tailieunhanh - Báo cáo toán học: "Bounds on the number of bound states for the Schroedinger equation in one and two dimensions "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Giới hạn về số lượng của các quốc gia bị ràng buộc đối với phương trình Schroedinger trong Một và Hai Kích thước. | Copyright by INCREST 1983 J. OPERATOR THEORY 10 1983 119- 125 BOUNDS ON THE NUMBER OF BOUND STATES FOR THE SCHRỠDINGER EQUATION IN ONE AND TWO DIMENSIONS ROGER G. NEWTON 1. INTRODUCTION It is well known that the Birman-Schwinger method 2 6 8 for estimating the number of bound states of the Schrodinger equation cannot be directly applied in R and R2. The reason is that in these cases the Green s function of the Lippmann--Schwinger equation possesses no finite limit as 0. In R it diverges as and in R2 as Inl-E l. As a consequence no bound on the number of bound states is explicitly known in R2. In this paper we prove such bounds by a suitable modification of the Birman-Schwinger method both for local and nonlocal potentials. The necessary modification was in fact introduced by this author 5 in 1962 in a context in which its relevance to R and R2 was not recognized. The bound there derived for the number of Regge trajectories for local central potentials in R3 that lead to z 1 2 as E 0 was 00 r ị dr ị dr rr tZ r Z7 r ln r r 1 0 1 i ------------------------ ị dr rt r 0 where 2 U x supio - V x X e R . This bound is also an upper limit for the number of rotationally invariant bound states for a local central potential in R2. 120 ROGER G. NEWTON The method of Reference 5 is applicable whenever the kernel K of the modified Lippmann-Schwinger equation for u . K - Ĩ71 2 t71 2 S -- E Ho 1 near E 0- is of the form 3 K P-rK where K is self-adjoint and in the trace-class and has a finite norm-limit as E - 0- p is an orthogonal projection on a one-dimensional subspace spanned by a unit vector p and Ẹ increases without bounds as E - 0-. Let a be the eigenvalues of K E . Then the crux of the Birman-Schwinger method is the recognition that the number n E of bound states of energies not greater than E is equal to the number of eigenvalues an E of K E that are not less than 1. Therefore trÁ a E n E . n However since 3 implies that as E - 0- the leading eigenvalue oq - oo this .