Relative oscillation theory, weighted zeros of the Wronskian, and the spectral shift function. (English) Zbl 1186.47009
An analog of classical oscillation theory is developed to measure the difference between the spectrum of two Sturm-Liouville operators. These operators \(H_j\), \(j=0,1\) have symbols \(\tau_j u=r^{-1}[(pu')'+q_ju]\), \(j=0,1\), on \(L^2((a,b),r \,dx)\), where \(p^{-1},r,q_0,q_1 \in L^{1}_{\text{loc}}(a,b)\), \(-\infty \leq a<b\leq \infty\), and boundary conditions for \(H_0\) and \(H_1\), if any, must be separated. The authors consider the (modified) Wronskian \(W_x(u_0,u_1)=u_0(x) p(x) u_1'(x)-p(x) u_0'(x) u_1(x)\), where \(u_j\), \(j=0,1\), are solutions of \(\tau_j u_j= \lambda_j u_j\).
Intuitively, one defines \(\#_{(a,b)}(u_0,u_1)\) to be the number of sign changes in \((a,b)\) of \(W_x(u_0,u_1)\) weighted so that a sign change counts as \(+1\) if \(q_0-\lambda_0r-(q_1-\lambda_1r)>0\) in a neighborhood of the sign change and is \(-1\) otherwise; moreover, \(\#_{(a,b)}(u_0,u_0)=-1\). This intuitive definition is inadequate for general \(q_j \in L^{1}_{\text{loc}}\); the general definition involves Prüfer variables. Significantly, \(\#_{(a,b)}(u_0,u_1)\) plays the role that the number of zeroes of solutions do in the classical Sturm-Liouville theory.
In order to state a result, some preparation is needed: let \(\psi_{0,-}(\lambda)\) be a solution \((\tau_0-\lambda)\psi_{0,-}(\lambda)=0\) which is \(L^2\) near \(a\) and satisfies any boundary condition at \(a\). Similarly, there is a solution \(\psi_{0,+}(\lambda)\) near \(b\); \(\psi_{1,\pm}(\lambda)\) are defined in the obvious manner. Recall that an endpoint is said to be regular for \(H_0\), if that \(p^{-1},r,q_0\) are integrable near the endpoint.
In the case of regular endpoints, the following result is proved: Suppose that \(H_0\) and \(H_1\) have the same boundary conditions at \(a\) and \(b\). Then
\[ \dim(P_{(-\infty,\lambda_1)}(H_1))-\dim(P_{(-\infty,\lambda_0]}(H_0))=\#_{(a,b)}(\psi_{0,\pm}(\lambda_0),\psi_{1,\mp}(\lambda_1)), \]
where \(P_{\Delta}(H_j)\), \(j=1,2\), denotes the spectral projection for \(H_j\) corresponding to \(\Delta \subseteq \mathbb{R}\).
The case where \(a\) and \(b\) need not be regular is also discussed, as is the case of spectral projections onto an interval \(\Delta\) whose closure avoids the essential spectrum of \(H_0\). Additionally, a result showing how the Krein spectral shift function \(\xi(\lambda)\) is related to \(\#_{(a,b)}(\psi_{0,\pm}(\lambda),\psi_{1,\mp}(\lambda))\) is established.
Intuitively, one defines \(\#_{(a,b)}(u_0,u_1)\) to be the number of sign changes in \((a,b)\) of \(W_x(u_0,u_1)\) weighted so that a sign change counts as \(+1\) if \(q_0-\lambda_0r-(q_1-\lambda_1r)>0\) in a neighborhood of the sign change and is \(-1\) otherwise; moreover, \(\#_{(a,b)}(u_0,u_0)=-1\). This intuitive definition is inadequate for general \(q_j \in L^{1}_{\text{loc}}\); the general definition involves Prüfer variables. Significantly, \(\#_{(a,b)}(u_0,u_1)\) plays the role that the number of zeroes of solutions do in the classical Sturm-Liouville theory.
In order to state a result, some preparation is needed: let \(\psi_{0,-}(\lambda)\) be a solution \((\tau_0-\lambda)\psi_{0,-}(\lambda)=0\) which is \(L^2\) near \(a\) and satisfies any boundary condition at \(a\). Similarly, there is a solution \(\psi_{0,+}(\lambda)\) near \(b\); \(\psi_{1,\pm}(\lambda)\) are defined in the obvious manner. Recall that an endpoint is said to be regular for \(H_0\), if that \(p^{-1},r,q_0\) are integrable near the endpoint.
In the case of regular endpoints, the following result is proved: Suppose that \(H_0\) and \(H_1\) have the same boundary conditions at \(a\) and \(b\). Then
\[ \dim(P_{(-\infty,\lambda_1)}(H_1))-\dim(P_{(-\infty,\lambda_0]}(H_0))=\#_{(a,b)}(\psi_{0,\pm}(\lambda_0),\psi_{1,\mp}(\lambda_1)), \]
where \(P_{\Delta}(H_j)\), \(j=1,2\), denotes the spectral projection for \(H_j\) corresponding to \(\Delta \subseteq \mathbb{R}\).
The case where \(a\) and \(b\) need not be regular is also discussed, as is the case of spectral projections onto an interval \(\Delta\) whose closure avoids the essential spectrum of \(H_0\). Additionally, a result showing how the Krein spectral shift function \(\xi(\lambda)\) is related to \(\#_{(a,b)}(\psi_{0,\pm}(\lambda),\psi_{1,\mp}(\lambda))\) is established.
Reviewer: Denis A. White (Toledo)
MSC:
| 47A55 | Perturbation theory of linear operators |
| 47E05 | General theory of ordinary differential operators |
| 34L05 | General spectral theory of ordinary differential operators |
| 34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |
| 34B24 | Sturm-Liouville theory |
| 47A10 | Spectrum, resolvent |
| 34D10 | Perturbations of ordinary differential equations |
Keywords:
oscillation theory; Sturm-Liouville operators; perturbations; spectral gaps; Krein spectral shift; WronskianReferences:
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