Multi-Well Potentials in Quantum Mechanics and Stochastic Processes
Victor P. Berezovoj, Glib I. Ivashkevych, Mikhail I. Konchatnij
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Symmetry, Integrability and Geometry: Methods and Applications SIGMA (2010), 098, 18 pages Multi-Well Potentials in Quantum Mechanicsand Stochastic Processes ⋆ Victor P. BEREZOVOJ, Glib I. IVASHKEVYCH and Mikhail I. KONCHATNIJA.I. Akhiezer Institute of Theoretical Physics, National Scientific Center“Kharkov Institute of Physics and Technology”, 1 Akademicheskaya Str., Kharkov, Ukraine
E-mail: [email protected], [email protected], [email protected]
Received October 06, 2010, in final form December 01, 2010; Published online December 18, 2010doi:10.3842/SIGMA.2010.098
Abstract.
Using the formalism of extended N = 4 supersymmetric quantum mechanicswe consider the procedure of the construction of multi-well potentials. We demonstrate theform-invariance of Hamiltonians entering the supermultiplet, using the presented relationfor integrals, which contain fundamental solutions. The possibility of partial N = 4 super-symmetry breaking is determined. We also obtain exact forms of multi-well potentials, bothsymmetric and asymmetric, using the Hamiltonian of harmonic oscillator as initial. Themodification of the shape of potentials due to variation of parameters is also discussed, aswell as application of the obtained results to the study of tunneling processes. We considerthe case of exact, as well as partially broken N = 4 supersymmetry. The distinctive featureof obtained probability densities and potentials is a parametric freedom, which allows tosubstantially modify their shape. We obtain the expressions for probability densities underthe generalization of the Ornstein–Uhlenbeck process. Key words: supersymmetry; solvability; partial breaking of N = 4 supersymmetry; stochas-tic processes Hamiltonians for systems with multi-well potentials are in the focus of classical and quantumdynamics. One of the most fascinating applications of multi-well potentials in classical dynamicsis the description of particles transitions from one local minimum to another under the influenceof different types of noise. This problem is general for many physical systems from Universeto microphysics. Calculations of the transition rates from one local minimum to another wereinitiated by Kramers [1] and after 70 years of development the problem is far from complete [2].From the modern perspective another effect of amplification of transitions over the barrier underthe action of weak time dependent periodic signal, which is called stochastic resonance [3, 4]attracts the special attention.The quantum mechanical dynamics in multi-well potentials is interesting for its own reasons.We just recall the tunneling effect, which is in focus again in respect to the wide-ranging researchof trapping of atoms of alkali metals in superfluid state [5, 6]. Moreover, when more thanone barrier exists, the resonant amplification of tunneling rate is possible, which is known asresonant tunneling [7, 8]. Experimental observation of this phenomenon in superconductiveheterostructures forms a basis for construction of resonant tunneling diode. We have to pointout the importance of having a mechanism of changing the parameters of the potential (locationsof minima, heights of barriers etc.), to determine the conditions of resonant tunneling. ⋆ V.P. Berezovoj, G.I. Ivashkevych and M.I. KonchatnijClassical and quantum systems mentioned in the above, are common in methods of theiranalysis. The dynamics of stochastic systems is described by the Fokker–Planck equation (FP)[9, 10]. One of the methods for solving the FP equation is the eigenfunction expansion method.Due to a formal similarity of the FP and the Schr¨odinger equations, the eigenfunction expansionmethod, which is analogous to the bound state expansion of the Schr¨odinger equation, can beapplied to the Fokker–Planck equation. For instance, in the case of bistable stochastic system,the knowledge of the full set of wave functions and eigenvalues of the corresponding quantumHamiltonian completely determines the time evolution of the solutions to the FP equation.Moreover, it allows one to make conclusions about the dynamics of the corresponding metastablesystem. Analysis of processes in multi-well potentials is complicated due to the fact that theexisting models deal usually with piecewise potentials (such as constructed from rectangular orparabolic wells and barriers), which are presumably far from the real potentials. That is whyanalytic wave functions and spectrum in such potentials are unknown, which implies the onlynumerical analysis of their properties.Solving the FP equation exactly in this approach is closely related to the existence of exactlysolvable quantum-mechanic problems, the number of which significantly increases during thelast years [11, 12]. Recall that the first meaningful example of construction of the significantlynonlinear models of diffusion in bistable system was constructed in [13, 14]. The formalism ofthe Darboux transformation, used in these papers, is the one of the basic methods in construct-ing the isospectral Hamiltonians in supersymmetric quantum mechanics (SUSY QM). Furtherdevelopment in the construction of the exactly solvable stochastic models was achieved in [15],where the Fokker–Planck models with prescribed properties were constructed by use of theDarboux–Grum procedure. Existence of exactly solvable [16] and partially solvable [17] quan-tum mechanical models with multi-well potentials could fill the gap in theoretical analysis ofthe above mentioned processes and may be considered as a more realistic approximation in theirresearch. Approach based on the accounting the instanton contribution in double-well-like po-tentials [18, 19] is commonly used in the description of tunneling processes and, in particular,in studying the features of the Bose–Einstein condencates in multi-well traps. The instantoncalculus is also well applied to the transition between the wells under the influence of noise,because the dynamics of these processes is mainly determined by the energy of the first excitedstate.This paper is aimed at determining the relations between different types of potentials in theframework of the extended supersymmetric quantum mechanics ( N = 4 SUSY QM) [20, 21, 22].We also study the possibility of changing parameters of potentials in wide range and obtainingthe exact expressions for spectrum and wave functions. The latter is especially important whenpotential is equipped with two and even more local minima, so the “resonant” tunneling maybe realized. The obtained expressions are used for derivation of new exactly solvable stochasticmodels. Here we develop the approach of [23] to the construction of exactly solvable stochasticmodels for potentials with several local minima. Contrary to the case of previously establishedconnection between the FP equation and N = 2 SUSY QM, the considered approach allows oneto extend the range of exactly solvable stochastic models, since the super Hamiltonian of N = 4SUSY QM contains large number of the isospectral Hamiltonians. A special characteristic ofthe approach is the existence of a parametric freedom in probability densities and new poten-tials entering the Langevin equation. It makes, in particular, possible to change the shape ofpotentials and densities without changing the time dependence of the probability density. Thisfact makes possible to study the characteristics of stochastic dynamics of particles in multi-wellpotentials with variable parameters. On the other hand, the development of new exactly solvablediffusional models leads to the construction of more adequate approximations to real processes.In Section 2 we briefly discuss the procedure of construction of isospectral Hamiltonians withadditional states below the ground state of the initial Hamiltonian in N = 4 SUSY QM. We es-ulti-Well Potentials in Quantum Mechanics and Stochastic Processes 3pecially emphasize that multi-well potentials can arise due to general properties of the solutionsto the Schr¨odinger type auxiliary equation. In particular, we discuss the phenomenon of partialsupersymmetry breaking in N = 4 SUSY QM. In the next section we give general expressions,that allows one to analyze the obtained potentials and wave functions, independently on theconcrete form of the initial Hamiltonian. It allows, for instance, to generalize the concept ofform-invariant potentials [24] and to calculate the normalization constants of zero-modes wavefunctions in the case of exact and partially broken supersymmetry. Section 4 is devoted tothe construction of stochastic models, by use of connections in the supermultiplet of isospectralHamiltonians of N = 4 SUSY QM. We obtain the expressions for corresponding probabilityfunctions and potentials, entering the Langevin equation. Using the harmonic oscillator Hamil-tonian as the initial one, we obtain the expressions of the potentials of isospectral Hamiltoniansand wave functions for the case of exact and partially broken supersymmetry. The analysis ofconditions for emergence of double and triple-well potentials and ways for varying of their formin wide range is provided in Section 5. The proposed scheme of construction of new stochasticmodels is demonstrated in Section 6 on the example of generalization of well known Ornstein–Uhlenbeck process. We give the results of the calculations of probability densities and discussthe ways of their modification when parameters of scheme vary. In conclusion we summarizethe main results and their possible applications. N = 4 SUSY QMwith additional states Extended N = 4 SUSY QM [20, 21, 22] is equivalent to the second-order polynomial SUSY QM(reducible case) [25, 26, 27, 28] and assumes the existence of complex operators of sypersym-metries Q ( ¯ Q ) and Q ( ¯ Q ), through which the Hamiltonians H σ σ can be expressed. Hamil-tonian of N = 4 SUSY QM has a form ( ~ = m = 1): H σ σ = 12 (cid:0) p + V ( x ) + σ (1)3 V ′ ( x ) (cid:1) ≡ (cid:0) p + V ( x ) + σ (2)3 V ′ ( x ) (cid:1) ,V i ( x ) = W ′ ( x ) + 12 σ ( i )3 W ′′ ( x ) W ′ ( x ) , where W ( x ) is a superpotential and σ ( i )3 are matrices, which commute with each other and haveeigenvalues ± σ (1)3 = σ ⊗ σ (2)3 = 1 ⊗ σ . Supercharges Q i of extended supersymmetricquantum mechanics form the algebra: (cid:8) Q i , ¯ Q k (cid:9) = 2 δ ik H, { Q i , Q k } = (cid:8) ¯ Q i , ¯ Q k (cid:9) = 0 , i, k = 1 , ,Q i = σ ( i ) − ( p + iV i +1 ( x )) , ¯ Q i = σ ( i )+ ( p − iV i +1 ( x )) , (2.1)where V ( x ) ≡ V ( x ), σ (1) ± = σ ± ⊗ σ (2) ± = 1 ⊗ σ ± . Hamiltonian and supercharges act onfour-dimensional internal space and Hamiltonian is diagonal on vectors ψ σ σ ( x, E ), where σ , σ are eigenvalues of σ (1)3 , σ (2)3 . Supercharges Q i ( ¯ Q i ) act as lowering (raising) operators for in-dices σ , σ . It is convenient to represent the Hamiltonian structure and the connection betweenwave functions in diagram form (see below).Obviously, due to commutativity of operators Q i and ¯ Q i with Hamiltonian, all ψ σ σ ( x, E ) areeigenfunctions of Hamiltonian with the same eigenvalue E . The only exception is the case, whenthe wave functions turn to 0 under the action of generators of supersymmetry.Construction of isospectral Hamiltonians within N = 4 SUSY QM is based on the factthat four Hamiltonians are combined into the supermultiplet H σ σ . Nevertheless, it has to be V.P. Berezovoj, G.I. Ivashkevych and M.I. Konchatnij ψ −− ( x, E ) ψ − + ( x, E ) ψ ++ ( x, E ) ψ + − ( x, E ) Q ¯ Q Q ¯ Q Q ¯ Q Q ¯ Q noted that due to the symmetry under σ ↔ σ , i.e. H − σ σ ≡ H σ − σ only three of them arenontrivial. Let’s note, that such a relation in the case of higher-derivative SUSY [26] establishesthe correspondence between a quasi-Hamiltonian and operators of the Schr¨odinger type, and isidentical for any superpotentials. The procedure of construction of isospectral Hamiltonians,when ground state is removed from the initial Hamiltonian, is considered in [22] in detail.We will consider the construction of isospectral Hamiltonians by adding the states above theground state of initial Hamiltonian. Similar procedure was already performed (e.g. in [23]), butthe distinctive feature of the present research is to obtain of general results without specificationof concrete form of the initial Hamiltonian. Let’s consider the auxiliary equation: Hϕ ( x ) = εϕ ( x ) . (2.2)Let’s take one of the Hamiltonians as the initial one H σ σ = 12 ( p − iσ V ( x ))( p + iσ V ( x )) + ε ≡
12 ( p − iσ V ( x ))( p + iσ V ( x )) + ε, (2.3)where ε is the so-called factorization energy. Hereafter the energy is measured from ε . Strictlyspeaking, the supersymmetry relations with supercharges (2.1) expressed through superpoten-tial W ( x ) are satisfied for shifted by ε Hamiltonian H − ε . However, due to commutativityof supercharges with constant ε , the relations between wave functions of H and H − ε remainthe same. When fixing operator H σ σ , the form of W ( x ) depends on the choice of factorizationenergy ε , hence the Hamiltonians H σ − σ , H − σ σ , H − σ − σ also have nontrivial dependence on ε .When ε < E (where E is the ground state energy of initial Hamiltonian), the auxiliaryequation (2.2) has two linear independent solutions ϕ i ( x, ε ), i = 1 ,
2, which are nonnegativeand have the following asymptotics [15]: ϕ ( x ) → + ∞ ( ϕ ( x ) →
0) under x → −∞ , and ϕ ( x ) → ϕ ( x ) → + ∞ ) under x → + ∞ , i.e. the general solution has the form ϕ ( x, ε, c ) = N ( ϕ ( x, ε ) + cϕ ( x, ε )) with appropriately chosen constants ( N is the normalization constant)and has no zeros. Thus, the function ˜ ϕ ( x, ε, c ) = N − ϕ ( x,ε,c ) is finite and can be normalized at everyconcrete choice of ε and c . Let’s note that, for some values of ε and c , ϕ ( x, ε, c ) can have localextrema. In this case the natural choice of the initial Hamiltonian is H + − or H − + (which areidentical due to the symmetry of H σ σ under σ ↔ σ ). Then the superpotential has the form: W ( x, ε, λ ) = −
12 ln (cid:18) λ Z xx i dt ˜ ϕ ( t, ε, c ) (cid:19) , (2.4)with two new arbitrary parameters λ , x i , but one of them is inessential, because it gives an ad-ditional contribution to W ( x ). All the Hamiltonians forming the supermultiplet have nontrivialdependence on these parameters.To consider the connection between Hamiltonians from the supermultiplet, let’s take H − + asthe initial one. Denoting the solution to H − + ψ − + ( x, E ) = Eψ − + ( x, E )ulti-Well Potentials in Quantum Mechanics and Stochastic Processes 5as ψ − + ( x, E ) and using the first representation of the Hamiltonian H σ σ (the l.h.s. of (2.3)), weobtain the following relation between H −− , ψ −− ( x, E ) and the initial expressions: H −− = H − + + d dx ln ˜ ϕ ( x, ε, c ) ,ψ −− ( x, E i ) = 1 p E i − ε ) W (cid:8) ψ − + ( x, E i ) , ϕ ( x, ε, c ) (cid:9) ϕ ( x, ε, c ) ,ψ −− ( x, E = 0) = N − ϕ ( x, ε, c ) = ˜ ϕ ( x, ε, c ) . (2.5)The new state with E = 0 and by definition normalized wave function (remind that energiesare measured from ε ) appears in the Hamiltonian. For the discrete spectrum the normalizationof the excited states wave functions is conserved. Using the second representation (the r.h.s.of (2.3)) H σ σ = 12 ( p − iσ V σ ( x ))( p + iσ V σ ( x )) + ε and identity H − + ≡ H + − the relation between H ++ , ψ ++ ( x, E ) and initial Hamiltonian and wavefunctions can be obtained: H ++ = H − + + d dx ln ˜ ϕ ( x, ε, c )1 + λ R xx i dt ˜ ϕ ( t, ε, c ) ! ,ψ ++ ( x, E = 0) = N − λ ˜ ϕ ( x, ε, c )(1 + λ R xx i dt ˜ ϕ ( t, ε, c )) ,ψ ++ ( x, E i ) = 1 p E i − ε ) ddx + ddx ln ˜ ϕ ( x, ε, c )(1 + λ R xx i dt ˜ ϕ ( t, ε, c )) ! ψ − + ( x, E i ) . (2.6)It is worth mentioning that the normalization of the wave function ψ ++ ( x, E = 0), as in the caseof one-well potentials, can always be performed at any ˜ ϕ ( x, ε, c ) and λ by use of the followingexpression in the normalization condition: N − λ ˜ ϕ ( x, ε, c )(1 + λN − R x −∞ dt ˜ ϕ ( t, ε, c )) = − N − λ λN − ddx λN − R x −∞ dt ˜ ϕ ( t, ε, c )) . From this relation it is easy to derive the relation between the normalization constants: N − λ =(1+ λ ) N − . The normalization of ψ ++ ( x, E i ) is the same as for ψ − + ( x, E i ) for any of ˜ ϕ ( x, ε, c ). Theusage of superpotential (2.4) with ˜ ϕ ( x, ε, c ) corresponds to exact supersymmetry, which leadsto existence of zero-modes in H −− and H ++ . Existence of two zero-modes in super Hamiltonianof N = 4 SUSY QM is caused by the fact that the Witten index theorem has to be modifiedwhen intertwining conditions are nonlinear, as discussed in detail in [25].Let’s consider the case when expression (2.4) contains one of the particular solutions, e.g. ϕ ( x, ε ), instead of ϕ ( x, ε, c ). If one of the particular solutions of second order differentialequation is known, the second solution can be obtained from the relation ϕ ( x, ε ) = ϕ ( x, ε ) Z x −∞ dt ϕ ( t, ε ) . Thus, the superpotential (2.4) becomes W ( x, ε, λ ) = −
12 ln (cid:18) λ Z x −∞ dt ϕ ( t, ε ) (cid:19) V.P. Berezovoj, G.I. Ivashkevych and M.I. Konchatnij ≡ −
12 ln (cid:18) ϕ ( x, ε ) ϕ ( x, ε ) + λϕ ( x, ε ) (cid:19) = −
12 ln (cid:18) ϕ ( x, ε ) ϕ ( x, ε, λ ) (cid:19) . It is easy to show, that the state with the energy E = 0 in the spectrum of H −− is absent (i.e.,the wave function ψ −− ( x, E = 0) is nonnormalizable) hence the spontaneously broken N = 2supersymmetry exists. On the one hand, the zero energy state has the wave function ψ ++ ( x, E = 0) ∼ ϕ ( x, ε, λ ) , normalizable for certain values of λ >
0, appears in the spectrum of H ++ . Here the exact N = 2supersymmetry takes place. Furthermore, it is known [29, 30], that the partial supersymmetrybreaking is impossible in N = 4 SUSY QM without central charges. This contradiction resolveswith taking into consideration that the employment of a factorization energy ε in the constructionof isospectral Hamiltonians is the simplest way of incorporation of central charges in N = 4SUSY QM. The similar situation occurs in consideration of form-invariant potentials [31]. Morecomplete and consistent consideration of N = 4 SUSY QM with central charges is given in [32],but this point is over the scope of the current paper. We get started the consideration of properties of isospectral Hamiltanians, derived in previoussection without appealing to the form of the initial Hamiltonian, from the following usefullexpressions. Let y and y be two linear independent solutions of a homogeneous second orderdifferential equation. Then the following expression holds [33]: Z xx i W { y , y } ( A y ( t )+ A y ( t )) dt = − A + A (cid:20)(cid:18) A y ( x ) − A y ( x ) A y ( x )+ A y ( x ) (cid:19) − (cid:18) A y ( x i ) − A y ( x i ) A y ( x i )+ A y ( x i ) (cid:19)(cid:21) . Here W { y , y } = y y ′ − y ′ y is the Wronskian, which for the second order differential equation,reduced to canonical form, as the Schr¨odinger equation, is independent on x and thus could betaken out of the integral. This expression is very useful for calculations of integrals in expressions(2.4)–(2.6). First of all, it is natural to set x i = −∞ in (2.4), because˜ ϕ ( x, ε, c ) = N − ϕ ( x, ε ) + cϕ ( x, ε )tends to 0 in the limit and the function with asymptotic ϕ i ( x,ε ) → x → −∞ alwaysexists, when we use a particular solution ϕ i ( x, ε ). Therefore N − Z x −∞ dt ( ϕ ( t, ε ) + c ϕ ( t, ε )) = − N − (1 + c ) W { ϕ , ϕ } [∆( x, ε, c ) − ∆( −∞ , ε, c )] ,N − = − c ) W { ϕ , ϕ } [∆(+ ∞ , ε, c ) − ∆( −∞ , ε, c )] , ∆( x, ε, c ) = cϕ ( x, ε ) − ϕ ( x, ε ) ϕ ( x, ε ) + cϕ ( x, ε ) . (3.1)Let’s fix c = 1 in (2.6). This choice allows one to simplify the consideration, but nevertheless,reveals the fundamental features of Hamiltonians and wave functions of N = 4 SUSY QM,using only general properties of the solutions to the auxiliary equation under ε < E . Usingulti-Well Potentials in Quantum Mechanics and Stochastic Processes 7relations (3.1), it is easy to obtain the following modulo constant term expression for the super-potential: W ( x, ε, λ ) = −
12 ln (cid:18) λN − Z x −∞ dt ( ϕ ( x, ε ) + ϕ ( x, ε )) (cid:19) = −
12 ln (cid:18) ϕ ( x, ε ) + Λ( ε, λ ) ϕ ( x, ε ) ϕ ( x, ε ) + ϕ ( x, ε ) (cid:19) , Λ( ε, λ ) = ∆( ∞ , ε, − λ − ( λ + 1)∆( −∞ , ε, ∞ , ε,
1) + λ − ( λ + 1)∆( −∞ , ε, . (3.2)Let’s note that ∆( ±∞ , ε, λ ), entering Λ( ε, λ ), are determined by the asymptotics of solutionsto the auxiliary equation. Due to this fact, and since for H −− the potential is determined bythe symmetric combination ϕ ( x, ε ) + ϕ ( x, ε ), while in the case of H ++ – by the asymmetriccombination ϕ ( x, ε ) + Λ( ε, λ ) ϕ ( x, ε ), we get: H −− = H − + − d dx ln( ϕ ( x, ε ) + ϕ ( x, ε )) ,H ++ = H − + − d dx ln ( ϕ ( x, ε ) + Λ( ε, λ ) ϕ ( x, ε )) . (3.3)In some sense, the potentials ¯ U −− ( x, ε ) and ¯ U ++ ( x, ε, λ ) are form-invariant [15], i.e. potentialsand wave functions transform to each other by changing of parameters and their spectra areidentical and this holds independently on the choice of the initial Hamiltonian. As it will beshown in the next section, if the potential in H −− is a multi-well symmetrical potential, then in H ++ it should be asymmetrical. Moreover, varying ε , as well as λ , the form of ¯ U ++ ( x, ε, λ ) can change.Relations (3.1) and (3.2) are also useful to derive the exact form of the wave functions ψ −− ( x, E )and ψ ++ ( x, E ). Thus, the expression for the wave functions ψ ++ ( x, E ) comes from the similarexpression for ψ −− ( x, E ) by the substitution ϕ ( x, ε ) + ϕ ( x, ε ) → ( ϕ ( x, ε ) + Λ( ε, λ ) ϕ ( x, ε )). Inparticular, the normalization constant for ψ ++ ( x, E = 0) can be obtained from the correspondingexpression for ψ −− ( x, E = 0).To demonstrate the possibility of partial supersymmetry breaking in N = 4 SUSY QM wehave to show that the wave function ψ ++ ( x, E = 0) ∼ ϕ ( x,ε,λ ) is normalizable. The value of thenormalization constant can be derived from (3.1) under c = λ >
0. We will return in whatfollows to the calculation of this constant for the concrete form of the initial Hamiltonian. Inits turn, the potential in H ++ has the same form as the corresponding potentials in the case ofexact supersymmetry under replacement Λ( ε, λ ) → λ .As it follows from (2.6) and (3.3), the parametric dependence on λ is only present in H ++ and ψ ++ ( x, E ). It should be noted that the value of λ is not to be fixed by the normalizationcondition for the wave function. Existence of a parametric freedom is common in building theisospectral Hamiltonians, which is based on different versions of the inverse scattering prob-lem [34, 35, 36]. It is caused by the ambiguity in reconstructing quantum-mechanical potentialsfrom the spectral data. We will show latter that similar situation exists also in stochastic mo-dels. This is unexpectedly enough, since the potentials of stochastic models have direct physicalmeaning in contrast to quantum mechanical potentials. The FP equation is equivalent to the Langevin equation, however its application in physicsis more wide, since it is formulated in more appropriate language of the probability densities P ± ( x, t ; x , t ). According to [9, 12] the FP equation has the form: ∂∂t P ± ( x, t ; x , t ) = D ∂ ∂x P ± ( x, t ; x , t ) ∓ ∂∂x V ( x ) P ± ( x, t ; x , t ) , V.P. Berezovoj, G.I. Ivashkevych and M.I. Konchatnij P ± ( x, t ; x , t ) = h δ ( x − x ) i , U ± ( x ) = ± Z x dzV ( z ) , where U ± ( x ) is the potential entering the Langevin equation. Let’s set t = 0. The Fokker–Planck equation describes the stochastic dynamics of particles in potentials U + ( x ) and U − ( x ) = − U + ( x ). Substituting P ± ( x, t ; x ,
0) = exp (cid:26) − D [ U ± ( x ) − U ± ( x )] (cid:27) K ± ( x, t )the FP equation transforms into the imaginary time Schr¨odinger equation: − D ∂∂t K ± ( x, t ) = (cid:26) − D ∂ ∂x + 12 (cid:2) V ( x ) ± D V ′ ( x ) (cid:3)(cid:27) K ± ( x, t ) ,K ± ( x, t ) = (cid:28) x (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:26) − tH ± D (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) x (cid:29) , H ± = − D ∂ ∂x + 12 (cid:2) V ( x ) ± D V ′ ( x ) (cid:3) , (4.1)in which the diffusion constant D can be treated as the “Planck constant”, while H ± has theform of N = 2 SUSY QM Hamiltonian. Equations (4.1) are basic to apply of the eigenfunctionsexpansion method for construct exactly solvable stochastic models by use of the results of cor-responding quantum mechanical problems. Details of this method in the framework of N = 2SUSY QM can be found in [9, 12]. Utilization of the extended N = 4 SUSY QM formalism givesadditional possibilities to construct new exactly-solvable models of stochastic processes, sincethe Hamiltonian H σ σ of N = 4 SUSY QM includes four isospectral Hamiltonians. It allows toobtain new K σ σ ( x, t ) and U σ σ ( x ), hence P σ σ ( x, t ; x ).To obtain the expressions for probability densities P σ σ ( x, t ; x ) we should to know not onlythe wave functions and the spectrum of H σ σ , but those of the corresponding U σ σ ( x ). It is easyto see that U − + ( x, c = 1) = − D ln ( ϕ ( x, ε ) + ϕ ( x, ε )) , U −− ( x, c = 1) = − U − + ( x, c = 1) . (4.2)Further consideration is based on account of the symmetry of N = 4 SUSY QM H σ σ under σ ↔ σ , that leads to the relation: H − + ( x, p ) = 12 ¯ Q ( − )1 Q ( − )1 ≡ Q (+)2 ¯ Q (+)2 = H + − ( x, p ) . (4.3)First equality indicates the existence of the expression of H (+) in terms of Q ( ¯ Q ), whilethe second equality implies that H ( − ) is expressed in terms of Q ( ¯ Q ). At the same time, thesupercharges entering H (+) and H ( − ) are substantially different: H + − ( x, p ) = 12 Q (+)2 ¯ Q (+)2 = 12 (cid:2) p + (cid:0) V (+)1 ( x ) (cid:1) − DV (+)1 ′ ( x ) (cid:3) ,V (+)1 ( x ) = D ddx ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ ϕ ( x, ε, c )1 + λ R xx i dx ′ [ ˜ ϕ ( x ′ , ε, c )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = − D ddx ln | ϕ ( x, ε ) + Λ( ε, λ ) ϕ ( x, ε ) | . (4.4)According to (4.3), quantum H + − and H − + have the same spectrum and the wave functions,thus the corresponding K ( x, t ) are identical. But the corresponding stochastic models are de-scribed by essentially different potentials, therefore their P ( x, t ; x ) are different. Moreover, thepotential U + − ( x, ε, Λ) = − U ++ ( x, ε, Λ) = − D ln | ϕ ( x, ε ) + Λ( ε, λ ) ϕ ( x, ε ) | has non-trivial para-metric dependence on λ , which implies the existence of the family of stochastic models with thesame time dependence of the probability density, but substantially different coordinates depen-dence. Surprisingly, the parametric freedom allows for a modification of the shape of potential.ulti-Well Potentials in Quantum Mechanics and Stochastic Processes 9Recall that physical quantities, e.g. time of passing the potential maximum, considerably dependon the local modification of the potential form [37, 38, 39].It is easy to obtain the corresponding P σ σ ( x, t ; x ) using the expressions for wave functions ψ σ σ ( x, E n ) and potentials U σ σ ( x ) from (2.5), (2.6) and (4.2), (4.4). It is important to note,that transitional probability densities P + − ( x, t ; x ) and P ++ ( x, t ; x ) have also the parametricdependence on λ .Denoting ϕ ( x, ε, c = 1) = ϕ ( x, ε ) + ϕ ( x, ε ), the expression for calculation of distributionfunction P −− ( x, t ; x ) takes the form: P −− ( x, t ; x ) = N − ( ϕ ( x, ε, + (cid:18) ϕ ( x , ε, ϕ ( x, ε, (cid:19) N X n =0 e − tD ( E n − ε ) ψ −− ( x, E n ) ψ −− ( x , E n ) . (4.5)For simplicity let’s assume that the spectrum of the initial Hamiltonian has only the discretestates. When the continuous spectrum exists the summation should be replaced with integra-tion over the corresponding density of states. Substituting the expressions for wave functions ψ −− ( x, E n ) (2.5) into (4.5) we obtain: P −− ( x, t ; x ) = N − ( ϕ ( x, ε, + 12 ( ϕ ( x , ε, × N X n =0 e − tD ( E n − ε ) ( E n − ε ) d dxdx ψ − + ( x, E n ) ϕ ( x, ε, ! ψ − + ( x , E n ) ϕ ( x , ε, ! . (4.6)The expression for P ++ ( x, t ; x ) can be obtained from equation (4.6) by substituting ϕ ( x, ε, → ϕ ( x, ε, Λ( λ, ε )), N − → N − . Both of P −− ( x, t ; x ) and P ++ ( x, t ; x ) have the equilibrium distri-butions at t → ∞ , which coincide with square of the zero modes wave functions in Hamilto-nians H −− and H ++ . Moreover, it is easy to see that terms from the excited states of H −− and H ++ do not contribute to the normalization condition of these density functions, i.e. the normaliza-tion of these functions is guaranteed by their equilibrium values. Dependence on parameter λ in P ++ ( x, t ; x ) is not eliminated by the normalization condition.The distribution P − + ( x, t ; x ) ( P + − ( x, t ; x )) describes the stochastic dynamics of a particle inthe metastable state of the potential, inverted to U − + ( x, ε ) ( − U ++ ( x, ε, Λ( ε, λ )). Thus, it has notthe equilibrium limit and is determined by the expression: P − + ( x, t ; x ) = ϕ ( x, ε, ϕ ( x , ε, N X n =0 e − ( En − ε ) tD ψ − + ( x, E n ) ψ − + ( x , E n ) . (4.7)Similarly to the previous case P + − ( x, t ; x ) can be obtained from P − + ( x, t ; x ) by substitution ϕ ( x, ε, → ϕ ( x, ε, Λ( λ, ε )). The parametric dependence on λ can be eliminated by the nor-malization condition for P + − ( x, t ; x ), but as it will be demonstrated below, choosing the initialHamiltonian of harmonic oscillator, the parametric arbitrariness in P + − ( x, t ; x ) remains.To summarize, we should note that in the framework of N = 4 SUSY QM four models ofstochastic dynamics emerge. Two of them correspond to the motion of particle in double-wellpotentials (symmetric and asymmetric) under the action of the Gaussian noise, the other onesdescribe the dynamics of the particle in the metastable state, when N = 4 supersymmetryis exact. For the case of the partial supersymmetry breaking P −− ( x, t ; x ), P − + ( x, t ; x ) and P + − ( x, t ; x ) describe stochastic processes in the metastable state, while P ++ ( x, t ; x ) – in bistable.Some of them possess the parametric freedom, that allows to change the parameters of externalfield.0 V.P. Berezovoj, G.I. Ivashkevych and M.I. Konchatnij To construct the explicit expressions for potentials and wave functions we choose the initialHamiltonian with the harmonic oscillator ( HO ) potential. Let’s consider the solution to theauxiliary equation for ε < E = ω , ( ~ = m = 1): (cid:18) d dx + 2 (cid:18) ε − ω x (cid:19)(cid:19) ϕ ( x, ε ) = 0 . Introducing dimensionless variables ξ = √ ωx we obtain the equation for ϕ ( ξ, ¯ ε ), where ¯ ε = εω : (cid:18) d dξ + (cid:18) ν + 12 − ξ (cid:19)(cid:19) ϕ ( ξ, ¯ ε ) = 0 , ν = −
12 + ¯ ε. Now we consider two different cases. a) Exact N = 4 SUSY. This equation has two linear independent solutions: paraboliccylinder functions D ν ( √ ξ ), D ν ( −√ ξ ). According to terminology in the above, we denote ϕ ( ξ, ¯ ε ) = D ν ( √ ξ ), ϕ ( ξ, ¯ ε ) = D ν ( −√ ξ ) and the Wronskian becomes W { ϕ , ϕ } = √ π Γ( − ν ) [40],with gamma-function Γ( − ν ). Following the procedure from the previous section, the generalsolution to the auxiliary equation is chosen to be: ϕ ( ξ, ¯ ε,
1) = D ν (cid:0) √ ξ (cid:1) + D ν (cid:0) − √ ξ (cid:1) . (5.1)As one can see from (5.1), ϕ ( x, ¯ ε,
1) is an even function of ξ . To obtain the exact form of thesuperpotential, the integral, entering the definition of the W ( x, ε, λ ) as well as the normalizationconstant N − have to be calculated from (3.1), (3.2). Due to the symmetry of ϕ ( ξ, ¯ ε ), expressionof the integral simplifies and takes the form:1 + λN − Z ξ −∞ dt ( ϕ + ϕ ) = 1 − λN − W { ϕ , ϕ } (cid:20)(cid:18) ϕ ( ξ, ¯ ε ) − ϕ ( ξ, ¯ ε ) ϕ ( ξ, ¯ ε ) + ϕ ( ξ, ¯ ε ) (cid:19) + ∆(+ ∞ , ¯ ε, (cid:21) . Hence, the superpotential is: W ( ξ, ¯ ε, λ ) = ln (1 + λ + λ ∞ , ¯ ε, ) ϕ + (1 + λ − λ ∞ , ¯ ε, ) ϕ ϕ + ϕ ! , (5.2) N − = − W { ϕ , ϕ } ∆(+ ∞ , ¯ ε,
1) = 2 √ π Γ( − ν ) . Using the asymptotic value of the parabolic cylinder function, we have ∆(+ ∞ , ¯ ε,
1) = − W ( ξ, ¯ ε, λ ) = −
12 ln (cid:18) ϕ ( ξ, ¯ ε ) + (1 + λ ) ϕ ( ξ, ¯ ε ) ϕ ( ξ, ¯ ε ) + ϕ ( ξ, ¯ ε ) (cid:19) = −
12 ln (cid:18) ϕ ( ξ, ¯ ε, λ ) ϕ ( ξ, ¯ ε, (cid:19) . (5.3)From (5.3) it follows, that the value of λ is restricted to λ > −
1. Using (2.5) for H −− and ψ −− ( x, E ), we can derive the form of the Hamiltonian and the corresponding wave functions(here ψ − + ( ξ, E i ) are the wave functions of the HO): H −− = H − + − d dx ln (cid:16) D ν ( √ ξ ) + D ν ( −√ ξ ) (cid:17) ,ψ −− ( ξ, E i ) = 1 p E i − ¯ ε ) W (cid:8) ψ − + ( ξ, E i ) , ϕ ( ξ, ¯ ε, (cid:9) ϕ ( ξ, ¯ ε, , ulti-Well Potentials in Quantum Mechanics and Stochastic Processes 11 Figure 1.
Shape of the potential ¯ U −− ( ξ, ¯ ε ). Figure 2.
Potential ¯ U ++ ( ξ, ¯ ε = 0 . , λ ). ψ −− ( x, E = 0) = N − (cid:0) D ν ( √ ξ ) + D ν ( −√ ξ ) (cid:1) = N − ϕ ( x, ¯ ε, , N − = 2 √ π Γ( − ν ) . (5.4)Analysis of (5.4) reveals that there exist several local minima only for 0 < ¯ ε < . With valuesof ¯ ε to be close to the right boundary, the third local minimum appears (Fig. 1) and depth ofthe outside minima increases. It should be noted that in terms of the dimensionless variable ξ the only way to vary the form of the potential is by varying ¯ ε and λ . In the case of naturalunits, additionally, the form of the potential (in particular, positions of the local minima) canbe changed by variation of ω .Connection between H ++ , ψ ++ ( ξ, E n ) and H − + , ψ − + ( ξ, E n ) can be viewed in the same manner.Expressions for H ++ and ψ ++ ( ξ, E n ) can be obtained from (5.4) by substituting ϕ ( ξ, ¯ ε, → ϕ ( ξ, ¯ ε, λ + 1): H ++ = H − + − d dx ln (cid:16) D ν ( √ ξ ) + (1 + λ ) D ν ( −√ ξ ) (cid:17) ,ψ ++ ( ξ, E i ) = 1 p E i − ¯ ε ) W (cid:8) ψ − + ( ξ, E i ) , ϕ ( ξ, ¯ ε, λ + 1) (cid:9) ϕ ( ξ, ¯ ε, λ + 1) ,ψ −− ( ξ, E = 0) = N − λ +1 (cid:0) D ν ( √ ξ ) + ( λ + 1) D ν ( −√ ξ ) (cid:1) = N − λ +1 ϕ ( x, ¯ ε, λ + 1) ,N − λ +1 = 2( λ + 1) √ π Γ( − ν ) . From these relations it follows the restriction − < λ . As it can be seen from Fig. 2, thepotential ¯ U ++ ( ξ, ¯ ε, λ ) possesses the well indicated asymmetry, which increases under λ → −
1. Thevalue ¯ ε = 0 .
47 corresponds to the region, where ¯ U ++ ( ξ, ¯ ε, λ ) has three local minima. It should benoted, that the depth of the central minima also increases with λ → −
1. The obtained potentialsand the corresponding wave functions may be applied for the resonant tunneling phenomenon [4]studies. b) Partial N = 4 SUSY breaking. The utilization of the particular solution of auxiliaryequation for derivation of superpotential (5.2) realizes the situation of partial supersymmetrybreaking. As it was mentioned above, the spectrum of H −− does not contain states with E = 0,because their wave function is nonnormalizable. The shape of ¯ U −− ( ξ, ¯ ε ) is presented in Fig. 3.At the same time, zero state appears in the spectrum of H ++ with the wave function: ψ ++ ( ξ, E = 0) = N − λ ϕ ( ξ, ¯ ε, λ ) = N − λ ( ϕ ( ξ, ¯ ε ) + λϕ ( ξ, ¯ ε )) . Figure 3.
Potential ¯ U −− ( ξ, ¯ ε ) (0 < ¯ ε < . The normalization constant N − λ is calculated by use of (5.4) and has the form N − λ = λ √ π Γ( − ν ) .Thus, it is restricted to λ >
0. The Hamiltonian becomes H ++ = H − + − d dξ ln ( ϕ ( ξ, ¯ ε, λ ). Thepotential shape (¯ ε = 0 .
47) is presented in Fig. 2 with λ > ε = − ( D − ( √ ξ ) = e ξ p π (1 − Φ( ξ )), with Φ( ξ ) to be the error function), potentials¯ U −− ( x, ¯ ε ) correspond to that of [41] (the only difference is in additional constant term)¯ U −− (cid:0) ξ, ¯ ε = − (cid:1) = (cid:18) ξ − (cid:19) + 4 √ π e − ξ (1 − Φ( ξ )) " e − ξ √ π (1 − Φ( ξ )) − ξ . (5.5)This potential is single-well, as well as corresponding ¯ U ++ ( ξ, ¯ ε = − , λ ), because existence ofseveral local minima is possible only for 0 < ¯ ε < . Using the form-invariance property of ¯ U ++ :¯ U ++ (cid:0) ξ, ¯ ε = − , λ (cid:1) = ξ − − d dξ ln ((1 + λ ) − (1 − λ )Φ( ξ )) . (5.6)The spectrum of the Hamiltonian with (5.6) contains, in contrast to (5.5), the state with E = 0and the wave function of the form: ψ ++ ( ξ, E = 0) = (2 λ √ π ) / e − ξ ((1 + λ ) − (1 − λ )Φ( ξ )) . Let us give an example of the construction of new stochastic models. We choose the harmonicoscillator (HO) Hamiltonian as the initial one. Then, we consider the auxiliary equation with ε < E = Dω (recall, that D plays the role of the “Planck constant” in the FP equation and¯ ε = εDω ). Here we consider two cases. a) Exact N = 4 SUSY. For many problems of stochastic dynamics it is reasonable touse symmetric double-well potentials, so we consider first the case c = 1. As it can be seenfrom (5.1), ϕ ( ξ, ¯ ε,
1) is even function, and the normalization constant N − can be calculatedfrom (5.2). One of the main characteristics of the stochastic process is the potential entering theLangevin equation. As it follows from (4.2), U −− ( x ) = − U − + ( x ) = D ln ϕ ( x, ¯ ε,
1) is symmetric,while U ++ ( x, Λ) = − U + − ( x, Λ) = D ln ϕ ( x, ¯ ε, λ + 1) is asymmetric, that depends on the value of λ .The existence of the parametric freedom in U ++ ( x, Λ) is quite surprising: contrary to quantummechanics, where potentials reconstructed from the spectral data indeed have a parametricfreedom, but give the same observables, in stochastic mechanics local properties of the potentialulti-Well Potentials in Quantum Mechanics and Stochastic Processes 13a) b) - - x - U Figure 4.
Potentials U σ σ . a) exact N = 4 SUSY: U ++ ( ξ, λ ) ( ω = D = 1, ¯ ε = 0 . N = 4 SUSY: U − + ( ξ ) – lilac, U −− ( ξ ) – blue, U + − ( ξ, λ ) – dashed, U ++ ( ξ, λ ) – red (¯ ε = 0 . λ = 1). substantially influence on many characteristics, for example, on the rates of the potential barriercrossing [37, 38, 39]. U + − ( x, Λ) = − U ++ ( x, Λ) = − D ln | ϕ ( x, ¯ ε ) + ( λ + 1) ϕ ( x, ¯ ε ) | , obtained within the frameworkof proposed model, is presented in Fig. 4. Changes of the U ++ ( ξ, λ ) shape due to varying λ become more and more sharp under λ → −
1. The analysis of the expressions of U ++ ( ξ, λ ) shows,that several local minima are possible only when 0 < ¯ ε < . Variation of ¯ ε inside this rangechanges the height of the barrier, which significantly increases with ¯ ε → . Moreover, the shapeof the potential can be changed (especially the minima locations) by varying ω , when switchingto natural variables. To summarize, the existence of ( ω, ¯ ε, λ ) significantly change the shape ofthe U ++ ( x, λ ).Substituting the HO wave functions to (4.7) and using the Mehler formula [40], we get P − + ( ξ, z ; ξ ): P − + ( ξ, z ; ξ ) = (cid:16) ωπD (cid:17) / e − ξ / ϕ ( ξ, ¯ ε, e − ξ / ϕ ( ξ , ¯ ε, z − ν √ − z e − ( ξz − ξ − z , (6.1)where z = e − ωt . As it was noted in the above P − + ( ξ, z ; ξ ) does not have the equilibrium valueand could be normalized (the proof of this claim is given in Appendix), i.e. Z + ∞−∞ dξ P − + ( ξ, z ; ξ ) = 1 . The probability density P −− ( ξ, z ; ξ ) is characterized by the existence of the equilibrium value,which is determined by the zero energy wave function of H −− , i.e. P −− ( x, t → ∞ ; x ) → N − ϕ ( x, ¯ ε, , N − = 2 √ π Γ( − ν ) . After some calculations (see Appendix) we obtain the expression for P −− ( ξ, z ; ξ ): P −− ( ξ, z, ξ ) = 2 √ π Γ( − ν ) ϕ ( ξ, ¯ ε,
1) + 12 D (cid:16) ωDπ (cid:17) / ddξ ϕ ( ξ, ¯ ε, Z z dτ τ − ( +¯ ε ) Φ( ξ, ξ , τ ) , (6.2)Φ( ξ, ξ , τ ) = (cid:18) ddξ [ F ( ξ, ξ , τ )] ϕ ( ξ , ˜ ε, − F ( ξ, ξ , τ ) ddξ [ ϕ ( ξ , ¯ ε, (cid:19) ,F ( ξ, ξ , τ ) = exp (cid:18) ξξ − τ − ( ξ + ξ )2 1 + τ − τ (cid:19) . λ = 0 λ = 0 . λ = − . λ = − . Figure 5. P ++ ( ξ, z ; ξ ) as a function of ξ and z (¯ ε = 0 . λ = 0, λ = 0 . λ = − . λ = − . ω = 1); z = e − ν . The probability density P + − ( ξ, z ; ξ ) can be obtained from P − + ( ξ, z ; ξ ) (6.1) by substituting ϕ ( x, ¯ ε, c = 1) → ϕ ( x, ¯ ε, λ + 1). Note that the normalization condition does not eliminatethe λ -freedom (see Appendix). As it has been noted in previous section, the expression for P ++ ( ξ, z ; ξ ) is obtained from P −− ( ξ, z ; ξ ) with replacing ϕ ( x, ¯ ε, c = 1) → ϕ ( x, ¯ ε, λ + 1), N − → N − λ = λ +1) √ π Γ( − ν ) .The dependencies of P ++ ( ξ, z ; ξ ) on spatial and time variables, presented in Fig. 5, demon-strate sharp modifications (with fixed ¯ ε , ω , D ) at different values of λ , especially with λ → − P ++ ( ξ, z ; ξ ) as well as decreasing the time of passing tobimodality are observed. Moreover, modifications of the shape of potential U ++ ( ξ, λ ), as well as P ++ ( ξ, z ; ξ ), can take place with variation of ω , when we pass from dimensionless variables ξ tophysical x . This leads to the shift of the local minima and change the height of the barriers.When ¯ ε → , the barrier between local minima significantly increases, that allows to study thetransition time from one local minimum to another as a function of ∆ UD . b) Partial N = 4 SUSY breaking. Let’s briefly discuss partial N = 4 SUSY breakingand its consequences to stochastic dynamics. In this case three models for the description of thestochastic dynamics of particles in metastable states ( U − + ( ξ, ¯ ε ), U −− ( ξ, ¯ ε ), U + − ( ξ, ¯ ε, λ )) and one inthe bistable state ( U ++ ( ξ, ¯ ε, λ ) ) arise in the considered approach. The corresponding potentialshave the form: U −− ( ξ, ¯ ε ) = − U − + ( ξ, ¯ ε ) = D ln (cid:12)(cid:12) D ν ( √ ξ ) (cid:12)(cid:12) ,U ++ ( ξ, ¯ ε, λ ) = − U + − ( ξ, ¯ ε, λ ) = D ln (cid:12)(cid:12) D ν ( √ ξ ) + λD ν ( −√ ξ ) (cid:12)(cid:12) . The range of λ variation in U ++ ( ξ, ¯ ε, λ ) and U + − ( ξ, ¯ ε, λ ) is restricted to be λ >
0. Fig. 4bpresents the dependencies of the potentials on spatial variables at fixed ¯ ε and λ . Stochasticdynamics in mentioned metastable states satisfies to different boundary conditions: reflectiveulti-Well Potentials in Quantum Mechanics and Stochastic Processes 15boundary at left (right) and absorbing at right (left) for U −− ( ξ, ¯ ε ) ( U − + ( ξ, ¯ ε )) and absorbingboundaries at both sides for U + − ( ξ, ¯ ε, λ ). The corresponding probability densities can be calcu-lated with taking into account previously obtained expressions with appropriate modifications. In the paper we consider the construction and the study, within the framework of N = 4SUSY QM, of the general properties of Hamiltonians with multi-well potentials. These resultshave been used to obtain exactly solvable models of stochastic dynamics. The special attentionwas made to studies of features of these Hamiltonians without concretization of the form ofinitial Hamiltonian. Relations for certain type of integrals, containing the fundamental solutionsto the Schr¨odinger type equations, allows one to show the form-invariance of the isospectralHamiltonians with multi-well potentials, obtained within of N = 4 SUSY QM. In other words,having the identical spectra of H −− and H ++ , the corresponding potentials can be obtained fromeach other through replacing parameters of the problem. Moreover, this relation allows toanalytically calculate the normalization constants of zero modes, using only asymptotic valuesof fundamental solutions. The construction of isospectral Hamiltonians within N = 4 SUSY QMalso implies the partial supersymmetry breaking. Taking the model of harmonic oscillator as anexample we derive the exact form of the isospectral Hamiltonians with multi-well potentials andthe corresponding wave functions. The existence of a parameteric freedom gives a possibilityto vary the shape of the potential in the wide range. This becomes important in the researchof different phenomena, such as tunneling processes, which are sensitive to the structure of themulti-well potentials.The obtained results have been used for the construction of exactly-solvable stochastic models.Obtained potentials, entering the Langevin equations, and probability functions are characteri-zed by the parametric dependence, which allows sufficiently modify their shape. A parametricfreedom is typical for quantum mechanical isospectral Hamiltonians, where different versions ofthe inverse scattering problem are used [34, 35, 36]. Such a freedom arise in view of an ambiguityreconstructing the potential from the spectral data. On the other hand, the parametric free-dom in stochastic models is quite surprising, since their potentials, in contrast to the quantummechanics, have direct physical meaning. Many characteristics of stochastic dynamics, such aspotential peak passage times and the metastable state lifetime [37, 38, 39], substantially dependon the shape of the potential. Therefore, it is interesting to investigate the characteristics ofstochastic processes in multi-well potentials, such as the Kramers problem, the stochastic reso-nance etc., and their dependence on a modification of the shape of the potential. We note thatour results admit a generalization to non-Markovian processes, by use of the approach of [42].Furthermore, the obtained results can be generalized to the stochastic models of polymer dy-namics [43]. We have also considered non-trivial consequences of the partial supersymmetrybreaking in N = 4 SUSY QM to the description of stochastic dynamics. In this case threeprobability density functions with the same time dependence (i.e. with the identical spectrumof the Fokker–Planck operator) describe stochastic dynamics of particles in metastable states,and one function corresponds to the dynamics in the bistable state. A Appendix
Let’s consider details of obtaining the expression of P −− ( ξ, z ; ξ ) and P ++ ( ξ, z ; ξ ). Accordingto (4.6) the distribution function P −− ( ξ, z ; ξ ) has the form: P −− ( x, t ; x ) = N − ( ϕ ( x, ε, + 12 ( ϕ ( x , ε, × N X n =0 e − tD ( E n − ε ) ( E n − ε ) d dxdx ψ − + ( x, E n ) ϕ ( x, ε, ! ψ − + ( x , E n ) ϕ ( x , ε, ! . (A.1)For the generalized Ornstein–Uhlenbeck process, substituting (5.1) and the wave function ofharmonic oscillator ψ − + ( ξ, E n ) = (cid:0) ωπD (cid:1) / D n ( √ ξ ) √ n ! into (A.1) we get the expression: P −− ( x, t ; x ) = N ϕ ( ξ, ¯ ε,
1) + 12 D (cid:16) ωπD (cid:17) / ddξ ϕ ( ξ, ¯ ε, ∞ X n =0 z n + − ¯ ε ( n + − ¯ ε ) n ! × D n ( √ ξ ) (cid:18) ddξ D n ( √ ξ ) ϕ ( ξ , ¯ ε, − D n ( √ ξ ) ddξ ϕ ( ξ , ¯ ε, (cid:19) , where ξ = p ωD x . Using the Mehler formula [40]: ∞ X n =0 n ! D n ( x ) D n ( y ) z n = 1 √ − z exp (cid:26) xyz − z − x + y z − z (cid:27) ≡ F ( x, y ; z )and the relation ∞ X n =0 z n + − ¯ ε n ! (cid:0) n + − ¯ ε (cid:1) D n ( x ) D n ( y ) = Z z dτ τ n − ( +¯ ε ) F ( x, y, τ )we obtain the expression for P −− ( ξ, z ; ξ ) (6.2). It should be noted that presence of spatial deriva-tives eliminate the contribution from excited states of the Hamiltonian H −− to the normalizationcondition for P −− ( ξ, z ; ξ ).Let’s consider the question of normalizability of P − + ( x, t ; x ) and P + − ( x, t ; x ) as well as thepossibility of elimination of the λ -freedom in P + − ( x, t ; x ) by the normalization condition. Asit was noted in the above, P + − ( x, t ; x ) is obtained from P − + ( x, t ; x ) by replacing ϕ ( x, ¯ ε, → ϕ ( x, ¯ ε, λ + 1) and has a form P + − ( ξ, z ; ξ ) = (cid:16) ωπD (cid:17) / e − ξ / ϕ ( ξ, ¯ ε, λ + 1) e − ξ / ϕ ( ξ , ¯ ε, λ + 1) z − ν √ − z e − ( ξz − ξ − z . The normalization condition for the distribution function: Z + ∞−∞ dx P + − ( x, t ; x ) = r Dω Z + ∞−∞ dξ P + − ( ξ, z ; ξ ) ≡ . Substituting the integral representation of the parabolic cylinder function D p ( z ) = e − z Γ( − p ) Z ∞ e − zx − x x − p − dx, Re p < , results in Z + ∞−∞ dξ e − ( ξz − ξ − z e − ξ / D ν ( ±√ ξ ) = √ π z ν (1 − z ) / e − ξ / D ν ( ±√ ξ ) . (A.2)Substituting (A.2) to the normalization condition for P + − ( x, t ; x ) and P − + ( x, t ; x ) it is easyto verify, that they are equal to one, thus the probability densities, which correspond to thestochastic dynamics in a metastable state, are normalizable and the normalization conditiondoes not eliminate the λ -freedom in P + − ( x, t ; x ).ulti-Well Potentials in Quantum Mechanics and Stochastic Processes 17 Acknowledgements
Authors thank to M. Plyushchay for helpful discussions and BVP Conference Organizers fora stimulating environment. We are thankful to A. Nurmagambetov for reading the manuscript,suggestions and improvings.
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