sl(2, ) as a complex Lie algebra and the associated non-Hermitian Hamiltonians with real eigenvalues

Abstract

The powerful group theoretical formalism of potential algebras is extended to non-Hermitian Hamiltonians with real eigenvalues by complexifying so(2,1), thereby getting the complex algebra sl(2,C) or A 1 . This leads to new types of both PTsymmetric and non-PT-symmetric Hamiltonians.

Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBqj3BWbIqubWexLMBb50ujbqegm0B % 1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqr % Ffpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0F % irpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaa % GcbaWefv3ySLgznfgDOfdarCqr1ngBPrginfgDObYtUvgaiuaacqWF % pepucqWFtepvaaa!46A4! $$ \mathcal{P}\mathcal{

“…We may also inquire into what would happen for the corresponding hyperbolic Scarf II potential (20) for which the corresponding variable z is given by z = i sinh x It is interesting to note that this problem can be easily coped with by going to the PT -symmetric Scarf II potential, obtained by replacing B by iB in (20) [22,23,24,25,26].…”

The Klein–Gordon equation of generalized Hulthén potential in complex quantum mechanics

“…Following the early studies of Bender and his co-workers [3], the PT-symmetric formulation has been succesfully utilized by many authors [4][5][6][7][8][9][10][11][12]. The PT-symmetric but non-Hermitian…”

“…NonHermitian Hamiltonians with real or complex spectra have also been analyzed by using different methods [5][6][7][8][12][13][14].…”

PT-Symmetric Solutions of Schrödinger Equation with Position-Dependent Mass via Point Canonical Transformation

“…Much attentioan has also been focused on PT -symmetric Hamiltonians. Following the early studies of Bender et al [29], the PT -symmetry formulation has been successfully utilized by many authors [30][31][32][33][34][35][36]. The PT -symmetric but non-Hermitian Hamiltonians have real spectra whether the Hamiltonians are Hermitian or not.…”

“…The PT -symmetric but non-Hermitian Hamiltonians have real spectra whether the Hamiltonians are Hermitian or not. Non-Hermitian Hamiltonians with real or complex spectra have also been analyzed by using different methods [31][32][33][34]37].…”