 Advances in Pure Mathematics, 2013, 3, 178-182 http://dx.doi.org/10.4236/apm.2013.31A025 Published Online January 2013 (http://www.scirp.org/journal/apm) Cyclic Operator Decomposition for Solving the Differential Equations Ivan Gonoskov Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod, Russia Email: ivan.gonoskov@gmail.com Received October 19, 2012; revised November 23, 2012; accepted December 6, 2012 ABSTRACT We present an approach how to obtain solutions of arbitrary linear operator equation for unknown functions. The par-ticular solution can be represented by the infinite operator series (Cyclic Operator Decomposition), which acts the gen-erating function. The method allows us to choose the cyclic operators and corresponding generating function selectively, depending on initial problem for analytical or numerical study. Our approach includes, as a particular case, the pertur-bation theory, but generally does not require inside any small parameters and unperturbed solutions. We demonstrate the applicability of the method to the analysis of several differential equations in mathematical physics, namely, classi-cal oscillator, Schrödinger equation, and wave equation in dispersive medium. Keywords: Operator Decomposition; Spectral Theory; Propagator 1. Introduction Various classical and quantum-mechanical problems in theoretical physics lead to the necessity of solving the linear operator equations for unknown functions and, in particular, the differential equations. Exact non-trivial analytical solutions of these equations, which include finite combinations of elementary operations and special functions, are known only for a number of specific cases. However, there are many actual and important cases for which such exact solutions were not still obtained even by using severe approximations for the corresponding interaction operators. For the cases when exact solutions are unknown, some approximate methods are usually used. They can be conventionally divided into two types: 1) varieties of perturbation theory and 2) numerical cal-culations (which generally are also based on perturbation theory). In spite of significant usefulness and applicabil-ity, these methods are not free from various limitations and disadvantages. The perturbation theory approaches may lead to divergent series, they need sometimes suit-able unperturbed solutions, and, finally, they do not pro-vide even estimations for precision in most cases (see [1,2] and references therein). On the other hand, numeri-cal schemes, which are from the very beginning ap-proximate, usually also do not give reliable estimations for the precision (some reasonings can be found in [3,4]). Moreover, they can hardly give an asymptotic behavior of the solutions at infinity. Thus, the development of the general method which allows to overcome some of the above-mentioned difficulties is the main object of our study. In this manuscript we develop an approach based on the theory of Cyclic Operator Decomposition (COD), which gives the opportunities to obtain solutions (exact or approximate) of the differential equations with arbi-trary operators. The particular solution can be repre-sented by the infinite cyclic operators series, which acts the previously determined generating function. The cy-clic operators and the corresponding generating function (COD components) can be specified through the given operators in the differential equation. Under the conver-gence requirement, these COD components can be cho-sen in different ways depending on the certain problem statement. The procedure differs from the using of Born series (or corresponding Neumann series) in the pertur-bation theory [5-8] and S-matrix theory of Heisenberg, Feynmann and Dyson . It can be understood easily by studying, for example, the difference between the formal definition of the generating function and Green’s func-tion (the last one is derived in some cases by using the operator resolvent formalism) [5,7,10-12]. Generally, the proposed series does not require any small parameters or unperturbed solutions for the convergence. But, as a matter of fact, the procedure can be transformed, under some certain choice of COD components, to the “stan-dard” perturbation theory with small parameters. For the potentials without strong singularities, with reasonable Copyright © 2013 SciRes. APM I. GONOSKOV 179choice of the cyclic operators and generating function, the corresponding series usually has uniform conver-gence. Some additional features and advantages of our approach for analytical and numerical solving the differ-ential equations are demonstrated in sections below. 2. Theory of Cyclic Operator Decomposition Let us start from the general case of operator equation for unknown function: ˆ0.DˆD (1) Here is an arbitrary given linear operator and  is an unknown function, which can be a vector or matrix of arbitrary dimensionality. This equation can lead in particular cases to arbitrary linear differential equations, which are considered in examples below. Let us consider a pair of operators and V, which are determined by the following condition: ˆGˆˆˆˆ.DGVˆG11ˆˆˆˆat: ;GGGIˆhat: 0,ggG ˆ (2) Since the choice of this pair is partly optional, we impose additional conditions on the operator : ,th (3) 0, t (4) where I is the identity operator. Any function g, which satisfies Equation (4), will be called generating function. Now we can write the following equation: ˆ1ˆˆ.gIGVˆG (5) As we can check, under the above-mentioned condi-tions for and g, any solution of Equation (5) ful-fills Equation (1). Equation (5) can be solved in terms of the following Cyclic Operator Decomposition: 1111ˆˆˆˆˆˆˆ1ˆˆˆ.gnnIGVGVIGVgGV1ˆˆGV  (6) This is the exact particular solution of Equation (1) with corresponding particular COD components deter-mined by Equations (2) and (3). The solution makes sense only if the obtained series is convergent. This can be achieved in different cases depending on and g, for example, if we work in Banach space and corre-sponding operator norm is 1ˆˆ1GVˆ,D. The convergence of some similar operator series was considered also in [5,13]. The theory can be easily generalized also to the case of the equations with given sources:  (7) where an arbitrary given function  describes the arbi-trary sources. The unknown function  could be found naturally if the inverse operator is known: 1ˆD1ˆD 111ˆˆˆˆngnIGV G. However, the inverse operator can not be easily found for a number of problems. Then, we can write a solution of this equation analogously by using COD: . 0g (8) In contrast to the case of Equation (1), now we can choose  for some non-trivial particular solutions. Then we can obtain the particular solution of Equation (7), which corresponds to the following particular deter-mination of the inverse operator in terms of COD: 1111ˆˆˆˆ.nnDI GVGˆG (9) Important feature of the proposed theory is that, while the conditions Equations (3) and (4) should be fulfilled and the convergence of the series is necessary, we still have a great freedom of choosing and corresponding . Generally, it gives us opportunities to obtain all the gpossible solutions of Equation (1). Sometimes we can naturally choose and ˆGg in accordance, for exam- ple, with the corresponding initial conditions for Cauchy problem or boundary conditions for boundary-value problems. In some cases, the exact solution Equation (6) can be used naturally for obtaining the approximate solution with finite number of terms. It can be done, for example, when, starting from certain number n, the following con- 111ˆˆˆˆnnggGV GVditions are satisfied: . These are the sufficient conditions enabling one to derive the approximate solution with the prescribed accuracy. Further, the proposed method provides another advantage if one performs numerical calculations. According to the exact solution Equation (6), we can use recurrent rela-tions when calculating numerically the approximate solu-tions. In this case, the calculation of any next term in the corresponding series does not require more numerical resources than the calculation of the previous one. 3. Examples In this section we apply the proposed theory of Cyclic Operator Decomposition for the various cases of differ-ential equations. Let us first consider the Cauchy prob-lem for the equation of classical oscillator. Note that this equation, if written in other variables, is the stationary one-dimensional Schrödinger equation with given energy, and it can be transformed also to the Riccati equation by using logarithmic substitution. Copyright © 2013 SciRes. APM I. GONOSKOV 180 20,,,abftfabftft (10) where ft2t is an unknown function, is an ar-bitrary time-dependent frequency and a, b are arbitrary constants. Here, it is natural to choose the components for COD as follows: 12,,ˆ, .gabttGVtdddaaa12212dˆdˆddabtttGat (11) If we fix (by our local convention, which we will use below) that we write for brevity the same variable upper limit of integration as the integration variable and deter-mine the successive integration (step by step from right to left), we can write a simple expression for the solution:  222dddabab abtttttt tttt ttftabttt tattttta    btttbttt2t (12) It is important to note now, that the presented series (Equation (12)) has rapid uniform convergence at least in any interval, where is bounded. For example, in the limited interval 0,t the rate of convergence for ft0 can be estimated in the following way (we assume here for simplicity, that abtt0b, , and the maximum of 2t in the corresponding interval is ): maxC2max2!nnCtnan-thtermofseries. (13) In the same way, the convergence can be demonstrated for other different COD’s, when the cyclic operators are bounded for the given generating functions in the given relevant interval. Moreover, if additionally t 220t is a real function, and , we have a decreasing alternating series for the above example, and we can estimate the precision of partial sum of the series by the value of the last term. t00Now we focus on some particular cases of 2t. To demonstrate that it is possible to obtain a solution with any prescribed precision, we consider a case when 211sin 02tt 1a0b and , , . Calculation of the first two terms in Equation (12) gives 0abtt211sin2ftt tt. 0.0273 (14) By calculating the third term in Equation (12), we obtain  if we consider t in the interval [0,1]. Sometimes we can find also the asymptotic behavior of the solution. As an example, we consider the case 2tt0 and abtt1a0b, , , where  is an arbitrary constant, 1  . Using again Equation (12) we obtain exact solution in the following form:  224112122324tftt   (15)  0t  By analyzing the corresponding series we can obtain a simple upper estimate for the solution : 112221exp,12 2ttft   :t (16) which gives us the following asymptotic behavior at 1122exp .2tft1 (17) In this case, the same asymptotic can be found also from WKB theory (see, for example, quasiclassical ap-proximation in [14,15]). Let us now demonstrate the selective choice of cyclic operators. For that we consider stationary one-dimen- sional Schrödinger equation (we use below the units where 1pm, ): 22d2e0,dxEA xx  (18) xwhere 22Em1 is an unknown function, which describes quantum state with energy E in the continuum; A, β are arbitrary real constants. Without loss of generality (one can use scale transformations of Equation (18)) we can assume , . To find the solution of this equation, we can choose the components for COD in different ways. For example, if one interests in the be-havior of x0x near  and in small values of E, he can choose 22dˆdGx1ˆG and use nearly the same tech- nique as in Equations (11) and (12). However, this choice can be inconvenient for the analysis of the long-range behavior. Another variant of choosing the components for COD is the following (we use also Equation (9) for the particular determination of ): Copyright © 2013 SciRes. APM I. GONOSKOV 1812221100011200dˆ,edˆˆˆˆˆ1ˆdd,gkkxxGmx CxGGVGxxV  121e,,ˆˆ, e,imx imxxCGmVAeimxgxp (19) where C1 and C2 are arbitrary constants. To obtain the general solution, we consider the particular case of gen- erating function and corresponding par-ticular solution x. Then we derive by using a rule of infinite geometric series: 12122e1e11im xkimxAimmAim11001211ˆˆˆˆˆ11e.12kgKKim xGV xGVimAim  (20) From here we obtain 212kimk2pp11e1 enimxn nxpnkxA  (21) and the general solution in the following form:  1xCx Cx (22) The corresponding series converges at any A and real m. In a similar way we can obtain solutions for multi- dimensional equations. Let us consider the stationary Schrödinger equation with potential surface Ur, m- dimensional Laplace operator ∆, and energy E: ulti20.rrEU  (23) Then we can choose ,2EUrˆˆGV , and g is any solution of 0g. From corresponding COD we can obtain a solution: 111VV r10.gV 1 2ˆ (24) The inverse Laplace operator can be written, for ex-ample, as 1ˆFkFˆ, where F is the Fourier transform operator and k is an absolute value of the wave vector in this transform. Another choice of COD components can be better for the finding of the bound states with . We can 0E2Urchoose: , ˆGEˆ2,Vg is any solution of 2gE 0, and write the inverse operator 1ˆG for COD as follows: 11121ˆˆˆ2.2GE FFEk (25) In this way, we can calculate in some cases the terms in the corresponding COD by evaluating the poles at imaginary values 2PkiE . Now we consider time-dependent three-dimensional Schrödinger equation to demonstrate other applications of the proposed method. Let us consider propagation of charged particle with arbitrary electromagnetic interactions (below ,tAr1qc is the vector potential, and ): 21,,,0.2ii tUttt Arr r (26) Here, we can choose the components for COD in a variety of ways depending on peculiar properties of the interactions. One special choice is the following: 00012ˆ,(,) ,ˆd,1ˆ,,,2gttGi ttGitVi tUt   rrAr rˆV 00020ˆˆˆ,1 ddd.tttttttitVitVtVrrˆP (27) where corresponds to the time-dependent Hamilto-nian. It gives the following solution: (28) This solution can be useful for the numerical calcula-tions, namely, for the finding the propagator , which gives: ˆ,,nttP tOt rrˆ,tr,t, see also  and references therein. If we use additionally internal time- ordering, we can transform this expression to Dyson se-ries (see [9,16]). Finally, we consider the wave equation for electro-magnetic waves in dispersive medium. We assume that an arbitrarily given operator  which describes electric dispersion does not depend nonlinearly on the field, i.e. we still have linear problem. In this case the equation for unknown vector potential Ar is the following (see for example ):  2ˆ,,0,tttt  rAr (29) with the initial conditions 00,.tttAASrRr (30) We can choose the following components for COD: Copyright © 2013 SciRes. APM I. GONOSKOV Copyright © 2013 SciRes. APM 182 011ˆˆˆˆˆd,tgtGtttGtr0012,,d, ,ˆd,ttttttttVSrrrRr012,dd, .ntttttttrr Rr (31) Then, we can find the solution, which follows from the corresponding COD series: 0011ˆ,1 dˆnttttt ArSr (32) For this solution the initial magnetic field is equal to SrRr and the initial electric field is equal to . 4. Conclusion In summary, we propose the theory of Cyclic Operator Decomposition, which allows one to obtain particular solutions of linear operator equations for unknown func-tions. In most cases it is possible to obtain all the possi-ble solutions, which satisfy the given conditions. We demonstrate by some reasonings and particular examples that our approach has the following remarkable proper-ties: 1) there is a freedom in choosing the COD compo-nents depending on the certain problem; 2) there is a rapid uniform convergence for most of the considered cases; 3) it is possible to find the asymptotic behavior of the solutions; 4) in many cases when one is analyzing the approximate solution, it is possible to estimate the accu-racy; 5) the proposed approach gives good opportunities for efficient implementation of numerical calculations due to the recurrent relations that can be used in COD. 5. Acknowledgements Author would like to thank academician L. D. Faddeev, M. Yu. Emelin, M. Yu. Ryabikin, and A. A. Gonoskov for the useful and stimulating discussions. REFERENCES  F. J. 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