arXiv 0710.40622007-10-28hep-th

双复SUSY变换：实势的形变及其谱特征

Double complex SUSY-transformations: deformations of real potentials and their spectral characteristics

Sergei P. Maydanyuk

备注：: 23 pages, 4 files of figures in EPS format; corrections of figures

摘要：

本文提出了复变换能不重合的双复超对称变换方法，使给定的实势$V_{1}$变形，得到精确解。得到了势的形状变形、势在任意能量下的波函数、势在变换能下的波函数、势在变换能下的波函数的显式解，确定了解的连续性条件和等谱条件。以有限宽无限高矩形势阱为起始$V_{1}$，采用离散能谱方法，得到了势阱的新的变形形式和能谱的整体变形.新势包含矩形势作为其部分情形（同时将新势的形状、能谱、所有束缚态的波函数、任意能量下的波函数变换为所需参数下矩形势的相应特征）。以零势为起始$V_{1}$，构造了具有连续能谱的无反射实势的新形式。该势推广了著名的$V_{\rm ref}(x) = A^{2}(1-2 {\rm sech}^{2}{Ax})$型无反射势，允许：在渐近区域中将势$V_{\rm ref}$的尾部拉低到零（保持深度不为零）;继续拉低空穴深度;沿$x$轴任意位移空穴并使其通过零点;产生并增加第二个空穴，使$V_{\rm ref}$转化为双阱势;以连续且简单地控制这种无反射电势的形状的不对称性。

英文摘要：

In paper approach of double complex SUSY-transformations with not coincident complex energies of transformation is developed, allowing to deform given real potential $V_{1}$ with obtaining exact solutions. The explicit solutions of the deformation of shape of the potential, its wave function at arbitrary energy, not coincident with energies of transformation, wave functions at the energies of transformation are obtained, condition of keeping of continuity of the solutions and isospectral condition are determined. Using a rectangular well of finite width with infinitely high walls as the starting $V_{1}$ with discrete energy spectrum, by the proposed approach new types of deformation of this potential with deformation of the energy spectrum as a whole have been obtained. The new potential contains the rectangular well as own partial case (with simultaneous transformation of the shape of this new potential, energy spectrum, wave functions of all bound states, wave function at arbitrary energy into corresponding characteristics of the rectangular well at needed choice of parameters). Using null potential as the starting $V_{1}$ with continuous energy spectrum, new form of reflectionless real potential has been constructed. This potential generalizes well-known reflectionless potential of the type $V_{\rm ref}(x) = A^{2}(1-2 {\rm sech}^{2}{Ax})$, allowing: to pull down tails of the potential $V_{\rm ref}$ in the asymptotic regions up to zero (with keeping of nonzero depth); to pull down continuously the depth of the hole; to displace arbitrary along axis $x$ the hole with its passing through zero; to create and to increase the second hole, transforming $V_{\rm ref}$ into double-well potential; to control continuously and simply the asymmetry of the shape of such reflectionless potential.

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英文摘要：

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