric operators: Operator-theoretic criteria

We generalize some recently established criteria for the reality and non-reality of the spectrum of some classes of \(\mathcal{P}\mathcal{T}\)-symmetric Schrödinger operators. The criteria include cases of discrete spectra and continuous ones.     

Gauge theory in deformed $$ \mathcal{N} $$ = (1, 1) superspace

We review the non-anticommutative Q-deformations of = (1, 1) supersymmetric theories in four-dimensional Euclidean harmonic superspace. These deformations preserve chirality and harmonic Grassmann analyticity. The associated field theories arise as a low-energy limit of string theory in specific backgrounds and generalize the Moyal-deformed supersymmetric field theories. A characteristic feature of the Q-deformed theories is the half-breaking of supersymmetry in the chiral sector of the Euclidean superspace. Our main focus is on the chiral singlet Q-deformation, which is distinguished by preserving the SO(4) ∼ Spin(4) “Lorentz” symmetry and the SU(2) R-symmetry. We present the superfield and component structures of the deformed = (1, 0) supersymmetric gauge theory as well as of hypermultiplets coupled to a gauge superfield: invariant actions, deformed transformation rules, and so on. We discuss quantum aspects of these models and prove their renormalizability in the Abelian case. For the charged hypermultiplet in an Abelian gauge superfield background we construct the deformed holomorphic effective action. The text was submitted by the authors in English.     

$${f(\mathcal R)}$$ quantum cosmology

We have quantized a flat cosmological model in the context of the metric models, using the causal Bohmian quantum theory. The equations are solved and then we have obtained how the quantum corrections influence the classical equations.