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It is conjectured that the component wave functions of the Rabi Hamiltonian in Bargmann's Hilbert space are terminating series of spheroidal wave functions and generalized spheroidal wave functions of Leitner and Meixner. Numerical calculations strongly support the conjecture.PACS numbers 03.65.Ge, 02.90. + pThe Rabi Hamiltonian has isolated exact solutions for particular values of the interaction constant. 1 In Bargmann's Hilbert space of analytical functions 2 " 4 the component wave functions are, in this case, terminating series of elementary transcendental functions. 1,5 " 7 This property is due to similarities in the pole structure of the differential equations defining the expansion functions and the differential equations of the component wave functions for the particular values of the interaction constant. We asked ourselves whether this observation could be a guide to more complicated but possibly known expansion functions, which allow for terminating expansions of the wave functions in the general case. By applying these principles (and a fair amount of intuition) we have in fact been able to guess the expansion functions and the expansion. We are, however, unable to produce a mathematical proof. However, we have a vast amount of numerical evidence, which, without any exception, supports our conjecture.The Rabi Hamiltonian in BargmamVs method 1,4 is a linear first-order matrix differential operator,whose eigenvalues A, in the excited state / (/ = 0,1,2 . . . ) are determined by the requirement that the up and down components of the wave functions */*»>> -(^/V2)^ + 1 / 2^/ ->(^)| T } + (f/V2)-* + 1 / 2 // (m) (f )l I > (2)(m= ± y) belong to the space of entire functions. We introduce a new independent variable z = y£ 2 , insert (1) and (2) in the Schrodinger equation, and collect the spin-up and -down components. We then obtain the following system of differential equations: d z-/ m) (z) dz (< \{m + \})4>l m Hz) + K ( z -m + l/2. 1 f \m _±U-m + 1/2 )f i (m Hz) + z+ m+ v 2^-rf m Hz) dz" = 0,Here A, = 2€/ + 1 = Vj + y ~ 2* 2and v t is the baseline parameter introduced by Judd. 8 For m = -y Eqs. (3) and (4) are identical with (2.28) and (2.29), for m = + y with (2.28a) and (2.29a), of Ref. 1. Equations (3) and (4) have an irregular singularity at infinity and two regular singularities at Z = K 2 (exponents 0 and v t ) and z = 0 (exponents 0 and -y). Therefore, this singularity is elementary in Ince's classification. 9 The conjecture is that the functions <£/ m) (z), f^m ) (z) can be expanded in / + 4 solutions H> 2 (3) (],V\A\Z) of a second-order differential equation with the same location and character of the singular points. The differential equation is given by dz 2 wPHz) + J±L + ± Z -K* ^2 (3) (Z) + dz K 2 -+• (Z~K 2 ) z\z Wj , <3) (z) = 0,

A theory, based on the method of moments, is described which predicts exact lineshapes for the optical absorption of bound small polarons. This theory explains the position, large width, and high oscillator strength of the optical absorption of the Vxl centre in MgO and the polarisation of V-in MgO, which has been observed under uniaxial stress.Eine Theorie, beruhend auf der Methode der Momente, wird beschrieben, die exakt die Linienform der optischen Absorption gebundener kleiner Polaronen herleitet. Diese Theorie erklart die Lage, grode Breite und hohe Oszillatorenstarke der optischen Absorption von Vgl Zentren in MgO und die Polarisation von V-in MgO, welche unter einachsigem Druck beobachtet wurden. Theory V-type centres in alkaline earth oxides (a hole trapped a t one of the 02-ions neighboring various negative defects [l])show broad absorption bands [ 2 to 51 in the visible, the origin of which has so far not been understood [ 3 ] . With V-[6] andVi1 [7] in MgO as examples (negative charge represented by a vacancy, isolated or associated with A13+, respectively), we will show that these bands can be explained if the distortion of the centres is taken into account.Even in V-, where the six 02-ions surrounding the vacancy are equivalent, the hole is localised at one ion a t low teperatures and the centre shows tetragonal symmetry. This can be attributed to a distortion of the centre by a coupling of the hole to a TI, phonon (coordinates Q,, Qy, Qz), competing with the tunneling of the hole. This exchange ( J > 0) between neighbouring 02-pz functions (Fig. l a ) respectively. If hole-T1,-phonon interaction is introduced in a manner analogous to that described by Ham [9, 101, it can be shown that to each af the six 02-p, functions there belongs a spherically symmetric oscillator, displaced in Q-space (Fig. lb), with the potential energies (in the limit J +-0)for the wells centred on the &,-axis and similarly for the four equivalent ones.EJT is an abbreviation for V2/(2pmE), where hm,, is the oscillator energy and V , the reduced matrix elementl) describing the strength of hole-phonon coupling.

Theoretical expressions for the complex electrical conductivity of small polarons are given for frequencies co~, where ~ is an average frequency of the optical phonons. In this frequency range the complex conductivity is independent of the temperature.As an application an analysis is carried out of optical and d.c. data for reduced SrTiO 3. It is shown that a consistent interpretation of these data as well with respect to their absolute magnitude as well with respect to their frequency dependence is possible in terms of the small polaron picture.

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