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The existence of a minimal measurable length is a common feature of variousapproaches to quantum gravity such as string theory, loop quantum gravity andblack-hole physics. In this scenario, all commutation relations are modifiedand the Heisenberg uncertainty principle is changed to the so-calledGeneralized (Gravitational) Uncertainty Principle (GUP). Here, we present aone-dimensional nonperturbative approach to quantum mechanics with minimallength uncertainty relation which implies X=x to all orders and P=p+(1/3)\betap^3 to first order of GUP parameter \beta, where X and P are the generalizedposition and momentum operators and [x,p]=i\hbar. We show that this formalismis an equivalent representation of the seminal proposal by Kempf, Mangano, andMann and predicts the same physics. However, this proposal reveals manysignificant aspects of the generalized uncertainty principle in a simple andcomprehensive form and the existence of a maximal canonical momentum ismanifest through this representation. The problems of the free particle and theharmonic oscillator are exactly solved in this GUP framework and the effects ofGUP on the thermodynamics of these systems are also presented. Although X, P,and the Hamiltonian of the harmonic oscillator all are formally self-adjoint,the careful study of the domains of these operators shows that only themomentum operator remains self-adjoint in the presence of the minimal lengthuncertainty. We finally discuss the difficulties with the definition ofpotentials with infinitely sharp boundaries.

New Approach to Nonperturbative Quantum Mechanics with Minimal Length Uncertainty