Quadratic Form Methods

Quadratic forms construct Hamiltonians from energy estimates when an operator sum is unavailable or its domain is difficult to identify. The central perturbation result is the KLMN theorem: a sufficiently small relative form perturbation preserves closedness and the form domain. The definitions and representation theorem belong to Quadratic Forms; this page applies that machinery to construction problems.

Helpful background. Quadratic Forms explains closedness and the associated operator. Domains of Hamiltonians supplies the stronger operator-bound criterion for comparison.

The KLMN form-perturbation theorem

Let $q_0$ be a densely defined, closed, nonnegative symmetric form on $Q\subset\mathcal H$. Let $v$ be a symmetric sesquilinear form defined on $Q$, and suppose

$$|v[\psi]|\leq a q_0[\psi]+b\|\psi\|^2, \qquad 0\leq a<1,\quad b\geq0.$$

Here $v[\psi]=v(\psi,\psi)$. Then

$$q=q_0+v,\qquad D(q)=Q,$$

is closed and bounded below by $-b$. Consequently it determines a unique self-adjoint semibounded operator $H$ with form domain $Q$. A semibounded $q_0$ can first be shifted to make it nonnegative.

The perturbation $v$ need not arise from a Hilbert-space operator, and it need not be closed or closable by itself. Smallness is measured against the reference energy form. This is weaker than an estimate on $\|V\psi\|$ and can apply when no expression $V\psi\in\mathcal H$ exists.

Proof through equivalence of form norms

The lower estimate follows immediately:

$$q[\psi]\geq(1-a)q_0[\psi]-b\|\psi\|^2.$$

Add $(b+1)\|\psi\|^2$ and use both sides of the relative estimate to obtain

$$\begin{aligned} (1-a)q_0[\psi]+\|\psi\|^2 &\leq q[\psi]+(b+1)\|\psi\|^2,\\ q[\psi]+(b+1)\|\psi\|^2 &\leq(1+a)q_0[\psi]+(2b+1)\|\psi\|^2. \end{aligned}$$

Thus the shifted form norm of $q$ is equivalent to the complete norm $(q_0[\psi]+\|\psi\|^2)^{1/2}$. It is therefore complete as well. Density has not changed, so the closed-form representation theorem supplies $H$.

This proof explains both the strict inequality $a<1$ and preservation of the form domain. It does not assert preservation of an operator domain. The operator is defined by the representation identity

$$q(\phi,\psi)=\langle\phi,H\psi\rangle \quad\text{for all }\phi\in Q,$$

on those $\psi\in Q$ for which this identity has a representing vector $H\psi\in\mathcal H$.

A point interaction defined by its energy form

Use units with $\hbar^2/(2m)=1$ on $L^2(\mathbb R)$. For real $g$, define

$$q_g[\psi]=\int_{\mathbb R}|\psi'(x)|^2\,dx+g|\psi(0)|^2, \qquad Q=H^1(\mathbb R).$$

The value at zero is the Sobolev trace of the continuous representative. Integrating the derivative of $|\psi|^2$ on the two half-lines and averaging gives

$$|\psi(0)|^2\leq\|\psi\|\,\|\psi'\| \leq\varepsilon\|\psi'\|^2+ \frac{1}{4\varepsilon}\|\psi\|^2.$$

The first step follows initially for smooth functions and then by density in $H^1$. For every fixed $g$, the point term has relative form bound zero. KLMN therefore constructs a self-adjoint Hamiltonian for either sign and any finite strength of the interaction.

To recover its operator domain, integrate the form pairing by parts separately on each half-line. Away from zero the representing vector is $-\psi''$. The remaining trace coefficient is

$$\overline{\phi(0)} \bigl[\psi'(0-)-\psi'(0+)+g\psi(0)\bigr].$$

Since $\phi(0)$ is arbitrary, the domain is

$$\begin{aligned} D(H_g)=\{\psi\in H^1(\mathbb R):\;& \psi|_{(-\infty,0)},\psi|_{(0,\infty)}\in H^2,\\ &\psi'(0+)-\psi'(0-)=g\psi(0)\}. \end{aligned}$$

The Sobolev membership enforces continuity of $\psi$ at zero; the derivative may jump. The operator acts as $-\psi''$ on the open half-lines. Necessity follows from the representation identity and one-dimensional weak regularity; conversely, these conditions cancel the trace term and prove the identity for every $\phi\in H^1$.

Writing $-d^2/dx^2+g\delta_0$ is a useful formal abbreviation, not an ordinary sum of two operators on $H^2(\mathbb R)$. The resulting realization and its solvable spectrum and scattering belong to Delta-Function Potential as Self-Adjoint Extension.

Positive terms can be added without smallness

For a measurable nonnegative potential $V$, consider

$$q[\psi]=\|\nabla\psi\|^2+\|V^{1/2}\psi\|^2, \quad Q=H^1(\mathbb R^d)\cap D(V^{1/2}).$$

Assume this intersection is dense in $L^2$. The form is closed without a relative smallness bound: a sequence Cauchy in the sum norm converges in the kinetic form space and in the multiplication-operator graph norm, with the same $L^2$ limit. This gives a complete intersection space.

Negative singular contributions require a lower-bound argument. Positivity of one term cannot compensate for an uncontrolled negative part by notation alone. The appropriate construction must specify the dense domain, prove a lower bound, and establish closedness before invoking the representation theorem.

Exercises

1. Show that the trace form $v[\psi]=|\psi(0)|^2$ on $H^1(\mathbb R)$ is not closable as a form on $L^2$, despite being an allowed KLMN perturbation.

Solution

Choose smooth compactly supported $\chi$ with $\chi(0)=1$ and put $\psi_n(x)=\chi(nx)$. Then $\|\psi_n\|^2=\|\chi\|^2/n\to0$, while $v[\psi_n]=1$ and $v[\psi_n-\psi_m]=0$. This violates the closability criterion. The kinetic term prevents this sequence from being Cauchy in the full form norm.

2. For $g<0$, obtain a lower bound for $q_g$ directly from the first trace estimate.

Solution

Set $X=\|\psi'\|$ and $Y=\|\psi\|$. Then $q_g[\psi]\geq X^2-|g|XY=(X-|g|Y/2)^2-g^2Y^2/4$. Thus $H_g\geq-g^2/4$. This is a lower-bound proof; identifying an eigenvector that attains it is an additional spectral calculation.

References

• K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265, Springer, 2012, Chapter 10, publisher edition — closed semibounded forms and operator construction.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, Section 6.5 and Theorem 6.24, author’s text — relatively form-bounded perturbations and KLMN.