LETTERS TO THE E D I T O R

O N T H E P E R T U R B A T I O N P R O C E D U R E W I T H N O N - H E R M I T I A N

P S E U D O - H A M I L T O N I A N

The orthogonalizat ion pseudo-Hamil tonian - - a theoretical concept currently used in Solid state physics - - is one of the cases when a non-Hermit ian operator appears in quan tum mechan- ics. As shown e.g. in [1], its general form conserving the valence energy values of the original Hermit ian Hamil tonian H is Hp = H + QA where Q is a projection operator in the subspace of the core states of H an& A is an arbitrary operator. The question under consideration is the influence of the possible non-Hermiticity of Hp on the per turbat ion theory.

I f g%, H o are the " t rue" unper turbed wave functions and Hamil tonian respectively, there are two possibilities. One can apply the per turbat ion R directly to Vo, H0 and then construct the , ,perturbed pseudo-Hamil tonian" H o + R + QA; sometimes it is effective to consider as a per- turbat ion the sum R + QA, when the core states of H o + R are known (see e.g, the diffraction model of metal [2]). This is certainly a good scheme no matter whether the QA is non-Hermit ian, if the per turbat ion series converges, of course.

Another possibility is to construct the pseudo-Hamil tonian from H o and only then apply the perturbation. A situation like that cart arise when the physical system is described not by ~0, Ho, but with the help of Hop = H o + Q B (B is an arbitrary operator again) and pseudo-wave func- tions 9o. As in the per turbat ion theory the system of unperturbed wave functions is used as a basis for expansion, it is reasonable to ask, what is the role of the Hermiticity of an operator in have a complete and or thonormal system of eigen-functions, which is equivalent to a basis in Hilbert space.

First of all the b ior thonormal set of adjoint eigen-functions will exist in the non-Hermit ian case instead of the simple or thonormal system. The general question of completeness is not usually investigated carefully in physics, thus let us refer to mathematical l i terature for some rigorous conditions.

According to [3], pp. 189 and 193, the self-ad]oint compact operator T has a finite o r infinite or thonormal system of eigen-functions {Xn} corresponding to the non-zero eigen-values {2n} which is complete in the region of T. To every compact normal opera tor T 4: 0, there belongs an or thonormal system of functions {Xn} and a set of non-zero (complex) numbers {2n} , for which Tx n = 2nXn, T + x n = 2*nx n. The system {Xn} is complete, so that any y of the form Th or T*h can be expanded in a series y = ~ ( y , Xn) x n (the y, h are functions, 0 denotes the scalar product).

n

Moreover, according to [4], let T be a symmetr ic operator with compact inverse T - 1 and let 2 n, x n be the eigen-values and eigen-functions of T - 1 respectively, an - - 1/2n. The or thonormal system {Xn} is complete and Tx n =~ItnX n. All the operators are assumed to be linear.

Condit ions based on a comparison of the system under consideration with another complete or thonormal system can be found in [5] and some more general theorems for dissipative operators in [6].

It is obvious that f rom the purely mathematical view-point the Hermiticity of an operator is by far not sufficient for eigen-functions to form a basis, as is sometimes stated in physical arguments. As for the pseudo-Hamiltonian, the mathematical requirements ment ioned above are not very easy to verify in practice and, on the other hand, they do not yield any convenient conditions for an arbitrary operator A. For instance, the well-known pseudo-Hamil tonian with A = -- V, V being the potential energy operator, is neither Hermit ian nor normal (unless V = const).

One can conclude that the mathematically correct application of the per turba t ion theory on the non-Hermit ian pseudo-Hamil tonian is more complicated than on many other quantum-mechani-

1602 Cz~h. J. Phys. B 19 (1969)

Letters to the Editor

cal operators and always requires special investigation. In addition, the biorthonormal system must be taken in computations, so that summation rules like in [7] cannot be used in perturbation series.

Received 16. 6. 1969. J. (~ADA

Institute of Solid State Physics, Czech. Acad. ScL, Prayue* )

References

[1] Cada J.: Czech. J. Phys. B 19 (1969), 1061. [2] H a r r i s o n W. A.: Pseudopotentials in the Theory of Metals. W. A. Benjamin, Inc., New

York--Amsterdam 1966. [3] Aehiezer N. I., G l a z m a n I. M.: Theory of Linear Operators in Hilbert Space (in Russian).

Nauka, Moscow 1966. [4] Tay lo r A. E.: Introduction to Functional Analysis. John Wiley& Sons, Inc., New York

1958, 343. [5] Ka to T.: Perturbation Theory for Linear Operators. Springer, Berlin--Heidelberg--New

York 1966. [6] G o c h b e r g I. C., K re in M. G.: Introduction to the Theory of Linear Non-Self-Adjoint

Operators in Hilbert Space (in Russian). Nauka, Moscow 1965. [7] Chen J. C. Y., D a l g a r n o A.: Proc. Phys. Soe. 85 (1965), 399.

T H E B O L O M E T R I C D E T E C T I O N O F F E R R O M A G N E T I C R E S O N A N C E

I N M E T A L L I C T H I N F I L M S

The following methods are used for ferromagnetic resonance detection in metallic thin films: 1) measurement of perturbation effects caused by the film being introduced into a microwave cavity [1]; 2) recording of induced tension as a result of the magnetization variation of the film at resonance, the film being introduced in a small coil [2]; 3) detection of galvanomagnetic tension produced by the Hall extraordinary effect [3, 4].

Recently it has been shown that EPR signals can be detected by the bolometric method which measures to what extent the studied sample has heated up as a result of the absorption power at resonance [5, 6]'. We extended this method to the detection of ferromagnetic resonance in thin films and present here the results of these experiments.

As a bolometer we used the metallic ferromagnetic film itself. The latter was obtained by ther- mic evaporation in vacuum (3 • 10- s Torr) of 40yo Ni 60~ Fe ~loy fixed on a small wolfram

Fig. 1. Schematic diagram of the bolometer. 1; support, 2. ferromagnetic film, 3. copper electrodes (~ = 0"4 mm), 4. silver layer.

*) Cukrovarnickd 10, Praha 6, Czechoslovakia.

Czech J; Phys. B 19 (1969) 1603