局域模(实的及虚的)的一种新表述

本文给出了一种新的表述局域模(实的及虚的)的方法。我们引入了变形豫解式的迹P(z).它仅在z平面的实轴上有奇异性,孤立极点对应于实局域模。将P(E+iη)向下解析延拓至第二黎曼面,则其复极点对应于虚的(或不稳定的)局域模。复极点的实部是不稳定局域模的能量,虚部是它寿命的倒数,自然地得到了寿命为正的限制。此能量与寿命符合于相应共振散射的能量与宽度。本文还讨论了实局域模与虚局域模之间的关系。最后,比较了P(z)的奇异性与态密度的变化,并进一步阐明了寿命为正的及负的解之间的差别。指出,一般说来,只有寿命为正的解当作用强度增加时会变成实的局域模,负寿命的解是不会的。

A NEW REPRESENTATION OF (REAL AND VIRTUAL) LOCALIZED MODES

A new representation of (real and virtual) localized modes in the solid state problems is formulated by considering P(z), the modified resolvent operator. It has singularities on the real axis of the z plane only. The isolated poles correspond to real localized modes. When P(E+iη) is analytically continuated onto the second Riemann surface (η > 0), its complex poles correspond to virtual (or unstable) localized modes. The real part of the complex pole is the energy of an unstable localized mode, and the reciprocal of the imaginary part is its life-time. Naturally the life-time is positive. The energy and life-time derived in this way are in accordance with the energy and width of the corresponding resonance scattering. The relation between real and virtual localized modes is discussed. Finally, the singularities of P(z) are related with the change in density of state. The differences between solutions with positive life-times and that with negative life-times are further explained. It is pointed out that, in general, only solutions with positive life-times can change into localized modes when the interaction strength increases, while for solutions with a negative life-time this does not happen.