论文标题
用于量子机械引导程序的半决赛编程算法
A Semidefinite Programming algorithm for the Quantum Mechanical Bootstrap
论文作者
论文摘要
我们提出了一个半决赛程序（SDP）算法，以在量子力学的自举方法中找到Schrödinger操作员的特征值。引导程序方法涉及两种成分：变量上的一组非线性约束（能量特征态中运算符的期望值），以及需要满足的阳性约束（Unitarity）。通过固定能量，我们将所有约束线性线性化，并表明可行性问题可以作为优化问题，对于不通过约束和一个额外的松弛变量固定的变量的优化问题，可以测量阳性的失败。为了说明该方法，我们能够获得高精度，急剧的特征力范围，以期在1-D中使用任意限制多项式电位。
We present a semidefinite program (SDP) algorithm to find eigenvalues of Schrödinger operators within the bootstrap approach to quantum mechanics. The bootstrap approach involves two ingredients: a nonlinear set of constraints on the variables (expectation values of operators in an energy eigenstate), plus positivity constraints (unitarity) that need to be satisfied. By fixing the energy we linearize all the constraints and show that the feasability problem can be presented as an optimization problem for the variables that are not fixed by the constraints and one additional slack variable that measures the failure of positivity. To illustrate the method we are able to obtain high-precision, sharp bounds on eigenenergies for arbitrary confining polynomial potentials in 1-D.
