Summary
We study the construction and action of certain Lie algebras of second- and higher-order differential operators on spaces of solutions of well-known parabolic, hyperbolic and elliptic linear differential equations. The latter include theN-dimensional quadratic quantum Hamiltonian Schrödinger equations, the one-dimensional heat and wave equations and the two-dimensional Helmholtz equation. In one approach, the usual similarity first-order differential operator algebra of the equation is embedded in the larger one, which appears as a quantum-mechanical dynamic algebra. In a second approach, the new algebra is built as the time evolution of a finite-transformation algebra on the initial conditions. In a third approach, the inhomogeneous similarity algebra is deformed to a noncompact classical one. In every case, we can integrate the algebra to a Lie group of integral transforms acting effectively on the solution space of the differential equation.
Riassunto
Si studia la costruzione e l’azione di certe algebre di Lie di operatori differenziali del secondo ordine o di ordine più alto su spazi di soluzioni di ben note equazioni differenziali lineari paraboliche, iperboliche ed ellittiche. Quest’ultime comprendono le equazioni di Schrödinger hamiltoniane quantiche quadratiche aN dimensioni, equazioni d’onda e di calore ad una dimensione e l’equazione di Helmholtz bidimensionale. In un primo approccio la solita algebra dell’operatore differenziale del primo ordine disimilarità dell’equazione è immersa in quella più grande, che compare come un’algebra dinamica quantomeccanica. In un secondo approccio si costruisce la nuova algebra come evoluzione temporale di un’algebra a trasformazione finita sulle condizioni iniziali. In un terzo approccio l’algebra di similarità inomogenea è deformata in una classica non compatta. In ogni caso, si può integrare l’algebra ad un gruppo di Lie di trasformazioni integrali che agiscono effettivamente sullo spazio delle soluzioni dell’equazione differenziale.
Резюме
Мы исследуем констрирование и действие некоторых алгебр Ли дифференциальных операторов второго и более высоких порядков на пространствах решений хорошо известных параболических, гиперболических и эллиптических линейных дифференциальных уравнений. Последние включаютN-мерные квадратичные квантовые уравнения Шредингера, одномерные уравнения теплопроводности и волновые уравнения и двумерное уравнение Гельмгольца. В первом подходе, алгебра дифференциальных операторов первого порядка внедряется в большую алгебру, которая выступает как квантовомеханическая динамическая алгебра. Во втором подходе, новая алгебра строится, как временная эволюция алгебры конечных преобразований, исходя из начальных условий. В третьем подходе, алгебра деформируется в некомпактную классическую алгебру. В каждом случае мы можем проинтегрировать алгебру в группу Ли интегральных преобразований, действующих эффективно на пространстве решений дифференциального уравнения.
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References
L. V. Ovsjannikov:Gruppovye Svoystva Differentsialnikh Uravnyeni, Academy of Sciences of the USSR (Siberian Branch) (Novosibirsk, 1962) (translated byG. W. Bluman).
G. W. Bluman andJ. D. Cole:Similarity Methods for Differential Equations. Applied Mathematical Sciences, Vol.13 (Berlin, 1974).
W. Miller jr.:Symmetry and Separation of Variables. Encyclopedia of Mathematics and its Application, Vol.4 (New York, N. Y., 1977).
M. Moshinsky:Group Theory and the Many-Body Problem (New York, N. Y., 1967).
B. G. Wybourne:Classical Groups for Physicists (New York, N. Y., 1974).
V. A. Fock:Zeits. Phys.,98, 145 (1935).
V. Bargmann:Zeits. Phys.,99, 576 (1936).
M. Bander andC. Itzykson:Rev. Mod. Phys.,38, 330, 346 (1966).
C. Fronsdal:Phys. Rev.,156, 1665 (1967).
C. P. Boyer andK. B. Wolf:Lett. Nuovo Cimento,8, 458 (1973).
J. D. Louck, M. Moshinsky andK. B. Wolf:Journ. Math. Phys.,14, 692 (1973).
M. Moshinsky andT. H. Seligman:Ann. of Phys.,114, 243 (1978).
L. L. Armstrong:Phys. Rev. A,3, 1546 (1971).
M. J. Cunningham:Journ. Math. Phys.,13, 33 (1972).
E. G. Kalnins:SIAM Journ. Math. Anal.,6, 340 (1975).
M. Moshinsky andC. Quesne:Journ. Math. Phys.,12, 1772 (1971).
C. Quesne andM. Moshinsky:Journ. Math.,12, 193 (1971).
M. Moshinsky:SIAM Journ. Appl. Math.,25, 193 (1973).
M. Moshinsky, T. H. Seligman andK. B. Wolf:Journ. Math. Phys.,13, 901 (1972).
P. Kramer, M. Moshinsky andT. H. Seligman:Complex extensions of canonical transformations in quantum mechanics, inGroup Theory and its Applications, Vol.3, edited byE. M. Loebl (New York, N. Y., 1975).
K. B. Wolf:Journ. Math. Phys.,15, 1295 (1974).
K. B. Wolf:Journ. Math. Phys.,15, 2102 (1974).
K. B. Wolf:Integral Transforms in Science and Engineering (New York, N. Y., 1979).
C. P. Boyer andK. B. Wolf:Rev. Mex. Fís.,25, 31 (1976).
K. B. Wolf:Journ. Math. Phys.,17, 602 (1976).
K. B. Wolf:Journ. Math. Phys.,18, 1046 (1977).
R. Ya. Grabovskaya andS. G. Krein:Math. Nachr.,75, 9 (1976).
Sophus Lie’s 1880 Transformation Group Paper (translated byM. Ackermann, commented byR. Hermann) (Brookline, Mass., 1977).
V. Bargmann:Ann. Math.,48, 568 (1947).
S. Steinberg andF. Trèves:Journ. Diff. Eq.,8, 333 (1970).
C. P. Boyer:Helv. Phys. Acta,47, 589 (1974).
U. Niederer:Helv. Phys. Acta,45, 802 (1972).
J. G. Nagel:Ann. Inst. H. Poincaré,13 A, 1 (1970).
C. P. Boyer andK. B. Wolf:Journ. Math. Phys.,14, 1853 (1973).
K. B. Wolf andC. P. Boyer:Journ. Math. Phys.,15, 2096 (1974).
K. B. Wolf:The Heisenberg-Weyl ring in quantum mechanics, inGroup Theory and its Applications, Vol.3, edited byE. M. Loebl (New York, N. Y., 1975).
E. G. Kalnins andW. Miller jr.:Journ. Math. Phys.,15, 1728 (1974).
C. P. Boyer andK. B. Wolf:Journ. Math. Phys.,16, 2215 (1975).
F. Soto,La ecuación de Helmholtz y el grupo tridimensional de Lorentz, Tesis professional, Facultad de Ciencias UNAM (1977).
P. M. Morse andH. Feshbach:Methods of Theoretical Physics (New York, N. Y., 1953).
R. L. Anderson, S. Kumei andC. E. Wulfman:Rev. Mex. Fís.,21, 1, 35 (1972).
R. L. Anderson, S. Kumei andC. E. Wulfman:Phys. Rev. Lett.,28, 988 (1972).
R. L. Anderson andD. Peterson:Nonlinear Anal.,1, 481 (1977).
P. Chand, C. L. Mehta, N. Mukunda andE. C. G. Sudarshan:Journ. Math. Phys.,8, 2048 (1967).
N. H. Ibragimov andR. L. Anderson:Journ. Math. Anal. Appl.,59, 145 (1977).
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Steinberg, S., Wolf, K.B. Groups of integral transforms generated by Lie algebras of second- and higher-order differential operators. Nuov Cim A 53, 149–177 (1979). https://doi.org/10.1007/BF02776412
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DOI: https://doi.org/10.1007/BF02776412