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Fundamental Solutions for Hermite and Subelliptic Operators

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Fundamental Solutions for Hermite and Subelliptic Operators FUNDAMENTAL SOLUTIONS FOR HERMITE AND SUBELLIPTIC OPERATORS £ By OVIDIU CALIN,DER-CHEN CHANG AND JINGZHI TIE Abstract. In this article, we introduce a geometric method based on multi- pliers to compute heat kernels for operators with potentials. Using the heat kernel, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point on Euclidean space and on Heisenberg groups. As a conse- £ quence, we obtain the fundamental solutions for the sub-Laplacian  in a family of quadratic submanifolds. 1 Introduction À The Hermite operator « arises in many contexts and has been studied by mathematicians and physicists for quite some time (see e.g., [1], [4], [17] and À [22]). In this paper, we study the fundamental solution of the operator « , where Ò ¾ ¾ ¾ À « · Ü « ¾ Ü ½ Ò Ê Ã ´Ü Ý µ in , i.e., we seek a distribution « such that ¾ ¿ Ò ¾ ¾ ¾ « · à ´Ü Ý µÆ ´Ü Ý µ Ü (1) « ¾ Ü ½ À Since the operator « is not left invariant under the Euclidean group action, we have to compute the fundamental solution with singularity at any point Ý . One way to À achieve this goal is to use the eigenfunction expansion for the operator « (see [4] and [14]). Here we adopt a different approach. First, we use the Hamilton-Jacobi theory to find the heat kernel for the heat equation with potentials. The idea is to write down the Hamilton system of equations and study the system qualitatively £ The research is partially supported by a William Fulbright Research Grant and a Competitive Research Grant at Georgetown University. 223 JOURNAL D’ANALYSE MATHEMATIQUE,´ Vol. 100 (2006) 224 O. CALIN, D.-C. CHANG AND JINGZHI TIE, from the point of view of the conservation law of energy. In general, these systems cannot be solved explicitly. For simple equations, one may characterize the solutions by finding the first integrals of motions. We demonstrate this method ¾ ¾ ´Üµ ¬ Ü by solving the operator with potential Í . This is exactly the case of the Hermite operator. Then we give some examples and comments for general potentials at the end of Section 2. For more details, readers can consult the book [7]. Then we use the heat kernel to construct the fundamental solution for the Hermite operator. This method can also be used to study subelliptic partial differential equations and has been exploited extensively by many mathematicians; see, e.g., [3], [8], [9] and [11]. In Section 4, using results for Hermite operator, we construct the fundamental £ ¦ solution for the sub-Laplacian  on a family of quadratic submanifolds Ò Ñ ¢ ¦ which was first studied by Nagel, Ricci and Stein [21]; is a special case of the more general submanifold introduced by Beals, Gaveau and Greiner [2]. We can compute the kernel explicitly in terms of certain geometric invariants. In particular, our result recovers the formula on the Heisenberg group. We also £ calculate the heat kernel for the operator  in Section 5. Finally, in Section 6, we consider the Hermite operator on the Heisenberg group ¾Ò ¡ ¾ ¾ ¾ À « · ¬ Ü Ì « ½ ¾Ò ¾Ò ¾ ¾ ¾ ¾ ¾ ¾ « · ¬ Ü Ü ¾ ¾ Ü Ø ½ ½ Ò ·¾ Ü Ü ·Ò Ü Ü Ø Ø ·Ò ½ Ì ¾Ò ·Ò Here ½ is the basis of the Heisenberg Lie algebra and ½ Ò for . In order to achieve these goals, we take the Fourier transform Ê with respect to the Ø-variable in (which is the center of the group) and reduce ¾Ò Ê « ¬ the operator to a Hermite operator in with parameters , , and the dual Ø variable of . Using results for the Hermite operator (see, e.g., [1] and [14]) and the inverse Fourier transform, we solve the problem. The literature is short of explicit computations for the heat kernel and fundamental solution of the operators À À « « and , so it seems worthwhile to give a detailed discussion. We thank Professor Malcolm F. Adams for discussing the problem with us at the outset and reading an early draft of the paper. He made valuable suggestions which have been incorporated into the paper. FUNDAMENTAL SOLUTIONS FOR HERMITE OPERATORS 225 ¾ 2 Heat Kernel for the Hermite Operator in Ê We start with the Hermite operator ¾ ½ ¾ ¾ À Ü ¾ ¾ Ü ¾ Ê where · is a nonnegative real parameter. We associate the Hamiltonian function as half of the principal symbol ½ ¾ ¾ ¾ ´ Ü µ ´ ܵ (2) À ¾ The Hamiltonian system is ¾ À Ü Ü À Ü and Ü Ü ¾ Ê Ü´×µ As we are interested in finding the geodesic between the points ¼ , will satisfy the boundary problem ¾ Ü Ü Ü´¼µ Ü Ü´ µÜ with ¼ The conservation of energy law is ½ ½ ¾ ¾ ¾ Ü ´×µ Ü ´×µ ¾ ¾ where is the energy constant. This can be used to obtain an ODE for the solution ´×µ Ü : Ô Ü Ü ¾ ¾ Ô ¾ · Ü µ × ¾ ¾ × ¾ · Ü × ¼ × Ø Ü´¼µ Ü Ü´ µÜ Integrating between and , with ¼ and , yields Ü Ú Ù Ú Ô Ô ´µ ¾ ¾ ¾ ¾ · Ù ½·Ú Ü Ú ¼ ¼ Ü Ü ¼ Ô Ô Ú Ú with and ¼ . Integration yields ¾ ¾ ½ ½ ½ ½ × Ò ´Ú µ × Ò ´Ú µ ´µ × Ò ´Ú µ× Ò ´Ú µ· ¼ ¼ ½ Ú × Ò × Ò ´Ú µ· It follows that ¼ . This is equivalent to ½ Ú Ú Ó× ´ µ·Ó× ´× Ò ´Ú µµ × Ò ´ µ ¼ ¼ Õ ¾ ´µ Ú Ú Ó× ´ µ· ½·Ú × Ò ´ µ ¼ ¼ 226 O. CALIN, D.-C. CHANG AND JINGZHI TIE, Hence Ö ¾ ¾ Ü Ü Ü ¼ ¼ Ô Ô Ó×´ µ· ½· × Ò´ µ ¾ ¾ ¾ and Õ ¾ ¾ Ü Ü Ó×´ µ· ¾ · Ü × Ò´ µ ¼ ¼ It follows that Õ ´Ü Ü Ó×´ µµ ¼ ¾ ¾ ¾ · Ü ¼ × Ò´ µ Solving for yields ¾ ¾ ´Ü Ü Ó×´ µµ ¼ ¾ ¾ ¾ Ü ¼ ¾ × Ò´ µ ¾ ¾ ¾ ¾ ¾ ¾ Ü ¾ÜÜ Ó×´ µ·Ü Ó×´ µ Ü × Ò´ µ ¼ ¼ ¼ ¾ × Ò´ µ ¾ ¾ ¾ Ü · Ü ¾ÜÜ Ó×´ µ ¼ ¼ ¾ × Ò´ µ Proposition 2.1. The energy along a geodesic derived from the Hamiltonian Ü Ü (2) between the points ¼ and is ¾ ¾ ¾ Ü · Ü ¾ÜÜ Ó×´ µ ¼ ¼ (3) ¾ ¾ × Ò´ µ Ü ¼ Setting ¼ , we obtain the following result. Corollary 2.2. The energy along a geodesic derived from the Hamiltonian (2) joining the origin and Ü is given by ¾ ¾ Ü (4) ¾ ¾ × Ò´ µ ¼ We note that if we take the limit in (3), we obtain the Euclidean energy ¾ ¾ ¾ ¾ ¾ Ü · Ü ¾ÜÜ Ó×´ µ ¼ ¼ ´Ü Ü µ ¼ Ð Ñ Ð Ñ ¾ ¾ ¾ ¼ ¼ × Ò´ µ ¾ ¾ Ë Ë ´Ü ܵ 2.1 The action function. Let ¼ be the action with initial Ü Ü point ¼ and final point within time . The action satisfies the Hamilton-Jacobi equation Ë · À ´ÖË µ¼ Note that ½ ½ ½ ¾ ¾ ¾ ¾ ¾ ¾ À ´ ܵ ´ Ü µ Ü Ü ¾ ¾ ¾ FUNDAMENTAL SOLUTIONS FOR HERMITE OPERATORS 227 Ë and hence . Using (3), we have ¾ ¾ ¾ Ü · Ü ¾ÜÜ Ó×´ µ ¼ ¼ Ë ¾ ¾× Ò´ µ ½ ¾ ¾ ´Ü · Ü µ ÓØ´ µ ÜÜ ¼ ¼ ¾ × Ò´ µ ÜÜ ¼ ¾ ¾ µÓØ´ µ ´Ü · Ü ¼ ¾ × Ò´ µ Hence ¾ÜÜ ¼ ¾ ¾ Ë ´Ü ܵ ´Ü · Ü µ ÓØ´ µ ¼ ¼ ¾ × Ò´ µ ½ ¾ ¾ ´Ü · Ü µ Ó×´ µ ¾ÜÜ (5) ¼ ¼ ¾ × Ò´ µ It follows that ¾ ´Ü Ü µ ¼ Ð Ñ Ë ´Ü ܵ ¼ ¼ ¾ which is the Euclidean action. Lemma 2.3. We have ¾ ¾ ¾ ´ Ë µ Ü ·¾ (i) Ü ¾ ÓØ´ µ (ii) Ë Ü Proof. (i) Differentiating in (5) with respect to the Ü-variable yields Ë Ü Ó×´ µ Ü Ü (6) ¼ × Ò´ µ ¾ ¾ ¾ ¾ Ü Ó× ´ µ·Ü ¾ÜÜ Ó×´ µ ¼ ¼ ¾ ´ Ë µ Ü ¾ × Ò ´ µ ¾ ¾ ¾ ¾ ¾ Ü · Ü × Ò ´ µ·Ü ¾ÜÜ Ó×´ µ ¼ ¼ ¾ × Ò ´ µ ¾ ¾ ¾ ´Ü · Ü ¾ÜÜ Ó×´ µµ ¼ ¼ ¾ ¾ Ü · ¾ × Ò ´ µ ¾ ¾ Ü ·¾ (ii) Again, differentiating in (6) with respect to the Ü-variable yields ¾ Ë Ó×´ µ ÓØ´ µ Ü £ × Ò´ µ 228 O. CALIN, D.-C. CHANG AND JINGZHI TIE, Now we are in a position to construct the fundamental solution of the heat equation, which has a representation as follows. From now on, we use Ø as the time ¾ variable. Inspired by the fundamental solution of the usual heat equation Ø , Ü we seek a heat kernel of the form Ë ´Ü Üص ¼ à ´Ü ÜصΠ´Øµ (7) ¼ Ë ´Ü Üص Î ´Øµ where ¼ is the action function defined in (5). Here will satisfy a volume function equation and is a real constant. Hence ¼ Ë Ë Ã Î ´Øµ · Î ´Øµ Ë Ø Ø ¡ Ë ¼ Î ´Øµ Î ´Øµ Lemma 2.3 provides Ë Ë Ë Ü Ü ¢ £ ¾ Ë ¾ Ë ¾ Ë ¾ Ë ¾ ¾ ´ Ë µ · Ë ´ µ · Ë Ü Ü Ü Ü Ü ¢ £ Ë ¾ ¾ ´ Ü ·¾ µ· ÓØ ´ ص We determine the heat kernel using a multiplier method. Let ¾ ¾ ¾ È · « Ü (8) Ø Ü « ÈÃ´Ü Üص¼ where is a real multiplier, to be determined such that ¼ for any ¼ Ø . ¡ Ë ¼ ÈÃ´Ü Üص Î ´Øµ Î ´Øµ ¼ ¡ Ë ¾ ¾ ¾ ¾ Ë ´ Ü ·¾ µ· ÓØ ´ ص Î ´Øµ« Ü Î ´Øµ ¼ Î ´Øµ Ë ¾ ¾ ¾ ¾ ¾ Î ´Øµ ´ Ü ·¾ µ ÓØ ´ ص·« Ü Î ´Øµ ¼ Î ´Øµ ¾ ¾ ¾ Ë ´¾ ·½µ·´« µ Ü ÓØ ´ ص Î ´Øµ Î ´Øµ ½¾ In order to eliminate the middle two terms in the brackets, we choose ½ ¬ ¾ ¼ and « . Let . Then the operator (8) becomes ¾ ¾ ¾ · ¬ Ü È (9) Ø Ü and ¡ ¼ ÈÃ´Ü Üصà ´Ü Üص Î ´ØµÎ ´Øµ·¬ ÓØ ´¾¬Øµ ¼ ¼ FUNDAMENTAL SOLUTIONS FOR HERMITE OPERATORS 229 ´Øµ We choose Î such that ¼ Î ´Øµ ¬ ÓØ´¾¬Øµ ؼ Î ´Øµ Integration yields ¡ ½ Ô ÐÒ Î ´Øµ ÐÒ ×Ò´¾¬Øµ µ Î ´Øµ ¾ ×Ò´¾¬Øµ Using the action (5), the fundamental solution formula (7) becomes ¾¬ ¾ ¾ ½ ´Ü ·Ü µ Ó×´¾¬Øµ ¾ÜÜ ¼ ¼ ×Ò´¾¬Øµ Ô Ã ´Ü Üص ¼ ×Ò´¾¬Øµ × ¾¬Ø ¾ ¾ ½ ¾¬Ø ¡ ´Ü ·Ü µ Ó×´¾¬Øµ ¾ÜÜ ¼ ¼ Ø ×Ò´¾¬Øµ Ô ×Ò´¾¬Øµ ¾¬Ø ¬ ¼ We find the constant by investigating the limit case , in which case ¾ the operator (9) becomes the usual 1-dimensional heat operator Ø .As Ü ¾¬Ø ½ , the above fundamental solution becomes × Ò ´¾¬Øµ ¾ ½ ´Ü Ü µ ¼ Ø Ô Ã ´Ü Ü Øµ ¬ ¼ ¼ ¾¬Ø By comparison with the fundamental solution for the usual heat operator ¾ ½ ½ ´Ü Ü µ ¼ Ø Ô Ø Õ ¬ we find .
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