R S Kaushal

Some remarks on complex Hamiltonian systems

The analyticity property of the one-dimensional complex Hamiltonian system H.x;p/D H1.x1;x2;p1;p2/ C iH2.x1;x2;p1;p2/ withpDp1C ix2, xDx1C ip2 is exploited to obtain a new class of the corresponding two-dimensional integrable Hamiltonian systems where H1 acts as a new Hamiltonian and H2 is a second integral of motion. Also a possible connection betweenH1 andH2 is sought in terms of an auto-Backlund transformation.

An exact solution of the Schrödinger wave equation for a sextic potential

Abstract Using an ansatz for the eigenfunction, we obtain an exact analytic solution of the Schrodinger wave equation for the doubly anharmonic (sextic) potential, V(r) = ar2−br4+cr6, where a, b, c are positive and satisfy the constraint b 2 = 4c 3 2 (2l+5)+4ac , with l as an orbital quantum number.

On the quantum bound states for the potential V(r) = ar2+br−4+cr−6

Abstract An ansatz for the eigenfunction is used to obtain an exact closed form solution to the Schrodinger wave equation, for the potential V ( r ) = ar 2 + br −4 + cr −6 . The results arrived at in this framework are discussed in the light of those obtained by Znojil using the continued fraction method for a similar potential.

CONSTRUCTION OF THE SECOND CONSTANT OF MOTION FOR TWO-DIMENSIONAL CLASSICAL SYSTEMS

A general method for the construction of the second constant of motion of third and fourth orders is given for two‐dimensional systems in terms of z=q1+iq2, and z=q1−iq2. Correspondingly, the third‐ and fourth‐order potential equations are obtained whose solutions directly provide the integrable systems. Using the Holt ansatz, the potential equation corresponding to the third‐order invariants has been reduced to a pair of potential equations whose solutions yield a large class of integrable systems.

Quantum mechanics of complex Hamiltonian systems in one dimension

With a view to obtaining further insight into the nature of eigenvalues and eigenfunctions of a stationary state one-dimensional Schrodinger equation corresponding to a non-Hermitian Hamiltonian H(x, p) we investigate the ground-state solutions for a variety of potentials within the framework of an extended complex phase space characterized by x = x1 + ip2, p = p1 + ix2, where (x1, p1) and (x2, p2) are real and considered as canonical pairs. The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunctions for different systems. It is noted that the imaginary part of the eigenvalue, Ei, turns out to be zero for all potentials V(x) with real couplings whereas it turns out to be nonzero for the case when the couplings are complex. The prescription is also extended to study the excited states. The problems related to the normalization of the eigenfunction and the boundary conditions to be used within this framework are also discussed.

Dynamical invariants for two‐dimensional time‐dependent classical systems

General equations are formulated to determine all potentials for two‐dimensional systems of the type L= (1)/(2) ( p21 +p22) −V(q1,q2,t), which admits invariants of the form I=a0+aiξi + (1)/(2) aijξiξj, i, j=1,2, where ξ1 =z=q1+iq2, ξ2=z=q1−iq2, a0, ai, aij are arbitrary functions of t, z=q1+iq2, and z=q1−iq2. Simplifying restrictions reduce the general equation to a tractable form. The resulting equations are solved for a special class of time‐separable potentials and derive (i) the van der Waals‐type long‐range potential, V(r,t)=β(t)(b/r4+d) and (ii) the quark‐confining logarithmic potential, V(r,t)=β(t)λ(ln r+b1/r4+d1). Invariants I for the resulting dynamical systems are found. Some observations on the present method in the context of Katzin and Levine and of Lewis and Leach analyses have also been made.

Complex phase space formulation of supersymmetric quantum mechanics: analysis of shape-invariant potentials

With a view to increasing the scope of applications of supersymmetric quantum mechanics (SUSY QM), we formulate the same in a complex phase space. Within this framework, the concept of shape invariance is reinvestigated and an insight into the eigenvalue spectra of non-Hermitian Hamiltonians is sought. The results are applied to a variety of potentials. We claim that the shape invariance for these potentials in the complex phase space can be retrieved in terms of the prescriptions already proposed in the conventional SUSY QM, in that the transformation of potential parameters takes the form of a reflection in the parameter space. Interestingly, some of these features turn out to be the generalization of the concept of quasi-parity used recently in the context of SUSY QM of PT-symmetric potentials. Further, a correspondence of the present approach with other complex formulations of SUSY QM and also in two real dimensions is demonstrated.

Dynamical algebraic approach and invariants for time‐dependent Hamiltonian systems in two dimensions

Dynamical invariants for time‐dependent (TD) oscillator systems in two dimensions are derived. The applications of the dynamical algebraic approach known already for one‐dimensional TD systems [R. S. Kaushal and H. J. Korsch, J. Math. Phys. 22, 1904 (1981)] are now carried out to two‐dimensional TD systems. A possible generalization of Ermakov systems to higher dimensions is discussed.

Two-dimensional time-dependent classical systems with confining potentials

Abstract The construction of invariants for two-dimensional, time-dependent classical systems has been carried out with special reference to linearly and harmonically confining potentials.

Further examples of integrable systems in two dimensions

The construction of the second constant of motion of second order for two-dimensional classical systems is carried out in terms ofz=q1 +iq2 andq=q1 −iq2. As a result a class of Toda-type potentials admitting second order invariants is explored.