Birman–Schwinger principle

Not to be confused with the Schwinger variational principle or Schwinger's quantum action principle.

In mathematics the Birman–Schwinger principle is a useful technique to reduce the eigenvalue problem for an unbounded differential operator (such as a Schrödinger operator) to an eigenvalue problem for a bounded integral operator. It originates from independent work by M. Sh. Birman^[1] and J. Schwinger^[2] in 1961.

The Birman–Schwinger principle has found numerous applications in deriving bounds on the discrete eigenvalues of differential operators. One of its most prominent uses is in the original proof of the Lieb–Thirring inequality.

Statement of the principle for self-adjoint Schrödinger operators

The technique was developed for self-adjoint Schrödinger operators ${\displaystyle -\Delta -V}$ on ${\displaystyle \mathbb {R} ^{n}}$ with real-valued potentials ${\displaystyle V:\mathbb {R} ^{n}\to [0,\infty )}$. If ${\displaystyle \lambda <0}$ is a negative eigenvalue of the Schrödinger operator with corresponding eigenfunction ${\displaystyle \psi \in L^{2}(\mathbb {R} ^{n})}$, then the eigenequation

${\displaystyle -\Delta \psi -V\psi =\lambda \psi }$

can be formally rearranged to

${\displaystyle {\sqrt {V}}(-\Delta -\lambda )^{-1}{\sqrt {V}}\,\phi =\phi }$

with ${\displaystyle \phi ={\sqrt {V}}\psi }$. This equation shows that ${\displaystyle \phi }$ is an eigenfunction with eigenvalue 1 of the Birman–Schwinger operator

${\displaystyle K_{\lambda }={\sqrt {V}}(-\Delta -\lambda )^{-1}{\sqrt {V}}.}$

The Birman–Schwinger principle rigorously formalises this idea, establishing that ${\displaystyle \lambda <0}$ is an eigenvalue of the Schrödinger operator if and only if 1 is an eigenvalue of ${\displaystyle K_{\lambda }}$ of the same multiplicity. Additionally, the principle asserts that the number ${\displaystyle N_{\lambda }}$ of eigenvalues of the Schrödinger operator less than or equal to ${\displaystyle \lambda <0}$ coincides with the number of eigenvalues of ${\displaystyle K_{\lambda }}$ greater than or equal to 1, i.e.,

${\displaystyle N_{\lambda }=\#\{{\text{eigenvalues of }}K_{\lambda }{\text{ greater than or equal to }}1\}.}$

The statement requires regularity assumptions on the potential ${\displaystyle V}$ which ensure that ${\displaystyle K_{\lambda }}$ is bounded and compact.^[3]

The usefulness of this spectral equivalence lies in the fact that the integral kernel of ${\displaystyle K_{\lambda }}$ is well known. By the Birman–Schwinger principle, upper bounds on ${\displaystyle N_{\lambda }}$ can be obtained by considering Schatten norms of ${\displaystyle K_{\lambda }}$.

The Birman–Schwinger bound

The integral kernel of ${\displaystyle K_{\lambda }}$ takes the form

${\displaystyle K_{\lambda }(x,y)={\sqrt {V(x)}}G_{\lambda }(x-y){\sqrt {V(y)}}}$

where ${\displaystyle G_{\lambda }(x-y)}$ is the integral kernel of the free resolvent ${\displaystyle (-\Delta -\lambda )^{-1}}$. In dimension ${\displaystyle n=3}$, it is explicitly given by

${\displaystyle K_{\lambda }(x,y)={\sqrt {V(x)}}{\frac {e^{-{\sqrt {-\lambda }}|x-y|}}{4\pi |x-y|}}{\sqrt {V(y)}}.}$

Computing the Hilbert–Schmidt norm of ${\displaystyle K_{\lambda }}$, Birman^[1] and Schwinger^[2] independently proved the bound

${\displaystyle N_{0}\leq {\frac {1}{(4\pi )^{2}}}\int _{\mathbb {R} ^{3}}\int _{\mathbb {R} ^{3}}{\frac {V(x)V(y)}{|x-y|}}\,dx\,dy.}$

By the Hardy–Littlewood–Sobolev inequality, the term on the right is further bounded by

${\displaystyle {\frac {1}{4(4\pi )^{2/3}}}\left(\int _{\mathbb {R} ^{3}}V(x)^{3/2}\,dx\right)^{4/3}.}$

In contrast, Weyl asymptotics for large coupling involve only the term ${\displaystyle \int _{\mathbb {R} ^{3}}V(x)^{3/2}\,dx}$, without the power ${\displaystyle 4/3}$. An upper bound of this form on ${\displaystyle N_{0}}$ can also be derived using the Birman–Schwinger principle. The result corresponds to an endpoint case of the Lieb–Thirring inequality, commonly referred to as the Cwikel–Lieb–Rozenblum bound.

Extensions of the principle

In some applications, it is advantageous to define the Birman–Schwinger operator as

${\displaystyle K_{\lambda }=(-\Delta -\lambda )^{-1/2}V(-\Delta -\lambda )^{-1/2}}$

instead. In particular, in one dimension ${\displaystyle n=1}$, this approach allows for distributional potentials.^[4]

While originally established for self-adjoint Schrödinger operators, it is possible to extend the principle to the non self-adjoint case. For ${\displaystyle \lambda \in \mathbb {C} \setminus [0,\infty )}$, the Birman–Schwinger operator can then be defined as

${\displaystyle K_{\lambda }=\operatorname {sgn} (V){\sqrt {|V|}}(-\Delta -\lambda )^{-1}{\sqrt {|V|}}.}$

Again, ${\displaystyle \lambda }$ is an eigenvalue of the Schrödinger operator ${\displaystyle -\Delta -V}$ if and only if 1 is an eigenvalue of ${\displaystyle K_{\lambda }}$. Other factorisations of the potential ${\displaystyle V}$ can be considered as well.^[5]

More generally, the principle can be established for operators of the form ${\displaystyle A-B}$. In both the self-adjoint^[3] and non self-adjoint cases,^[5] the Birman–Schwinger principle connects eigenvalues ${\displaystyle \lambda }$ of ${\displaystyle A-B}$ to the eigenvalue 1 of ${\displaystyle B_{1}(A-\lambda )^{-1}B_{2}^{*}}$ for a suitable factorisation ${\displaystyle B=B_{2}^{*}B_{1}}$.