Resumo: This is a master’s degree thesis in Mathematical Physics and intends to find symmetry generators of 1 + 1-dimensional Schrödinger equation through the algebraic method of prolongationof vector fields in jet bundles. Basic notions of the prolongation method are presented and its offered a pedagogical dictionary to translate the matrix differential operator method, as well as a parallel between both methods is drawn, showing the advantage of working with prolongations. The potentials considered for a single equation are those of the free particle, linear potential, harmonic oscillator and the Calogero type potential. For a system with two equations we consider
the potentials of the Calogero type both pure and with oscillator deformation. We found similarity transformations that attest the equivalence of the equation of different potentials and allow transforming the generators obtained from one model to another. We have defined an associative product of vector fields to build generalized symmetry generators that allowed us to deduce Heisenberg’s universal enveloping algebra, as well as fixing the structure constants of its
higher spin algebra; out of simple symmetry considerations, how to extend this algebra to any spatial dimension. For two Schrödinger equations with pure and deformed Calogero type potentials, we find two equivalent families, each with an infinite number of higher spin (super)algebras,
which we called q(k, gk), that depend on a continuous parameter gk, which stems from the potential, and a discrete parameter k, which is given by the lowest order of generators in the odd sector of the $Z_2$-graded infinite space. The cases for k = 0, 1, 2, 3 of the deformed potential,
which has a discrete energy spectrum, are worked out in detail. We give the explicit form of all wave functions for any k and the condition imposed by their normalization on the parameters. In addition, we also reveal the exact contractions of the continuous parameters to combine two
different algebras into a single Hilbert space and draw the state diagrams. All commutations of generators that cannot be written as a product are computed, as well as the structure constants in the commutations with wave functions, which reveal that the Hilbert spaces are different for
each k. Comparing the algebras’ structure constants, we conclude that q(k, gk) q(k′, gk′) only if g$_k$ = gk′ = kk′, k and k′ have same parity and k > k′; all other possibilities culminate in different structure constants. The higher-spin superalgebra q(1, g1) was used by Vasiliev in the construction of a Chern-Simons supergravity model and contains the generators of the universal
enveloping algebra of the finite superalgebra osp(2|2). In covariant form, the superalgebra osp(2|2) allows us to interpret the odd sector of q(k, gk) as two-component Majorana spinors in 2 + 1-dimensions and we give an indication of how to alternate the spin representations inside the algebras. Finally, we find the odd generators with simple negative root of the pure potential
for any k, allowing us to find all the other odd generators by commutation with the generatorof positive root in sl(2,R), and then map them in generators of the deformed potential through similarity transformation obtained.
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