January 2011
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Day January 16, 2011

The Black-Scholes Equation and Certain Quantum Hamiltonians

Juan M. Romero, O. Gonzalez-Gaxiola, J. Ruiz de Chavez, R. Bernal-Jaquez

Abstract: In this paper a quantum mechanics is built by means of a non-Hermitian momentum operator. We have shown that it is possible to construct two Hermitian and two non-Hermitian type of Hamiltonians using this momentum operator. We can construct a generalized supersymmetric quantum mechanics that has a dual based on these Hamiltonians. In addition, it is shown that the non-Hermitian Hamiltonians of this theory can be related to Hamiltonians that naturally arise in the so-called quantum finance.

Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)

An exactly solvable quantum-lattice model with a tunable degree of nonlocality

Miloslav Znojil

An array of N subsequent Laguerre polynomials is interpreted as an eigenvector of a non-Hermitian tridiagonal Hamiltonian $H$ with real spectrum or, better said, of an exactly solvable N-site-lattice cryptohermitian Hamiltonian whose spectrum is known as equal to the set of zeros of the N-th Laguerre polynomial. The two key problems (viz., the one of the ambiguity and the one of the closed-form construction of all of the eligible inner products which make $H$ Hermitian in the respective {\em ad hoc} Hilbert spaces) are discussed. Then, for illustration, the first four simplest, $k-$parametric definitions of inner products with $k=0,k=1,k=2$ and $k=3$ are explicitly displayed. In mathematical terms these alternative inner products may be perceived as alternative Hermitian conjugations of the initial N-plet of Laguerre polynomials. In physical terms the parameter $k$ may be interpreted as a measure of the “smearing of the lattice coordinates” in the model.

Mathematical Physics (math-ph); Quantum Physics (quant-ph)