Intertwining operator method and supersymmetry for effective mass Schrödinger equations

Intertwining relations and Darboux transformations for the wave equations

“…In the case with a constant weighted energy potential, e.g. q(x) = 1, from our supersymmetry approach we get the supersymmetry for the effective mass Schrödinger equation [23,25]. In the case with constant mass m(x) = m 0 from our approach we obtain the supersymmetry for Schrödinger equation with weighted energy [34].…”

“…(2) and (3) can then be written as one single matrix equation in the form (24) On defining H s = diag(Ᏼ, ) and Φ= (φ, ) T , the above matrix Schrödinger Eq. (24) can be written as (25) where I is the 2 × 2 unity matrix. Similar to the case of the standard Schrödinger equation, we define two supercharge operators Q, Q † as follows: (26) where ᏸ and ᏸ † are the operators given by (17) and (22), respectively.…”

“…(100) can be written as (101) where c 1 , c 2 are free constants and v 2 = (-1 + 4α 2 k 2 )/4α 2 . Recently in [25] we have constructed the potentials for effective mass Schrödinger equation (99) with creation of one and two bound states without investigation of potential forms. Here we would like to apply our technique to construction of double well and even triple well potentials with creation of two and three bound states, to generation of completely isospectral potentials and to investigation of the influ ence of position dependent mass on the form of con structed potentials.…”

“…Many interesting exactly solvable models have been constructed for Schrödinger equation using the Darboux and Barg mann transformation techniques [4][5][6][7][8][9][10][11][12][13][14][15]. In the last few years the research efforts on the topic of algebraic methods have been considerably intensified [16][17][18][19][20][21][22][23][24][25][26][27][28] due to the rapid development of nanoelec tronics, the basic elements of which are low dimen sional structures such as quantum wells, wires, dots and superlattices [29][30][31][32][33]. The Darboux transforma tions are one of the most powerful method permitting to extend the class of new exactly solvable models.…”

“…It has been constructed for a variety of linear and nonlin ear equations. Among the linear equations to which the Darboux transformations are applicable, there are Pauli and Dirac equations [4,7,8], conventional Schrödinger equations, see the reviews in [8][9][10][11] the class of Schrödinger equations with position depen dent effective mass [22][23][24][25][26] and Schrödinger type equations with energy dependent potentials [15,34]. The Darboux transformations is essentially based on intertwining relations and differ from other transfor 1 The article is published in the original.…”